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Page 1: Representation Theory of Algebraic Groups

Representation theory of algebraic groupsFrom Wikipedia, the free encyclopedia

Page 2: Representation Theory of Algebraic Groups

Contents

1 (B, N) pair 11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Properties of groups with a BN pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Adelic algebraic group 32.1 Ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Tamagawa numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 History of the terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Algebraic group 53.1 Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Algebraic subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Algebraic torus 74.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.4 Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.5 Arithmetic invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Approximation in algebraic groups 105.1 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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5.3 Formal definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Arason invariant 126.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

7 Arithmetic group 137.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

8 Borel subgroup 148.1 Parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.2 Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

9 Borel–de Siebenthal theory 169.1 Connected subgroups of maximal rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.2 Maximal connected subgroups of maximal rank . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.3 Closed subsystems of roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189.4 Applications to symmetric spaces of compact type . . . . . . . . . . . . . . . . . . . . . . . . . . 189.5 Applications to hermitian symmetric spaces of compact type . . . . . . . . . . . . . . . . . . . . . 209.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

10 Bott–Samelson variety 2310.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

11 Bruhat decomposition 2411.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2411.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2411.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.4 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

12 Cartan subgroup 2612.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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12.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

13 Chevalley’s structure theorem 2713.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

14 Cohomological invariant 2814.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2814.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2814.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

15 Complexification (Lie group) 3015.1 Universal complexification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

15.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015.1.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015.1.3 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015.1.4 Injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

15.2 Chevalley complexification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.2.1 Hopf algebra of matrix coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.2.2 Invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

15.3 Decompositions in the Chevalley complexification . . . . . . . . . . . . . . . . . . . . . . . . . . 3215.3.1 Cartan decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3215.3.2 Gauss decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3215.3.3 Bruhat decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3315.3.4 Iwasawa decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

15.4 Complex structures on homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3415.5 Noncompact real forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

15.5.1 Involutions of simply connected compact Lie groups . . . . . . . . . . . . . . . . . . . . . 3515.5.2 Conjugations on the complexification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.5.3 Cartan decomposition in a real form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.5.4 Iwasawa decomposition in a real form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

15.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3715.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3715.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

16 Cuspidal representation 4116.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

17 Diagonalizable group 4317.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

18 Dieudonné module 44

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18.1 Dieudonné rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4418.2 Dieudonné modules and groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

19 Differential algebraic group 4619.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

20 Differential Galois theory 4720.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4720.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4720.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4720.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

21 E6 (mathematics) 4921.1 Real and complex forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4921.2 E6 as an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5021.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

21.3.1 Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5021.3.2 Roots of E6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5021.3.3 Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5321.3.4 Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

21.4 Important subalgebras and representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.5 E6 polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.6 Chevalley and Steinberg groups of type E6 and 2E6 . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.7 Importance in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5621.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

22 E7 (mathematics) 5822.1 Real and complex forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5822.2 E7 as an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5922.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

22.3.1 Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5922.3.2 Root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5922.3.3 Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6122.3.4 Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

22.4 Important subalgebras and representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6222.4.1 E7 Polynomial Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

22.5 Chevalley groups of type E7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322.6 Importance in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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22.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

23 E8 (mathematics) 6523.1 Basic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6523.2 Real and complex forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6523.3 E8 as an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6623.4 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6623.5 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.6 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.7 E8 root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

23.7.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.7.2 Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.7.3 Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923.7.4 Simple roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7023.7.5 Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.7.6 E8 root lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.7.7 Simple subalgebras of E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

23.8 Chevalley groups of type E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.9 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.10Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7423.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

24 F4 (mathematics) 7624.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

24.1.1 Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.1.2 Weyl/Coxeter group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.1.3 Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.1.4 F4 lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.1.5 Roots of F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7724.1.6 F4 polynomial invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

24.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7924.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7924.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

25 Fixed-point subgroup 81

26 Formal group 8226.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8326.3 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8326.4 The logarithm of a commutative formal group law . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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26.5 The formal group ring of a formal group law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8426.6 Formal group laws as functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.7 The height of a formal group law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.8 Lazard ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.9 Formal groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.10Lubin–Tate formal group laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

27 Fundamental lemma (Langlands program) 8927.1 Motivation and history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8927.2 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8927.3 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

28 G2 (mathematics) 9228.1 Real forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

28.2.1 Dynkin diagram and Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.2.2 Roots of G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.2.3 Weyl/Coxeter group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.2.4 Special holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

28.3 Polynomial Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.4 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.6 Finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

29 Geometric invariant theory 9629.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9629.2 Mumford’s book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9729.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9929.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

30 Glossary of algebraic groups 10030.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

31 Good filtration 10131.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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32 Grosshans subgroup 10232.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10232.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

33 Group of Lie type 10333.1 Classical groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10333.2 Chevalley groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10333.3 Steinberg groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10333.4 Suzuki–Ree groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10433.5 Relations with finite simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.6 Small groups of Lie type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.7 Notation issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10633.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10733.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10733.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

34 Group scheme 10934.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10934.2 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10934.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11034.4 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11134.5 Finite flat group schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11134.6 Cartier duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11234.7 Dieudonné modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11234.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

35 Haboush’s theorem 11435.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11435.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11435.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

36 Hochschild–Mostow group 11636.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

37 Hyperspecial subgroup 11737.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

38 Inner form 11838.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

39 Iwahori subgroup 11939.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

40 Jordan–Chevalley decomposition 120

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40.1 Decomposition of endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12040.2 Decomposition in a real semisimple Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12040.3 Decomposition in a real semisimple Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12140.4 Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12140.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

41 Kazhdan–Lusztig polynomial 12241.1 Motivation and history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12241.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12241.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12341.4 Kazhdan–Lusztig conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

41.4.1 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12441.5 Relation to intersection cohomology of Schubert varieties . . . . . . . . . . . . . . . . . . . . . . 12541.6 Generalization to real groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12541.7 Generalization to other objects in representation theory . . . . . . . . . . . . . . . . . . . . . . . 12641.8 Combinatorial theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12641.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12641.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

42 Kempf vanishing theorem 12842.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

43 Kneser–Tits conjecture 12943.1 Fields for which the Whitehead group vanishes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12943.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12943.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

44 Kostant polynomial 13044.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13044.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13044.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13144.4 Steinberg basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13344.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

45 Lang’s theorem 13545.1 The Lang–Steinberg theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13545.2 Proof of Lang’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13545.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13645.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

46 Langlands decomposition 13746.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13746.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13746.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

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47 Lattice (discrete subgroup) 13847.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13947.2 Arithmetic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13947.3 S-arithmetic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13947.4 Adelic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13947.5 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13947.6 Tree lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14047.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14047.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

48 Lazard’s universal ring 14148.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

49 Lie–Kolchin theorem 14249.1 Triangularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14249.2 Lie’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14249.3 Counter-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14349.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

50 Mirabolic group 14450.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

51 Mumford–Tate group 14551.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14551.2 Mumford–Tate conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14651.3 Period conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14651.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14651.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14651.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

52 Observable subgroup 14752.1 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

53 Pseudo-reductive group 14853.1 Examples of pseudo reductive groups that are not reductive . . . . . . . . . . . . . . . . . . . . . 14853.2 Classification and exotic phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14853.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

54 Quasi-split group 15054.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15054.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

55 Radical of an algebraic group 15155.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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55.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

56 Rational representation 15256.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

57 Reductive group 15357.1 Lie group case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15357.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15357.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15457.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

58 Restricted Lie algebra 15558.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15558.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15558.3 Restricted universal enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15558.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15558.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

59 Root datum 15759.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15759.2 The root datum of an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15759.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

60 Rost invariant 15960.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15960.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

61 Semisimple algebraic group 16161.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16161.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16161.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16161.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

62 Serre group 16262.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16262.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

63 Severi–Brauer variety 16363.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16463.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16463.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

64 Siegel parabolic subgroup 16564.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

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65 Spaltenstein variety 16665.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

66 Special group (algebraic group theory) 167

67 Springer resolution 16867.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

68 Steinberg representation 16968.1 The Steinberg representation of a finite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16968.2 The Steinberg representation of a p-adic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16968.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

69 Superstrong approximation 17169.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17169.2 Proofs of superstrong approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17169.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17169.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

70 Taniyama group 17370.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

71 Tannakian category 17471.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17471.2 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17471.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

72 Thin group (algebraic group theory) 17672.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

73 Unipotent 17773.1 Unipotent algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17773.2 Unipotent radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17773.3 Jordan decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17873.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17873.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

74 Weil conjecture on Tamagawa numbers 17974.1 Tamagawa measure and Tamagawa numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17974.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17974.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18074.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18074.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

75 Weyl module 181

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75.1 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

76 Witt vector 18276.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

76.1.1 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18276.2 Construction of Witt rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18376.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18476.4 Universal Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18476.5 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

76.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18576.5.2 Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18576.5.3 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

76.6 Ring schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18676.7 Commutative unipotent algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18676.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18676.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

77 Wonderful compactification 18777.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

78 Étale group scheme 18878.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18878.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 189

78.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18978.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19278.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

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Chapter 1

(B, N) pair

In mathematics, a (B, N) pair is a structure on groups of Lie type that allows one to give uniform proofs of manyresults, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups aresimilar to the general linear group over a field. They were invented by the mathematician Jacques Tits, and are alsosometimes known as Tits systems.

1.1 Definition

A (B, N) pair is a pair of subgroups B and N of a group G such that the following axioms hold:

• G is generated by B and N.

• The intersection, H, of B and N is a normal subgroup of N.

• The groupW = N/H is generated by a set S of elements wi of order 2, for i in some non-empty set I.

• If wi is an element of S and w is any element ofW, then wiBw is contained in the union of BwiwB and BwB.

• No generator wi normalizes B.

The idea of this definition is that B is an analogue of the upper triangular matrices of the general linear groupGLn(K),H is an analogue of the diagonal matrices, and N is an analogue of the normalizer of H.The subgroup B is sometimes called the Borel subgroup, H is sometimes called the Cartan subgroup, and W iscalled theWeyl group. The pair (W,S) is a Coxeter system.The number of generators is called the rank.

1.2 Examples• Suppose that G is any doubly transitive permutation group on a set X with more than 2 elements. We let B bethe subgroup of G fixing a point x, and we let N be the subgroup fixing or exchanging 2 points x and y. Thesubgroup H is then the set of elements fixing both x and y, and W has order 2 and its nontrivial element isrepresented by anything exchanging x and y.

• Conversely, if G has a (B, N) pair of rank 1, then the action of G on the cosets of B is doubly transitive. SoBN pairs of rank 1 are more or less the same as doubly transitive actions on sets with more than 2 elements.

• Suppose thatG is the general linear groupGLn(K) over a field K. We take B to be the upper triangular matrices,H to be the diagonal matrices, and N to be the monomial matrices, i.e. matrices with exactly one non-zeroelement in each row and column. There are n − 1 generators wi, represented by the matrices obtained byswapping two adjacent rows of a diagonal matrix.

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2 CHAPTER 1. (B, N) PAIR

• More generally, any group of Lie type has the structure of a BN-pair.

• A reductive algebraic group over a local field has a BN-pair where B is an Iwahori subgroup.

1.3 Properties of groups with a BN pair

The map taking w to BwB is an isomorphism from the set of elements ofW to the set of double cosets of B; this isthe Bruhat decomposition G = BWB.If T is a subset of S then let W(T) be the subgroup of W generated by T : we define and G(T) = BW(T)B to be thestandard parabolic subgroup for T. The subgroups of G containing conjugates of B are the parabolic subgroups; con-jugates of B are called Borel subgroups (or minimal parabolic subgroups). These are precisely the standard parabolicsubgroups.

1.4 Applications

BN-pairs can be used to prove that many groups of Lie type are simple modulo their centers. More precisely, if Ghas a BN-pair such that B is a solvable group, the intersection of all conjugates of B is trivial, and the set of generatorsofW cannot be decomposed into two non-empty commuting sets, then G is simple whenever it is a perfect group. Inpractice all of these conditions except for G being perfect are easy to check. Checking that G is perfect needs someslightly messy calculations (and in fact there are a few small groups of Lie type which are not perfect). But showingthat a group is perfect is usually far easier than showing it is simple.

1.5 References• Bourbaki, Nicolas (2002). Lie Groups and Lie Algebras: Chapters 4–6. Elements of Mathematics. Springer.ISBN 3-540-42650-7. Zbl 0983.17001. The standard reference for BN pairs.

• Serre, Jean-Pierre (2003). Trees. Springer. ISBN 3-540-44237-5. Zbl 1013.20001.

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Chapter 2

Adelic algebraic group

In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over anumber field K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the definition ofthe appropriate topology is straightforward only in caseG is a linear algebraic group. In the case ofG an abelian varietyit presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawanumbers. Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphicrepresentations, and the arithmetic of quadratic forms.In case G is a linear algebraic group, it is an affine algebraic variety in affine N-space. The topology on the adelicalgebraic groupG(A) is taken to be the subspace topology in AN , the Cartesian product of N copies of the adele ring.

2.1 Ideles

An important example, the idele group I(K), is the case of G = GL1 . Here the set of ideles (also idèles /ɪˈdɛlz/)consists of the invertible adeles; but the topology on the idele group is not their topology as a subset of the adeles.Instead, considering that GL1 lies in two-dimensional affine space as the 'hyperbola' defined parametrically by

(t, t−1),

the topology correctly assigned to the idele group is that induced by inclusion in A2; composing with a projection, itfollows that the ideles carry a finer topology than the subspace topology from A.Inside AN , the product KN lies as a discrete subgroup. This means that G(K) is a discrete subgroup of G(A), also. Inthe case of the idele group, the quotient group

I(K)/K×

is the idele class group. It is closely related to (though larger than) the ideal class group. The idele class group is notitself compact; the ideles must first be replaced by the ideles of norm 1, and then the image of those in the idele classgroup is a compact group; the proof of this is essentially equivalent to the finiteness of the class number.The study of the Galois cohomology of idele class groups is a central matter in class field theory. Characters of theidele class group, now usually called Hecke characters, give rise to the most basic class of L-functions.

2.2 Tamagawa numbers

See also: Weil conjecture on Tamagawa numbers

For more general G, the Tamagawa number is defined (or indirectly computed) as the measure of

G(A)/G(K).

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4 CHAPTER 2. ADELIC ALGEBRAIC GROUP

Tsuneo Tamagawa's observation was that, starting from an invariant differential form ω on G, defined over K, themeasure involved was well-defined: while ω could be replaced by cω with c a non-zero element of K, the productformula for valuations in K is reflected by the independence from c of the measure of the quotient, for the productmeasure constructed from ω on each effective factor. The computation of Tamagawa numbers for semisimple groupscontains important parts of classical quadratic form theory.

2.3 History of the terminology

Historically the idèles were introduced by Chevalley (1936) under the name "élément idéal”, which is “ideal element”in French, which Chevalley (1940) then abbreviated to “idèle” following a suggestion of Hasse. (In these papers healso gave the ideles a non-Hausdorff topology.) This was to formulate class field theory for infinite extensions in termsof topological groups. Weil (1938) defined (but did not name) the ring of adeles in the function field case and pointedout that Chevalley’s group of Idealelemente was the group of invertible elements of this ring. Tate (1950) defined thering of adeles as a restricted direct product, though he called its elements “valuation vectors” rather than adeles.Chevalley (1951) defined the ring of adeles in the function field case, under the name “repartitions”. The term adèle(short for additive idèles, and also a French woman’s name) was in use shortly afterwards (Jaffard 1953) and mayhave been introduced by André Weil. The general construction of adelic algebraic groups by Ono (1957) followedthe algebraic group theory founded by Armand Borel and Harish-Chandra.

2.4 References• Chevalley, Claude (1936), “Généralisation de la théorie du corps de classes pour les extensions infinies.”,Journal de Mathématiques Pures et Appliquées (in French) 15: 359–371, JFM 62.1153.02

• Chevalley, Claude (1940), “La théorie du corps de classes”, Annals of Mathematics. Second Series 41: 394–418, ISSN 0003-486X, JSTOR 1969013, MR 0002357

• Chevalley, Claude (1951), Introduction to the Theory of Algebraic Functions of One Variable, MathematicalSurveys, No. VI, Providence, R.I.: American Mathematical Society, MR 0042164

• Jaffard, Paul (1953), Anneaux d'adèles (d'après Iwasawa), Séminaire Bourbaki, Secrétariat mathématique,Paris, MR 0157859

• Ono, Takashi (1957), “Sur une propriété arithmétique des groupes algébriques commutatifs”, Bulletin de laSociété Mathématique de France 85: 307–323, ISSN 0037-9484, MR 0094362

• Tate, John T. (1950), “Fourier analysis in number fields, and Hecke’s zeta-functions”, Algebraic Number The-ory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–347, ISBN 978-0-9502734-2-6, MR 0217026

• Weil, André (1938), “Zur algebraischen Theorie der algebraischen Funktionen.”, Journal für Reine und Ange-wandte Mathematik (in German) 179: 129–133, doi:10.1515/crll.1938.179.129, ISSN 0075-4102

2.5 External links• Rapinchuk, A.S. (2001), “Tamagawa number”, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

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Chapter 3

Algebraic group

In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that themultiplication and inversion operations are given by regular functions on the variety.In terms of category theory, an algebraic group is a group object in the category of algebraic varieties.

3.1 Classes

Several important classes of groups are algebraic groups, including:

• Finite groups

• GL(n, C), the general linear group of invertible matrices over C

• Jet group

• Elliptic curves.

Two important classes of algebraic groups arise, that for the most part are studied separately: abelian varieties (the'projective' theory) and linear algebraic groups (the 'affine' theory). There are certainly examples that are neither onenor the other — these occur for example in the modern theory of integrals of the second and third kinds such as theWeierstrass zeta function, or the theory of generalized Jacobians. But according to Chevalley’s structure theorem anyalgebraic group is an extension of an abelian variety by a linear algebraic group. This is a result of Claude Chevalley:if K is a perfect field, and G an algebraic group over K, there exists a unique normal closed subgroup H in G, suchthat H is a linear group and G/H an abelian variety.According to another basic theorem, any group in the category of affine varieties has a faithful finite-dimensionallinear representation: we can consider it to be a matrix group over K, defined by polynomials over K and with matrixmultiplication as the group operation. For that reason a concept of affine algebraic group is redundant over a field —we may as well use a very concrete definition. Note that this means that algebraic group is narrower than Lie group,when working over the field of real numbers: there are examples such as the universal cover of the 2×2 special lineargroup that are Lie groups, but have no faithful linear representation. A more obvious difference between the twoconcepts arises because the identity component of an affine algebraic group G is necessarily of finite index in G.When one wants to work over a base ring R (commutative), there is the group scheme concept: that is, a group objectin the category of schemes over R. Affine group scheme is the concept dual to a type of Hopf algebra. There is quitea refined theory of group schemes, that enters for example in the contemporary theory of abelian varieties.

3.2 Algebraic subgroup

An algebraic subgroup of an algebraic group is a Zariski closed subgroup. Generally these are taken to be connected(or irreducible as a variety) as well.

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Another way of expressing the condition is as a subgroup which is also a subvariety.This may also be generalized by allowing schemes in place of varieties. The main effect of this in practice, apartfrom allowing subgroups in which the connected component is of finite index > 1, is to admit non-reduced schemes,in characteristic p.

3.3 Coxeter groups

Main article: Coxeter groupFurther information: Field with one element

There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number ofelements of the symmetric group is n! , and the number of elements of the general linear group over a finite fieldis the q-factorial [n]q! ; thus the symmetric group behaves as though it were a linear group over “the field with oneelement”. This is formalized by the field with one element, which considers Coxeter groups to be simple algebraicgroups over the field with one element.

3.4 See also• Algebraic topology (object)

• Borel subgroup

• Tame group

• Morley rank

• Cherlin–Zilber conjecture

• Adelic algebraic group

• Glossary of algebraic groups

3.5 Notes

3.6 References• Chevalley, Claude, ed. (1958), Séminaire C. Chevalley, 1956-−1958. Classification des groupes de Lie al-gébriques, 2 vols, Paris: Secrétariat Mathématique, MR 0106966, Reprinted as volume 3 of Chevalley’s col-lected works.

• Humphreys, James E. (1972), Linear Algebraic Groups, Graduate Texts in Mathematics 21, Berlin, New York:Springer-Verlag, ISBN 978-0-387-90108-4, MR 0396773

• Lang, Serge (1983), Abelian varieties, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90875-5

• Milne, J. S., Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups

• Mumford, David (1970),Abelian varieties, OxfordUniversity Press, ISBN978-0-19-560528-0, OCLC138290

• Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics 9 (2nd ed.), Boston, MA:Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713

• Waterhouse, William C. (1979), Introduction to affine group schemes, Graduate Texts in Mathematics 66,Berlin, New York: Springer-Verlag, ISBN 978-0-387-90421-4

• Weil, André (1971), Courbes algébriques et variétés abéliennes, Paris: Hermann, OCLC 322901

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Chapter 4

Algebraic torus

In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named byanalogy with the theory of tori in Lie group theory (see maximal torus). The theory of tori is in some sense oppositeto that of unipotent groups, because tori have rich arithmetic structure but no deformations.

4.1 Definition

Given a base scheme S, an algebraic torus over S is defined to be a group scheme over S that is fpqc locally isomorphicto a finite product of copies of the multiplicative group schemeGm/S over S. In other words, there exists a faithfullyflat map X → S such that any point in X has a quasi-compact open neighborhood U whose image is an open affinesubscheme of S, such that base change to U yields a finite product of copies of GL₁,U = Gm/U. One particularlyimportant case is when S is the spectrum of a field K, making a torus over S an algebraic group whose extension tosome finite separable extension L is a finite product of copies of Gm/L. In general, the multiplicity of this product(i.e., the dimension of the scheme) is called the rank of the torus, and it is a locally constant function on S.If a torus is isomorphic to a product of multiplicative groupsGm/S, the torus is said to be split. All tori over separablyclosed fields are split, and any non-separably closed field admits a non-split torus given by restriction of scalars over aseparable extension. Restriction of scalars over an inseparable field extension will yield a commutative group schemethat is not a torus.

4.2 Weights

Over a separably closed field, a torus T admits two primary invariants. The weight lattice X•(T ) is the group ofalgebraic homomorphisms T → G , and the coweight lattice X•(T ) is the group of algebraic homomorphisms G→ T. These are both free abelian groups whose rank is that of the torus, and they have a canonical nondegeneratepairingX•(T )×X•(T ) → Z given by (f, g) 7→ deg(f g) , where degree is the number n such that the compositionis equal to the nth power map on the multiplicative group. The functor given by taking weights is an antiequivalenceof categories between tori and free abelian groups, and the coweight functor is an equivalence. In particular, mapsof tori are characterized by linear transformations on weights or coweights, and the automorphism group of a torusis a general linear group over Z. The quasi-inverse of the weights functor is given by a dualization functor from freeabelian groups to tori, defined by its functor of points as:

D(M)S(X) := Hom(M,Gm,S(X)).

This equivalence can be generalized to pass between groups of multiplicative type (a distinguished class of formalgroups) and arbitrary abelian groups, and such a generalization can be convenient if one wants to work in a well-behaved category, since the category of tori doesn't have kernels or filtered colimits.When a field K is not separably closed, the weight and coweight lattices of a torus over K are defined as the respectivelattices over the separable closure. This induces canonical continuous actions of the absolute Galois group of K on

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8 CHAPTER 4. ALGEBRAIC TORUS

the lattices. The weights and coweights that are fixed by this action are precisely the maps that are defined over K. Thefunctor of taking weights is an antiequivalence between the category of tori over K with algebraic homomorphismsand the category of finitely generated torsion free abelian groups with an action of the absolute Galois group of K.Given a finite separable field extension L/K and a torus T over L, we have a Galois module isomorphism

X•(ResL/KT ) ∼= IndGK

GLX•(T ).

If T is the multiplicative group, then this gives the restriction of scalars a permutation module structure. Tori whoseweight lattices are permutation modules for the Galois group are called quasi-split, and all quasi-split tori are finiteproducts of restrictions of scalars.For a general base scheme S, weights and coweights are defined as fpqc sheaves of free abelian groups on S. Theseprovide representations of fundamental groupoids of the base with respect the fpqc topology. If the torus is locallytrivializable with respect to a weaker topology such as the etale topology, then the sheaves of groups descend to thesame topologies and these representations factor through the respective quotient groupoids. In particular, an etalesheaf gives rise to a quasi-isotrivial torus, and if S is locally noetherian and normal (more generally, geometricallyunibranched), the torus is isotrivial. As a partial converse, a theorem of Grothendieck asserts that any torus of finitetype is quasi-isotrivial, i.e., split by an etale surjection.Given a rank n torus T over S, a twisted form is a torus over S for which there exists a fpqc covering of S for whichtheir base extensions are isomorphic, i.e., it is a torus of the same rank. Isomorphism classes of twisted forms ofa split torus are parametrized by nonabelian flat cohomology H1(S,GLn(Z)) , where the coefficient group forms aconstant sheaf. In particular, twisted forms of a split torus T over a field K are parametrized by elements of the Galoiscohomology pointed setH1(GK , GLn(Z))with trivial Galois action on the coefficients. In the one-dimensional case,the coefficients form a group of order two, and isomorphism classes of twisted forms of G are in natural bijectionwith separable quadratic extensions of K.Since taking a weight lattice is an equivalence of categories, short exact sequences of tori correspond to short exactsequences of the corresponding weight lattices. In particular, extensions of tori are classified by Ext1 sheaves. Theseare naturally isomorphic to the flat cohomology groupsH1(S,HomZ(X

•(T1), X•(T2))) . Over a field, the extensions

are parametrized by elements of the corresponding Galois cohomology group.

4.3 Example

Let S be the restriction of scalars of G over the field extension C/R. This is a real torus whose real points formthe Lie group of nonzero complex numbers. Restriction of scalars gives a canonical embedding of S into GL2, andcomposition with determinant gives an algebraic homomorphism of tori from S to G , called the norm. The kernelof this map is a nonsplit rank one torus called the norm torus of the extension C/R, and its real points form the Liegroup U(1), which is topologically a circle. It has no multiplicative subgroups (equivalently, the weight lattice has nononzero Galois fixed points), and such tori are called anisotropic. Its weight lattice is a copy of the integers, with thenontrivial Galois action that sends complex conjugation to the minus one map.

4.4 Isogenies

An isogeny is a surjective morphism of tori whose kernel is a finite flat group scheme. Equivalently, it is an injectionof the corresponding weight lattices with finite cokernel. The degree of the isogeny is defined to be the order of thekernel, i.e., the rank of its structure sheaf as a locally free OS -module, and it is a locally constant function on thebase. One can also define the degree to be order of the cokernel of the corresponding linear transformation on weightlattices. Two tori are called isogenous if there exists an isogeny between them. An isogeny is an isomorphism if andonly if its degree is one. Note that if S doesn't have a map to Spec Q, then the kernel may not be smooth over S.Given an isogeny f of degree n, one can prove using linear algebra on weights and faithfully flat descent that thereexists a dual isogeny g such that gf is the nth power map on the source torus. Therefore, isogeny is an equivalencerelation on the category of tori. T. Ono pointed out that two tori over a field are isogenous if and only if their weightlattices are rationally equivalent as Galois modules, where rational equivalence means we tensor over Z with Q andget equivalent vector spaces with Galois action. This extends naturally from Galois modules to fpqc sheaves, whereZ and Q are constant sheaves rather than plain groups.

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4.5. ARITHMETIC INVARIANTS 9

4.5 Arithmetic invariants

In his work on Tamagawa numbers, T. Ono introduced a type of functorial invariants of tori over finite separableextensions of a chosen field k. Such an invariant is a collection of positive real-valued functions fK on isomorphismclasses of tori over K, as K runs over finite separable extensions of k, satisfying three properties:

1. Multiplicativity: Given two tori T1 and T2 over K, fK(T1 × T2) = fK(T1) fK(T2)

2. Restriction: For a finite separable extension L/K, fL evaluated on an L torus is equal to fK evaluated on itsrestriction of scalars to K.

3. Projective triviality: If T is a torus over K whose weight lattice is a projective Galois module, then fK(T) = 1.

T. Ono showed that the Tamagawa number of a torus over a number field is such an invariant. Furthermore, he showedthat it is a quotient of two cohomological invariants, namely the order of the groupH1(Gk, X

•(T )) ∼= Ext1(T,Gm)(sometimes mistakenly called the Picard group of T, although it doesn't classify G torsors over T), and the order ofthe Tate–Shafarevich group.The notion of invariant given above generalizes naturally to tori over arbitrary base schemes, with functions takingvalues in more general rings. While the order of the extension group is a general invariant, the other two invariantsabove do not seem to have interesting analogues outside the realm of fraction fields of one-dimensional domains andtheir completions.

4.6 See also• Torus based cryptography

• Toric geometry

4.7 References• A. Grothendieck, SGA 3 Exp. VIII–X

• T. Ono, On Tamagawa Numbers

• T. Ono, On the Tamagawa number of algebraic tori Annals of Mathematics 78 (1) 1963.

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Chapter 5

Approximation in algebraic groups

In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraicgroups G over global fields k.

5.1 Use

They give conditions for the group G(k) to be dense in a restricted direct product of groups of the form G(ks) for ksa completion of k at the place s. In weak approximation theorems the product is over a finite set of places s, while instrong approximation theorems the product is over all but a finite set of places.

5.2 History

Eichler (1938) proved strong approximation for some classical groups. Strong approximation was established in the1960s and 1970s, for semisimple simply-connected algebraic groups over global fields. The results for number fieldsare due to Kneser (1966) and Platonov (1969); the function field case, over finite fields, is due to Margulis (1977) andPrasad (1977). In the number field case Platonov also proved a related a result over local fields called the Kneser–Titsconjecture.

5.3 Formal definitions and properties

Let G be a linear algebraic group over a global field k, and A the adele ring of k. If S is a non-empty finite set ofplaces of k, then we write AS for the ring of S-adeles and AS for the product of the completions ks, for s in the finiteset S; thus A = AS × AS. For any choice of S, G(k) embeds in G(AS) and G(AS).The question asked in weak approximation is whether the embedding of G(k) in G(AS) has dense image. If the groupG is connected and k-rational, then it satisfies weak approximation with respect to any set S (Platonov, Rapinchuk1994, p.402). More generally, for any connected group G, there is a finite set T of finite places of k such that Gsatisfies weak approximation with respect to any set S that is disjoint with T (Platonov, Rapinchuk 1994, p.415). Inparticular, if k is an algebraic number field then any group G satisfies weak approximation with respect to the set S =S∞ of infinite places.The question asked in strong approximation is whether the embedding of G(k) in G(AS) has dense image, or equiva-lently whether the set

G(k)G(AS)

is a dense subset in G(A). The main theorem of strong approximation (Kneser 1966, p.188) states that a non-solvablelinear algebraic group G over a global field k has strong approximation for the finite set S if and only if its radical Nis unipotent, G/N is simply connected, and each almost simple component H of G/N has a non-compact componentHs for some s in S (depending on H).

10

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5.4. SEE ALSO 11

The proofs of strong approximation depended on the Hasse principle for algebraic groups, which for groups of typeE8 was only proved several years later.Weak approximation holds for a broader class of groups, including adjoint groups and inner forms of Chevalleygroups, showing that the strong approximation property is restrictive.

5.4 See also• Superstrong approximation

5.5 References• Eichler, Martin (1938), “Allgemeine Kongruenzklasseneinteilungen der Ideale einfacher Algebren über alge-braischen Zahlkörpern und ihre L-Reihen.”, Journal für Reine und Angewandte Mathematik (in German) 179:227–251, doi:10.1515/crll.1938.179.227, ISSN 0075-4102

• Kneser, Martin (1966), “Strong approximation”, Algebraic Groups and Discontinuous Subgroups (Proc. Sym-pos. Pure Math., Boulder, Colo., 1965), Providence, R.I.: American Mathematical Society, pp. 187–196, MR0213361

• Margulis, G. A. (1977), “Cobounded subgroups in algebraic groups over local fields”, Akademija Nauk SSSR.Funkcional'nyi Analiz i ego Priloženija 11 (2): 45–57, 95, ISSN 0374-1990, MR 0442107

• Platonov, V. P. (1969), “The problem of strong approximation and the Kneser–Tits hypothesis for algebraicgroups”, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 33: 1211–1219, ISSN 0373-2436, MR0258839

• Platonov, Vladimir; Rapinchuk, Andrei (1994), Algebraic groups and number theory. (Translated from the1991 Russian original by Rachel Rowen.), Pure and Applied Mathematics 139, Boston, MA: Academic Press,Inc., ISBN 0-12-558180-7, MR 1278263

• Prasad, Gopal (1977), “Strong approximation for semi-simple groups over function fields”, Annals of Mathe-matics. Second Series 105 (3): 553–572, ISSN 0003-486X, JSTOR 1970924, MR 0444571

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Chapter 6

Arason invariant

In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank andtrivial discriminant and Clifford invariant over a field k of characteristic not 2, taking values in H3(k,Z/2Z). It wasintroduced by (Arason 1975, Theorem 5.7).The Rost invariant is a generalization of the Arason invariant to other algebraic groups.

6.1 Definition

Suppose thatW(k) is theWitt ring of quadratic forms over a field k and I is the ideal of forms of even dimension. TheArason invariant is a group homomorphism from I3 to the Galois cohomology group H3(k,Z/2Z). It is determinedby the property that on the 8-dimensional diagonal form with entries 1, –a, –b, ab, -c, ac, bc, -abc (the 3-fold Pfisterform«a,b,c») it is given by the cup product of the classes of a, b, c in H1(k,Z/2Z) = k*/k*2. The Arason invariantvanishes on I4, and it follows from the Milnor conjecture proved by Voevodsky that it is an isomorphism from I3/I4to H3(k,Z/2Z).

6.2 References• Arason, Jón Kr. (1975), “Cohomologische Invarianten quadratischer Formen”, J. Algebra (in German) 36 (3):448–491, doi:10.1016/0021-8693(75)90145-3, ISSN 0021-8693, MR 0389761, Zbl 0314.12104

• Esnault, Hélène; Kahn, Bruno; Levine, Marc; Viehweg, Eckart (1998), “The Arason invariant and mod 2algebraic cycles”, J. Amer. Math. Soc. 11 (1): 73–118, doi:10.1090/S0894-0347-98-00248-3, ISSN 0894-0347, MR 1460391, Zbl 1025.11009

• Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003), Cohomological invariants in Galois coho-mology, University Lecture Series 28, Providence, RI: American Mathematical Society, ISBN 0-8218-3287-5,MR 1999383, Zbl 1159.12311

• Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions,Colloquium Publications 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, p.436, ISBN 0-8218-0904-0, Zbl 0955.16001

12

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Chapter 7

Arithmetic group

In mathematics, an arithmetic group is a subgroup of a linear algebraic group with simple algebraic properties.

7.1 Formal definition

An arithmetic group (arithmetic subgroup) in a linear algebraic groupG defined over a number field K is a subgroupΓ of G(K) that is commensurable with G(O), where O is the ring of integers of K. Here two subgroups A and B of agroup are commensurable when their intersection has finite index in each of them. It can be shown that this conditiondepends only on G, not on a given matrix representation of G.

7.2 Examples

Examples of arithmetic groups include the groups GLn(Z). The idea of arithmetic group is closely related to that oflattice in a Lie group. Lattices in that sense tend to be arithmetic, except in well-defined circumstances.

7.3 History

The exact relationship of arithmetic groups and lattices in Lie groups was established by the work of Margulis onsuperrigidity. The general theory of arithmetic groups was developed by Armand Borel and Harish-Chandra; thedescription of their fundamental domains was in classical terms the reduction theory of algebraic forms.

7.4 References• Hazewinkel, Michiel, ed. (2001), “Arithmetic group”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

13

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Chapter 8

Borel subgroup

In the theory of algebraic groups, aBorel subgroup of an algebraic groupG is amaximal Zariski closed and connectedsolvable algebraic subgroup. For example, in the group GLn (n x n invertible matrices), the subgroup of invertibleupper triangular matrices is a Borel subgroup.For groups realized over algebraically closed fields, there is a single conjugacy class of Borel subgroups.Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive)algebraic groups, in Jacques Tits' theory of groups with a (B,N) pair. Here the group B is a Borel subgroup and N isthe normalizer of a maximal torus contained in B.The notion was introduced by Armand Borel, who played a leading role in the development of the theory of algebraicgroups.

8.1 Parabolic subgroups

Subgroups between a Borel subgroup B and the ambient group G are called parabolic subgroups. Parabolic sub-groups P are also characterized, among algebraic subgroups, by the condition thatG/P is a complete variety. Workingover algebraically closed fields, the Borel subgroups turn out to be the minimal parabolic subgroups in this sense.Thus B is a Borel subgroup when the homogeneous space G/B is a complete variety which is “as large as possible”.For a simple algebraic group G, the set of conjugacy classes of parabolic subgroups is in bijection with the set of allsubsets of nodes of the corresponding Dynkin diagram; the Borel subgroup corresponds to the empty set and G itselfcorresponding to the set of all nodes. (In general each node of the Dynkin diagram determines a simple negativeroot and thus a one-dimensional 'root group' of G---a subset of the nodes thus yields a parabolic subgroup, generatedby B and the corresponding negative root groups. Moreover any parabolic subgroup is conjugate to such a parabolicsubgroup.)

8.2 Lie algebra

For the special case of a Lie algebra g with a Cartan subalgebra h , given an ordering of h , the Borel subalgebra is thedirect sum of h and the weight spaces of g with positive weight. A Lie subalgebra of g containing a Borel subalgebrais called a parabolic Lie algebra.

8.3 See also

• Hyperbolic group

14

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8.4. REFERENCES 15

8.4 References• Gary Seitz (1991). “Algebraic Groups”. In B. Hartley et al. Finite and Locally Finite Groups. pp. 45–70.

• J. Humphreys (1972). Linear Algebraic Groups. New York: Springer. ISBN 0-387-90108-6.

• A. Borel (2001). Essays in the History of Lie Groups and Algebraic Groups. Providence RI: AMS. ISBN0-8218-0288-7.

8.5 External links• Popov, V.L. (2001), “Parabolic subgroup”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Platonov, V.P. (2001), “Borel subgroup”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

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Chapter 9

Borel–de Siebenthal theory

In mathematics, Borel–de Siebenthal theory describes the closed connected subgroups of a compact Lie group thathavemaximal rank, i.e. contain a maximal torus. It is named after the Swiss mathematicians Armand Borel and Jeande Siebenthal who developed the theory in 1949. Each such subgroup is the identity component of the centralizer ofits center. They can be described recursively in terms of the associated root system of the group. The subgroups forwhich the corresponding homogeneous space has an invariant complex structure correspond to parabolic subgroupsin the complexification of the compact Lie group, a reductive algebraic group.

9.1 Connected subgroups of maximal rank

Let G be connected compact Lie group with maximal torus T. Hopf showed that the centralizer of a torus S ⊆ T isa connected closed subgroup containing T, so of maximal rank. Indeed, if x is in CG(S), there is a maximal toruscontaining both S and x and it is contained in CG(S).[1]

Borel and de Siebenthal proved that the connected closed subgroups of maximal rank are precisely the identity com-ponents of the centralizers of their centers.[2]

Their result relies on a fact from representation theory. The weights of an irreducible representation of a connectedcompact semisimple group K with highest weight λ can be easily described (without their multiplicities): they areprecisely the saturation under the Weyl group of the dominant weights obtained by subtracting off a sum of simpleroots from λ. In particular, if the irreducible representation is trivial on the center of K (a finite Abelian group), 0 isa weight.[3]

To prove the characterization of Borel and de Siebenthal, let H be a closed connected subgroup of G containing Twith center Z. The identity component L of CG(Z) contains H. If it were strictly larger, the restriction of the adjointrepresentation of L to H would be trivial on Z. Any irreducible summand, orthogonal to the Lie algebra of H, wouldprovide non-zero weight zero vectors for T / Z ⊆ H / Z, contradicting the maximality of the torus T / Z in L / Z.[4]

9.2 Maximal connected subgroups of maximal rank

Borel and de Siebenthal classified the maximal closed connected subgroups of maximal rank of a connected compactLie group.The general classification of connected closed subgroups of maximal rank can be reduced to this case, because anyconnected subgroup of maximal rank is contained in a finite chain of such subgroups, each maximal in the next one.Maximal subgroups are the identity components of any element of their center not belonging to the center of thewhole group.The problem of determining the maximal connected subgroups of maximal rank can be further reduced to the casewhere the compact Lie group is simple. In fact the Lie algebra g of a connected compact Lie group G splits as adirect sum of the ideals

16

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9.2. MAXIMAL CONNECTED SUBGROUPS OF MAXIMAL RANK 17

g = z⊕ g1 ⊕ · · · ⊕ gm,

where z is the center and the other factors gi are simple. If T is a maximal torus, its Lie algebra t has a correspondingsplitting

t = z⊕ t1 ⊕ · · · ⊕ tm,

where ti is maximal abelian in gi . If H is a closed connected of G containing T with Lie algebra h , the complexi-fication of h is the direct sum of the complexification of t and a number of one-dimensional weight spaces, each ofwhich lies in the complexification of a factor gi . Thus if

hi = h ∩ gi,

then

h = z⊕ h1 ⊕ · · · ⊕ hm.

If H is maximal, all but one of the hi 's coincide with gi and the remaining one is maximal and of maximal rank.For that factor, the closed connected subgroup of the corresponding simply connected simple compact Lie group ismaximal and of maximal rank.[5]

Let G be a connected simply connected compact simple Lie group with maximal torus T. Let g be the Lie algebra ofG and t that of T. Let Δ be the corresponding root system. Choose a set of positive roots and corresponding simpleroots α1, ..., αn. Let α0 the highest root in gC and write

α0 = m1α1 + · · ·+mnαn

with mi ≥ 1. (The number of mi equal to 1 is equal to |Z| – 1, where Z is the center of G.)TheWeyl alcove is defined by

A = T ∈ t : α1(T ) ≥ 0, . . . , αn(T ) ≥ 0, α0(T ) ≤ 1.

Élie Cartan shouwed that it is a fundamental domain for the affine Weyl group. If G1 = G / Z and T1 = T / Z, itfollows that the exponential mapping from g to G1 carries 2πA onto T1.The Weyl alcove A is a simplex with vertices at

v0 = 0, vi = m−1i Xi,

where αi(Xj) = δij.The main result of Borel and de Siebenthal is as follows.THEOREM. The maximal connected subgroups of maximal rank in G1 up to conjugacy have the form• CG1 (Xi) for mi = 1• CG1(vi) for mi a prime.The structure of the corresponding subgroup H1 can be described in both cases. It is semisimple in the second casewith a system of simple roots obtained by replacing αi by −α0. In the first case it is the direct product of the circlegroup generated by Xi and a semisimple compact group with a system of simple roots obtained by omitting αi.This result can be rephrased in terms of the extended Dynkin diagram of g which adds an extra node for the highestroot as well as the labelsmi. The maximal subalgebras h of maximal rank are either non-semisimple or semimsimple.The non-semisimple ones are obtained by deleting two nodes from the extended diagram with coefficient one. Thecorresponding unlabelled diagram gives the Dynkin diagram semisimple part of h , the other part being a one-dimensional factor. The Dynkin diagrams for the semisimple ones are obtained by removing one node with coefficienta prime. This leads to the following possibilities:

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18 CHAPTER 9. BOREL–DE SIEBENTHAL THEORY

• An: Ap × A n ₋ p ₋ ₁ × T (non-semisimple)

• Bn: Dn or Bp × Dn ₋ p (semisimple), Bn ₋ ₁ × T (non-semisimple)

• Cn: Cp × Cn ₋ p (SS), An - ₁ × T (NSS)

• Dn: Dp × Dn - p (SS), Dn - ₁ × T, A -₁ × T (NSS)

• E6: A1 × A5, A2 × A2 × A2 (SS), D5 × T (NSS)

• E7: A1 × D6, A2 × A5, A7 (SS), E6 × T (NSS)

• E8: D8, A8, A4 × A4, E6 × A2, E7 × A1 (SS)

• F4: B4, A2 × A2, A1 × C3 (SS)

• G2: A2, A1 × A1 (SS)

All the corresponding homogeneous spaces are symmetric, since the subalgebra is the fixed point algebra of an innerautomorphism of period 2, apart fromG2/A2, F4/A2×A2, E6/A2×A2×A2, E7/A2×A5 and all the E8 spaces other thanE8/D8 and E8/E7×A1. In all these exceptional cases the subalgebra is the fixed point algebra of an inner automorphismof period 3, except for E8/A4×A4 where the automorphism has period 5. The homogeneous spaces are then calledweakly symmetric spaces.To prove the theorem, note that H1 is the identity component of the centralizer of an element exp T with T in 2π A.Stabilizers increase in moving from a subsimplex to an edge or vertex, so T either lies on an edge or is a vertex. Ifit lies on an edge than that edge connects 0 to a vertex vi with mi = 1, which is the first case. If T is a vertex vi andmi has a non-trivial factor m, then mT has a larger stabilizer than T, contradicting maximality. So mi must be prime.Maximality can be checked directly using the fact that an intermediate subgroup K would have the same form, so thatits center would be either (a) T or (b) an element of prime order. If the center of H1 is 'T, each simple root with miprime is already a root of K, so (b) is not possible; and if (a) holds, αi is the only root that could be omitted with mj= 1, so K = H1. If the center of H1 is of prime order, αj is a root of K for mj = 1, so that (a) is not possible; if (b)holds, then the only possible omitted simple root is αi, so that K = H1.[6]

9.3 Closed subsystems of roots

A subset Δ1 ⊂ Δ is called a closed subsystem if whenever α and β lie in Δ1 with α + β in Δ, then α + β lies in Δ1.Two subsystems Δ1 and Δ2 are said to be equivalent if σ( Δ1) = Δ2 for some σ inW = NG(T) / T, the Weyl group.Thus for a closed subsystem

tC ⊕⊕α∈∆1

is a subalgebra of gC containing tC ; and conversely any such subalgebra gives rise to a closed subsystem. Borel andde Siebenthal classified the maximal closed subsystems up to equivalence.[7]

THEOREM. Up to equivalence the closed root subsystems are given by mi = 1 with simple roots all αj with j≠ i or by mi > 1 prime with simple roots −α0 and all αj with j ≠ i.This result is a consequence of the Borel–de Siebenthal theorem for maximal connected subgroups of maximal rank.It can also be proved directly within the theory of root systems and reflection groups.[8]

9.4 Applications to symmetric spaces of compact type

Let G be a connected compact semisimple Lie group, σ an automorphism of G of period 2 and Gσ the fixed pointsubgroup of σ. Let K be a closed subgroup of G lying between Gσ and its identity component. The compact homo-geneous space G / K is called a symmetric space of compact type. The Lie algebra g admits a decomposition

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9.4. APPLICATIONS TO SYMMETRIC SPACES OF COMPACT TYPE 19

g = k⊕ p,

where k , the Lie algebra of K, is the +1 eigenspace of σ and p the –1 eigenspace. If k contains no simple summandof g , the pair ( g , σ) is called an orthogonal symmetric Lie algebra of compact type.[9]

Any inner product on g , invariant under the adjoint representation and σ, induces a Riemannian structure on G /K, with G acting by isometries. Under such an inner product, k and p are orthogonal. G / K is then a Riemanniansymmetric space of compact type.[10]

The symmetric space or the pair ( g , σ) is said to be irreducible if the adjoint action of k (or equivalently the identitycomponent of Gσ or K) is irreducible on p . This is equivalent to the maximality of k as a subalgebra.[11]

In fact there is a one-one correspondence between intermediate subalgebras h and K-invariant subspaces p1 of p givenby

h = k⊕ p1, p1 = h ∩ p.

Any orthogonal symmetric algebra ( g , σ) can be decomposed as an (orthogonal) direct sum of irreducible orthogonalsymmetric algebras.[12]

In fact g can be written as a direct sum of simple algebras

g = ⊕Ni=1gi,

which are permuted by the automorphism σ. If σ leaves an algebra g1 invariant, its eigenspace decomposition coin-cides with its intersections with k and p . So the restriction of σ to g1 is irreducible. If σ interchanges two simplesummands, the corresponding pair is isomorphic to a diagonal inclusion of K in K × K, with K simple, so is alsoirreducible. The involution σ just swaps the two factors σ(x,y)=(y,x).This decomposition of an orthogonal symmetric algebra yields a direct product decomposition of the correspondingcompact symmetric spaceG / K whenG is simply connected. In this case the fixed point subgroupGσ is automaticallyconnected (this is no longer true, even for inner involutions, if G is not simply connected).[13] For simply connectedG, the symmetric space G / K is the direct product of the two kinds of symmetric spaces Gi / Ki or H × H / H.Non-simply connected symmetric space of compact type arise as quotients of the sinply connected space G / K byfinite Abelian groups. In fact if

G/K = G1/K1 × · · · ×Gs/Ks,

let

Γi = Z(Gi)/Z(Gi) ∩Ki

and let Δi be the subgroup of Γi fixed by all automorphisms ofGi preserving Ki (i.e. automorphisms of the orthogonalsymmetric Lie algebra). Then

∆ = ∆1 × · · · ×∆s

is a finite Abelian group acting freely on G / K. The non-simply connected symmetric spaces arise as quotients bysubgroups of Δ. The subgroup can be identified with the fundamental group, which is thus a finite Abelian group.[14]

The classification of compact symmetric spaces or pairs ( g , σ) thus reduces to the case whereG is a connected simplecompact Lie group. There are two possibilities: either the automorphism σ is inner, in which case K has maximalrank and the theory of Borel and de Siebenthal applies; or the automorphism σ is outer, so that, because σ preserves amaximal torus, the rank of K is less than the rank ofG and σ corresponds to an automorphism of the Dynkin diagrammodulo inner automorphisms. Wolf (2010) determines directly all possible σ in the latter case: they correspond tothe symmetric spaces SU(n)/SO(n), SO(a+b)/SO(a)×SO(b) (a and b odd), E6/F4 and E6/C4.[15]

Victor Kac noticed that all finite order automorphisms of a simple Lie algebra can be determined using the corre-sponding affine Lie algebra: that classification, which leads to an alternative method of classifying pairs ( g , σ), isdescribed in Helgason (1978).

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20 CHAPTER 9. BOREL–DE SIEBENTHAL THEORY

9.5 Applications to hermitian symmetric spaces of compact type

The equal rank case with K non-semisimple corresponds exactly to the Hermitian symmetric spacesG / K of compacttype.In fact the symmetric space has an almost complex structure preserving the Riemannian metric if and only if there isa linear map J with J2 = −I on p which preserves the inner product and commutes with the action of K. In this case Jlies in k and exp Jt forms a one-parameter group in the center of K. This follows because if A, B, C, D lie in p , thenby the invariance of the inner product on g [16]

([[A,B], C], D) = ([A,B], [C,D]) = ([[C,D], B], A).

Replacing A and B by JA and JB, it follows that

[JA, JB] = [A,B].

Define a linear map δ on g by extending J to be 0 on k . The last relation shows that δ is a derivation of g . Since gis semisimple, δ must be an inner derivation, so that

δ(X) = [T +A,X],

with T in k and A in p . Taking X in k , it follows that A = 0 and T lies in the center of k and hence that K isnon-semisimple. [17]

If on the other hand G / K is irreducible with K non-semisimple, the compact group G must be simple and K ofmaximal rank. From the theorem of Borel and de Siebenthal, the involution σ is inner and K is the centralizer of atorus S. It follows that G / K is simply connected and there is a parabolic subgroup P in the complexification GC ofG such that G / K = GC / P. In particular there is a complex structure on G / K and the action of G is holomorphic.In general any compact hermitian symmetric space is simply connected and can be written as a direct product ofirreducible hermitian symmetric spaces Gi / Ki with Gi simple. The irreducible ones are exactly the non-semisimplecases described above.[18]

9.6 Notes[1] Helgason 1978

[2] Wolf 2010

[3] See:

• Wolf 2010• Bourbaki 1981• Humphreys 1997• Duistermaat & Kolk 2000

[4] Wolf 2010

[5] Wolf, p. 276

[6] See:

• Wolf 2010• Kane 2001

[7] Kane 2001, pp. 135–136

[8] Kane 2007

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9.7. REFERENCES 21

[9] Wolf 2010

[10] See:

• Helgason 1978• Wolf 2010

[11] See:

• Wolf 2010• Helgason 1978, p. 378

[12] See:

• Helgason 1978, pp. 378–379• Wolf 2010

[13] Helgason 1978, pp. 320–321

[14] See:

• Wolf 2010, pp. 244,263–264• Helgason 1978, p. 326

[15] Wolf 2010

[16] Kobayashi & Nomizu 1996, pp. 149–150

[17] Kobayashi & Nomizu 1996, pp. 261–262

[18] Wolf 2010

9.7 References• Borel, A.; De Siebenthal, J. (1949), “Les sous-groupes fermés de rang maximum des groupes de Lie clos”,Commentarii mathematici Helvetici 23: 200–221

• Borel, Armand (1952), Les espaces hermitiens symétriques, Exposé No. 62, Séminaire Bourbaki 2

• Bourbaki, N. (1981), Groupes et Algèbres de Lie (Chapitres 4,5 et 6), Éléments de Mathématique, Masson,ISBN 978-3-540-34490-2

• Bourbaki, N. (1982), Groupes et Algèbres de Lie (Chapitre 9), Éléments de Mathématique, Masson, ISBN978-3-540-34392-9

• Duistermaat, J.J.; Kolk, A. (2000), Lie groups, Universitext, Springer, ISBN 978-3-540-15293-4

• Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN978-0-8218-2848-9

• Humphreys, James E. (1981), Linear Algebraic Groups, Graduate texts in mathematics 21, Springer, ISBN978-0-387-90108-4

• Humphreys, James E. (1997), Introduction to Lie Algebras and Representation Theory, Graduate texts in math-ematics 9 (2nd ed.), Springer, ISBN 978-3-540-90053-5

• Kane, Richard (2001), Reflection Groups and Invariant Theory, Springer, ISBN 978-0-387-98979-2

• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of differential geometry 2, Wiley-Interscience,ISBN 978-0-471-15732-8

• Malle, Gunter; Testerman, Donna (2011), Linear Algebraic Groups and Finite Groups of Lie Type, CambridgeStudies in Advanced Mathematics 133, Cambridge University Press, ISBN 978-1-139-49953-8

• Wolf, Joseph A. (2010), Spaces of constant curvature, AMS Chelsea Publishing (6th ed.), American Mathe-matical Society, ISBN 978-0-8218-5282-8

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22 CHAPTER 9. BOREL–DE SIEBENTHAL THEORY

Extended Dynkin diagrams for the simple complex Lie algebras

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Chapter 10

Bott–Samelson variety

In mathematics, Bott–Samelson varieties were introduced independently by Hansen (1973) and Demazure (1974)(who named them Bott–Samelson varieties) as an algebraic group analogue of the spaces constructed for compactgroups by Bott and Samelson (1958, p. 970). They are sometimes desingularizations of Schubert varieties.A Bott–Samelson variety Z can be constructed as

P1 ×B P2 ×B × · · · ×B Pn/B

where B is a Borel subgroup of a reductive algebraic group G and the Ps are minimal parabolic subgroups containingB. (An element b of B acts on P on the right as right multiplication by b and acts on P on the left as left multiplicationby b−1.) Taking the product of its coordinates gives a proper map from the Bott–Samelson variety Z to the flagvariety G/B whose image is a Schubert variety. In some cases this map is birational and gives a desingularization ofthe Schubert variety.See also Bott–Samelson resolution.

10.1 References• Bott, Raoul; Samelson, Hans (1958), “Applications of the theory of Morse to symmetric spaces”, AmericanJournal of Mathematics 80: 964–1029, doi:10.2307/2372843, ISSN 0002-9327, MR 0105694

• Demazure, Michel (1974), “Désingularisation des variétés de Schubert généralisées”, Annales Scientifiques del'École Normale Supérieure. Quatrième Série 7: 53–88, ISSN 0012-9593, MR 0354697

• Hansen, H. C. (1973), “On cycles in flag manifolds”, Mathematica Scandinavica 33: 269–274, ISSN 0025-5521, MR 0376703

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Chapter 11

Bruhat decomposition

In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by ClaudeChevalley in general) G = BWB into cells can be regarded as a general expression of the principle of Gauss–Jordanelimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—butwith exceptional cases. It is related to the Schubert cell decomposition of Grassmannians: see Weyl group for this.More generally, any group with a (B,N) pair has a Bruhat decomposition.

11.1 Definitions

• G is a connected, reductive algebraic group over an algebraically closed field.

• B is a Borel subgroup of G

• W is a Weyl group of G corresponding to a maximal torus of B.

The Bruhat decomposition of G is the decomposition

G = BWB =⨿

w∈W

BwB

ofG as a disjoint union of double cosets of B parameterized by the elements of theWeyl groupW. (Note that althoughW is not in general a subgroup of G, the coset wB is still well defined.)

11.2 Examples

Let G be the general linear group GL of invertible n × n matrices with entries in some algebraically closed field,which is a reductive group. Then the Weyl group W is isomorphic to the symmetric group Sn on n letters, withpermutation matrices as representatives. In this case, we can take B to be the subgroup of upper triangular invertiblematrices, so Bruhat decomposition says that one can write any invertible matrix A as a product U1PU2 where U1 andU2 are upper triangular, and P is a permutation matrix. Writing this as P = U1

−1AU2−1, this says that any invertible

matrix can be transformed into a permutation matrix via a series of row and column operations, where we are onlyallowed to add row i (resp. column i) to row j (resp. column j) if i>j (resp. i<j). The row operations correspond toU1

−1, and the column operations correspond to U2−1.

The special linear group SL of invertible n × n matrices with determinant 1 is a semisimple group, and hencereductive. In this case,W is still isomorphic to the symmetric group Sn. However, the determinant of a permutationmatrix is the sign of the permutation, so to represent an odd permutation in SL , we can take one of the nonzeroelements to be −1 instead of 1. Here B is the subgroup of upper triangular matrices with determinant 1, so theinterpretation of Bruhat decomposition in this case is similar to the case of GL .

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11.3. GEOMETRY 25

11.3 Geometry

The cells in the Bruhat decomposition correspond to the Schubert cell decomposition of Grassmannians. The dimen-sion of the cells corresponds to the length of the word w in the Weyl group. Poincaré duality constrains the topologyof the cell decomposition, and thus the algebra of the Weyl group; for instance, the top dimensional cell is unique (itrepresents the fundamental class), and corresponds to the longest element of a Coxeter group.

11.4 Computations

The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the q-polynomial[1] ofthe associated Dynkin diagram.

11.5 See also• Lie group decompositions

• Birkhoff factorization, a special case of the Bruhat decomposition for affine groups.

11.6 Notes[1] This Week’s Finds in Mathematical Physics, Week 186

11.7 References• Borel, Armand. Linear Algebraic Groups (2nd ed.). NewYork: Springer-Verlag, 1991. ISBN 0-387-97370-2.

• Bourbaki, Nicolas, Lie Groups and Lie Algebras: Chapters 4-6 (Elements of Mathematics), Springer-Verlag,2008. ISBN 3-540-42650-7

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Chapter 12

Cartan subgroup

In mathematics, a Cartan subgroup of a Lie group or algebraic group G is one of the subgroups whose Lie algebrais a Cartan subalgebra. The dimension of a Cartan subgroup, and therefore of a Cartan subalgebra, is the rank of G.

12.1 Conventions

The identity component of a subgroup has the same Lie algebra. There is no standard convention for which one ofthe subgroups with this property is called the Cartan subgroup, especially in the case of disconnected groups.

12.2 Definitions

A Cartan subgroup of a compact connected Lie group is a maximal connected Abelian subgroup (a maximaltorus). Its Lie algebra is a Cartan subalgebra.For disconnected compact Lie groups there are several inequivalent definitions of a Cartan subgroup. The mostcommon seems to be the one given by David Vogan, who defines a Cartan subgroup to be the group of elements thatnormalize a fixed maximal torus and fix the fundamental Weyl chamber. This is sometimes called the large Cartansubgroup. There is also a small Cartan subgroup, defined to be the centralizer of a maximal torus. These Cartansubgroups need not be abelian in general.For connected algebraic groups over an algebraically closed field a Cartan subgroup is usually defined as thecentralizer of a maximal torus. In this case the Cartan subgroups are connected, nilpotent, and are all conjugate.

12.3 See also• Cartan subalgebra

• Carter subgroup

12.4 References• Armand Borel (1991-12-31). Linear algebraic groups. ISBN 3-540-97370-2.

• Anthony William Knapp; David A. Vogan (1995). Cohomological Induction and Unitary Representations.ISBN 978-0-691-03756-1.

• Popov, V. L. (2001), “C/c020560”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

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Chapter 13

Chevalley’s structure theorem

In algebraic geometry, Chevalley’s structure theorem states that a connected algebraic group over a perfect fieldhas a unique normal affine algebraic subgroup such that the quotient is an abelian variety. It was proved by Chevalley(1960) (though he had previously announced the result in 1953), Barsotti (1955), and Rosenlicht (1956).Chevalley’s original proof, and the other early proofs by Barsotti and Rosenlicht, used the idea of mapping thealgebraic group to its Albanese variety. The original proofs were based on Weil’s book Foundations of algebraicgeometry, but Conrad (2002) later gave an exposition of Chevalley’s proof in scheme-theoretic terminology.

13.1 References• Barsotti, Iacopo (1955), “Structure theorems for group-varieties”, Annali di Matematica Pura ed Applicata.Serie Quarta 38: 77–119, doi:10.1007/bf02413515, ISSN 0003-4622, MR 0071849

• Barsotti, Iacopo (1955), “Un teorema di struttura per le varietà gruppali”, Atti della Accademia Nazionale deiLincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 18: 43–50, MR 0076427

• Chevalley, C. (1960), “Une démonstration d'un théorème sur les groupes algébriques”, Journal de Mathéma-tiques Pures et Appliquées. Neuvième Série 39: 307–317, ISSN 0021-7824, MR 0126447

• Conrad, Brian (2002), “A modern proof of Chevalley’s theorem on algebraic groups”, Journal of the Ramanu-jan Mathematical Society 17 (1): 1–18, ISSN 0970-1249, MR 1906417

• Rosenlicht, Maxwell (1956), “Some basic theorems on algebraic groups”, American Journal of Mathematics78: 401–443, doi:10.2307/2372523, ISSN 0002-9327, MR 0082183

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Chapter 14

Cohomological invariant

In mathematics, a cohomological invariant of an algebraic group G over a field is an invariant of forms of G takingvalues in a Galois cohomology group.

14.1 Definition

Suppose that G is an algebraic group defined over a field K, and choose a separably closed field K containing K. Fora finite extension L of K in K let ΓL be the absolute Galois group of L. The first cohomology H1(L, G) = H1(ΓL, G)is a set classifying the forms of G over L, and is a functor of L.A cohomological invariant ofG of dimension d taking values in a ΓK-moduleM is a natural transformation of functors(of L) from H1(L, G) to Hd(L, M).In other words a cohomological invariant associates an element of an abelian cohomology group to elements of anon-abelian cohomology set.More generally, if A is any functor from finitely generated extensions of a field to sets, then a cohomological invariantof A of dimension d taking values in a Γ-module M is a natural transformation of functors (of L) from A to Hd(L,M).The cohomological invariants of a fixed group G or functor A, dimension d and Galois module M form an abeliangroup denoted by Invd(G,M) or Invd(A,M).

14.2 Examples

• Suppose A is the functor taking a field to the isomorphism classes of dimension n etale algebras over it. Thecohomological invariants with coefficients in Z/2Z is a free module over the cohomology of k with a basis ofelements of degrees 0, 1, 2, ..., m where m is the integer part of n/2.

• The Hasse−Witt invariant of a quadratic form is essentially a dimension 2 cohomological invariant of thecorresponding spin group taking values in a group of order 2.

• IfG is a quotient of a group by a smooth finite central subgroup C, then the boundary map of the correspondingexact sequence gives a dimension 2 cohomological invariant with values in C. If G is a special orthogonal groupand the cover is the spin group then the corresponding invariant is essentially the Hasse−Witt invariant.

• IfG is the orthogonal group of a quadratic form in characteristic not 2, then there are Stiefel–Whitney classes foreach positive dimension which are cohomological invariants with values in Z/2Z. (These are not the topologicalStiefel–Whitney classes of a real vector bundle, but are the analogues of them for vector bundles over a scheme.)For dimension 1 this is essentially the discriminant, and for dimension 2 it is essentially the Hasse−Witt in-variant.

• The Arason invariant e3 is a dimension 3 invariant of some even dimensional quadratic forms q with trivialdiscriminant and trivial Hasse−Witt invariant. It takes values in Z/2Z. It can be used to construct a dimension

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14.3. REFERENCES 29

3 cohomological invariant of the corresponding spin group as follows. If u is in H1(K, Spin(q)) and p is thequadratic form corresponding to the image of u in H1(K, O(q)), then e3(p−q) is the value of the dimension 3cohomological invariant on u.

• The Merkurjev−Suslin invariant is a dimension 3 invariant of a special linear group of a central simple algebraof rank n taking values in the tensor square of the group of nth roots of unity. When n=2 this is essentially theArason invariant.

• For absolutely simple simply connected groups G, the Rost invariant is a dimension 3 invariant taking values inQ/Z(2) that in some sense generalizes the Arason invariant and theMerkurjev−Suslin invariant to more generalgroups.

14.3 References• Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003), Cohomological invariants in Galois coho-mology, University Lecture Series 28, Providence, RI: American Mathematical Society, ISBN 0-8218-3287-5,MR 1999383

• Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involu-tions, Colloquium Publications 44, Providence, RI: American Mathematical Society, ISBN 0-8218-0904-0,Zbl 0955.16001

• Serre, Jean-Pierre (1995), “Cohomologie galoisienne: progrès et problèmes”, Astérisque, Séminaire Bourbaki,Vol. 1993/94. Exp. No. 783, 227: 229–257, MR 1321649

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Chapter 15

Complexification (Lie group)

In mathematics, the complexification or universal complexification of a real Lie group is given by a continuoushomomorphism of the group into a complex Lie group with the universal property that every continuous homomor-phism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphismbetween the complex Lie groups. The complexification, which always exists, is unique up to isomorphism. Its Liealgebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if theoriginal group has a quotient by a discrete normal subgroup which is linear.For compact Lie groups, the complexification, sometimes called theChevalley complexification after ClaudeCheval-ley, can be defined as the group of complex characters of the Hopf algebra of representative functions, i.e. the matrixcoefficients of finite-dimensional representations of the group. In any finite-dimensional faithful unitary representa-tion of the compact group it can be realized concretely as a closed subgroup of the complex general linear group. Itconsists of operators with polar decomposition g = u • exp iX, where u is a unitary operator in the compact group andX is a skew-adjoint operator in its Lie algebra. In this case the complexification is a complex algebraic group and itsLie algebra is the complexification of the Lie algebra of the compact Lie group.

15.1 Universal complexification

15.1.1 Definition

If G is a Lie group, a universal complexification is given by a complex Lie group GC and a continuous homomor-phism φ: G → GC with the universal property that, if f: G → H is an arbitrary continuous homomorphism into acomplex Lie group H, then there is a unique complex analytic homomorphism F: GC→ H such that f = F ∘ φ.

15.1.2 Uniqueness

The universal property implies that the universal complexification, if it exists, is unique up to complex analytic iso-morphism.

15.1.3 Existence

If G is connected with Lie algebra 𝖌, then its universal covering group G is simply connected. Let GC be the simplyconnected complex Lie group with Lie algebra 𝖌 ⊗ C. Let Φ: G→ GC be the natural homomorphism and π:G→ Gthe natural covering map. Then given a homomorphism f: G→H, there is a unique complex analytic homomorphismE: GC → H such that f ∘ π = E ∘ Φ. Let K be the intersection of the kernels of the homomorphisms E as f variesover all possibilities. Then K is a closed normal complex Lie subgroup of GC and the quotient group is a universalcomplexification. In particular if G is simply connected, its universal complexification is just GC.[1]

For non-connected Lie groups G with identity component Go and component group Γ = G / Go, the extension

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15.2. CHEVALLEY COMPLEXIFICATION 31

1 → Go → G→ Γ → 1

induces an extension

1 → (Go)C → GC → Γ → 1

and the complex Lie group GC is a complexification of G.[2]

15.1.4 Injectivity

If the original group is linear, so too is the universal complexification and the homomorphism between the two is aninclusion.[3] Onishchik&Vinberg (1994) give an example of a connected real Lie group for which the homomorphismis not injective even at the Lie algebra level: they take the product of T by the universal covering group of SL(2,R)and quotient out by the discrete cyclic subgroup generated by an irrational rotation in the first factor and a generatorof the center in the second.

15.2 Chevalley complexification

15.2.1 Hopf algebra of matrix coefficients

If G is a compact Lie group, the *-algebra A of matrix coefficients of finite-dimensional unitary representations is auniformly dense *-subalgebra of C(G), the *-algebra of complex-valued continuous functions on G. It is naturally aHopf algebra with comultiplication given by

∆f(g, h) = f(gh).

The characters of A are the *-homomorphisms of A into C. They can be identified with the point evaluations f ↦f(g) for g in G and the comultiplication allows the group structure on G to be recovered. The homomorphisms of Ainto C also form a group. It is a complex Lie group and can be identified with the complexification GC of G. The*-algebra A is generated by the matrix coefficients of any faithful representation σ of G. It follows that σ defines afaithful complex analytic representation of GC.[4]

15.2.2 Invariant theory

The original approach of Chevalley (1946) to the complexification of a compact Lie group can be concisely statedwithin the language of classical invariant theory, described in Weyl (1946). Let G be a closed subgroup of the unitarygroup U(V) where V is a finite-dimensional complex inner product space. Its Lie algebra consists of all skew-adjointoperatorsX such that exp tX lies inG for all real t. SetW =V ⊕Cwith the trivial action ofG on the second summand.The group G acts on W⊗N , with an element u acting as u⊗N . The commutant (or centralizer algebra) is denoted byAN = EndG W⊗N . It is generated as a *-algebra by its unitary operators and its commutant is the *-algebra spannedby the operators u⊗N . The complexification GC of G consists of all operators g in GL(V) such that g⊗N commuteswith AN and g acts trivially on the second summand inC. By definition it is a closed subgroup of GL(V). The definingrelations (as a commutant) show that G is an algebraic subgroup. Its intersection with U(V) coincides with G, sinceit is a priori a larger compact group for which the irreducible representations stay irreducible and inequivalent whenrestricted to G. Since AN is generated by unitaries, an invertible operator g lies in GC if the unitary operator u andpositive operator p in its polar decomposition g = u ⋅ p both lie in GC. Thus u lies in G and the operator p can bewritten uniquely as p = exp T with T a self-adjoint operator. By the functional calculus for polynomial functions itfollows that h⊗N lies in the commutant of AN if h = exp z T with z in C. In particular taking z purely imaginary, Tmust have the form iX with X in the Lie algebra of G. Since every finite-dimensional representation of G occurs asa direct summand of W⊗N , it is left invariant by GC and thus every finite-dimensional representation of G extendsuniquely to GC. The extension is compatible with the polar decomposition. Finally the polar decomposition impliesthat G is a maximal compact subgroup of GC, since a strictly larger compact subgroup would contain all integerpowers of a positive operator p, a closed infinite discrete subgroup.[5]

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32 CHAPTER 15. COMPLEXIFICATION (LIE GROUP)

15.3 Decompositions in the Chevalley complexification

15.3.1 Cartan decomposition

The decomposition derived from the polar decomposition

GC = G · P = G · exp ig,

where 𝖌 is the Lie algebra of G, is called the Cartan decomposition of GC. The exponential factor P is invariantunder conjugation byG but is not a subgroup. The complexification is invariant under taking adjoints, sinceG consistsof unitary operators and P of positive operators.

15.3.2 Gauss decomposition

The Gauss decomposition is a generalization of the LU decomposition for the general linear group and a special-ization of the Bruhat decomposition. For GL(V) it states that with respect to a given orthonormal basis e1, …, en anelement g of GL(V) can be factorized in the form

g = XDY

with X lower unitriangular, Y upper unitriangular and D diagonal if and only if all the principal minors of g arenon-vanishing. In this case X, Y and D are uniquely determined.In fact Gaussian elimination shows there is a unique X such that X−1 g is upper triangular.[6]

The upper and lower unitriangular matrices, N₊ andN₋, are closed unipotent subgroups of GL(V). Their Lie algebrasconsist of upper and lower strictly triangular matrices. The exponential mapping is a polynomial mapping from theLie algebra to the corresponding subgroup by nilpotence. The inverse is given by the logarithm mapping which byunipotence is also a polynomial mapping. In particular there is a correspondence between closed connected subgroupsof N± and subalgebras of their Lie algebras. The exponential map is onto in each case, since the polynomial functionlog ( eA eB ) lies in a given Lie subalgebra if A and B do and are sufficiently small.[7]

The Gauss decomposition can be extended to complexifications of other closed connected subgroups G of U(V) byusing the root decomposition to write the complexified Lie algebra as[8]

gC = n− ⊕ tC ⊕ n+,

where 𝖙 is the Lie algebra of a maximal torus T of G and 𝖓± are the direct sum of the corresponding positive andnegative root spaces. In the weight space decomposition of V as eigenspaces of T, 𝖙 acts as diagonally, 𝖓₊ acts aslowering operators and 𝖓₋ as raising operators. 𝖓± are nilpotent Lie algebras acting as nilpotent operators; they areeach other’s adjoints on V. In particular T acts by conjugation of 𝖓₊, so that 𝖙C ⊕ 𝖓₊ is a semidirect product of anilpotent Lie algebra by an abelian Lie algebra.By Engel’s theorem, if 𝖆 ⊕ 𝖓 is a semidirect product, with 𝖆 abelian and 𝖓 nilpotent, acting on a finite-dimensionalvector spaceW with operators in 𝖆 diagonalizable and operators in 𝖓 nilpotent, there is a vectorw that is an eigenvectorfor 𝖆 and is annihilated by 𝖓. In fact it is enough to show there is a vector annihilated by 𝖓, which follows by inductionon dim 𝖓, since the derived algebra 𝖓' annihilates a non-zero subspace of vectors on which 𝖓 / 𝖓' and 𝖆 act with thesame hypotheses.Applying this argument repeatedly to 𝖙C ⊕ 𝖓₊ shows that there is an orthonormal basis e1, …, en of V consisting ofeigenvectors of 𝖙C with 𝖓₊ acting as upper triangular matrices with zeros on the diagonal.If N± and TC are the complex Lie groups corresponding to 𝖓₊ and 𝖙C, then the Gauss decomposition states that thesubset

N−TCN+

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15.3. DECOMPOSITIONS IN THE CHEVALLEY COMPLEXIFICATION 33

is a direct product and consists of the elements in GC for which the principal minors are non-vanishing. It is openand dense. Moreover, if T denotes the maximal torus in U(V),

N± = N± ∩GC, TC = TC ∩GC.

These results are an immediate consequence of the corresponding results for GL(V).[9]

15.3.3 Bruhat decomposition

IfW = NG(T) / T denotes the Weyl group of T and B denotes the Borel subgroup TC N₊, the Gauss decompositionis also a consequence of the more precise Bruhat decomposition

GC =∪

σ∈W

BσB,

decomposing GC into a disjoint union of double cosets of B. The complex dimension of a double coset BσB isdetermined by the length of σ as an element of W. The dimension is maximized at the Coxeter element and givesthe unique open dense double coset. Its inverse conjugates B into the Borel subgroup of lower triangular matrices inGC.[10]

The Bruhat decomposition is easy to prove for SL(n,C).[11] Let B be the Borel subgroup of upper triangular matricesand TC the subgroup of diagonal matrices. So N(TC) / TC = Sn. For g in SL(n,C), take b in B so that bgmaximizesthe number of zeros appearing at the beginning of its rows. Because a multiple of one row can be added to another,each row has a different number of zeros in it. Multiplying by a matrix w in N(TC), it follows that wbg lies in B. Foruniqueness, if w1b w2 = b0, then the entries of w1w2 vanish below the diagonal. So the product lies in TC, provinguniqueness.Chevalley (1955) showed that the expression of an element g as g = b1σb2 becomes unique if b1 is restricted to lie inthe upper unitriangular subgroup Nσ = N₊ ∩ σ N₋ σ−1. In fact, ifMσ = N₊ ∩ σ N₊ σ−1, this follows from the identity

N+ = Nσ ·Mσ.

The group N₊ has a natural filtration by normal subgroups N₊(k) with zeros in the first k − 1 superdiagonals andthe successive quotients are Abelian. Defining Nσ(k) and Mσ(k) to be the intersections with N₊(k), it follows bydecreasing induction on k that N₊(k) = Nσ(k) ⋅ Mσ(k). Indeed Nσ(k)N₊(k + 1) and Mσ(k)N₊(k + 1) are specified inN₊(k) by the vanishing of complementary entries (i, j) on the kth superdiagonal according to whether σ preserves theorder i < j or not.[12]

The Bruhat decomposition for the other classical simple groups can be deduced from the above decomposition usingthe fact that they are fixed point subgroups of folding automorphisms of SL(n,C).[13] For Sp(n,C), let J be the n × nmatrix with 1’s on the antidiagonal and 0’s elsewhere and set

A =

(0 J−J 0

).

Then Sp(n,C) is the fixed point subgroup of the involution θ(g) =A (gt)−1 A−1 of SL(2n,C). It leaves the subgroupsN±,TC and B invariant. If the basis elements are indexed by n, n−1, …, 1, −1, …, −n, then the Weyl group of Sp(n,C)consists of σ satisfying σ(j) = −j, i.e. commuting with θ. Analogues of B, TC and N± are defined by intersectionwith Sp(n,C), i.e. as fixed points of θ. The uniqueness of the decomposition g = nσb = θ(n) θ(σ) θ(b) implies theBruhat decomposition for Sp(n,C).The same argument works for SO(n,C). It can be realised as the fixed points of ψ(g) = B (gt)−1 B−1 in SL(n,C) whereB = J.

15.3.4 Iwasawa decomposition

The Iwasawa decomposition

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34 CHAPTER 15. COMPLEXIFICATION (LIE GROUP)

GC = G ·A ·N

gives a decomposition forGC for which, unlike the Cartan decomposition, the direct factorA ⋅N is a closed subgroup,but it is no longer invariant under conjugation by G. It is the semidirect product of the nilpotent subgroup N by theAbelian subgroup A.For U(V) and its complexification GL(V), this decomposition can be derived as a restatement of the Gram–Schmidtorthonormalization process.[14]

In fact let e1, …, en be an orthonormal basis of V and let g be an element in GL(V). Applying the Gram–Schmidtprocess to ge1, …, gen, there is a unique orthonormal basis f1, …, fn and positive constants ai such that

fi = aigei +∑j<i

njigej .

If k is the unitary taking (ei) to (fi), it follows that g−1k lies in the subgroup AN, where A is the subgroup of positivediagonal matrices with respect to (ei) and N is the subgroup of upper unitriangular matrices.[15]

Using the notation for the Gauss decomposition, the subgroups in the Iwasawa decomposition for GC are defined by[16]

A = exp it = A ∩GC, N = exp n+ = N ∩GC.

Since the decomposition is direct for GL(V), it is enough to check that GC = GAN. From the properties of theIwasawa decomposition for GL(V), the map G × A × N is a diffeomorphism onto its image in GC, which is closed.On the other hand the dimension of the image is the same as the dimension of GC, so it is also open. So GC = GANbecause GC is connected.[17]

Zhelobenko (1973) gives a method for explicitly computing the elements in the decomposition.[18] For g in GC set h= g*g. This is a positive self-adjoint operator so its principal minors do not vanish. By the Gauss decomposition, itcan therefore be written uniquely in the form h = XDY with X in N₋, D in TC and Y in N₊. Since h is self-adjoint,uniqueness forces Y = X*. Since it is also positive D must lie in A and have the form D = exp iT for some unique Tin 𝖙. Let a = exp iT/2 be its unique square root in A. Set n = Y and k = g n−1 a−1. Then k is unitary, so is in G, and g= kan.

15.4 Complex structures on homogeneous spaces

The Iwasawa decomposition can be used to describe complex structures on theG-orbits in complex projective space ofhighest weight vectors of finite-dimensional irreducible representations of G. In particular the identification betweenG / T and GC / B can be used to formulate the Borel–Weil theorem. It states that each irreducible representation ofG can be obtained by holomorphic induction from a character of T, or equivalently that it is realized in the space ofsections of a holomorphic line bundle on G / T.The closed connected subgroups of G containing T are described by Borel–de Siebenthal theory. They are exactlythe centralizers of tori S ⊆ T. Since every torus is generated topologically by a single element x, these are the same ascentralizers CG(X) of elements X in 𝖙. By a result of Hopf CG(x) is always connected: indeed any element y is alongwith S contained in some maximal torus, necessarily contained in CG(x).Given an irreducible finite-dimensional representation Vλ with highest weight vector v of weight λ, the stabilizer ofC v in G is a closed subgroup H. Since v is an eigenvector of T, H contains T. The complexification GC also acts onV and the stabilizer is a closed complex subgroup P containing TC. Since v is annihilated by every raising operatorcorresponding to a positive root α, P contains the Borel subgroup B. The vector v is also a highest weight vector forthe copy of sl2 corresponding to α, so it is annihilated by the lowering operator generating 𝖌-α if (λ, α) = 0. The Liealgebra p of P is the direct sum of 𝖙C and root space vectors annihilating v, so that

p = b⊕⊕

(α,λ)=0

g−α.

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15.5. NONCOMPACT REAL FORMS 35

The Lie algebra of H = P ∩ G is given by p ∩ 𝖌. By the Iwasawa decomposition GC = GAN. Since AN fixes C v, theG-orbit of v in the complex projective space of Vλ coincides with the GC orbit and

G/H = GC/P.

In particular

G/T = GC/B.

Using the identification of the Lie algebra of T with its dual,H equals the centralizer of λ inG, and hence is connected.The group P is also connected. In fact the space G / H is simply connected, since it can be written as the quotient ofthe (compact) universal covering group of the compact semisimple group G / Z by a connected subgroup, where Zis the center of G.[19] If Po is the identity component of P, GC / P has GC / Po as a covering space, so that P = Po.The homogeneous space GC / P has a complex structure, because P is a complex subgroup. The orbit in complexprojective space is closed in the Zariski topology by Chow’s theorem, so is a smooth projective variety. The Borel–Weil theorem and its generalizations are discussed in this context in Serre (1954), Helgason (1994), Duistermaat &Kolk (2000) and Sepanski (2007).The parabolic subgroup P can also be written as a union of double cosets of B

P =∪

σ∈Wλ

BσB,

where Wλ is the stabilizer of λ in the Weyl group W. It is generated by the reflections corresponding to the simpleroots orthogonal to λ.[20]

15.5 Noncompact real forms

There are other closed subgroups of the complexification of a compact connected Lie group G which have the samethe complexified Lie algebra. These are the other real forms of GC.[21]

15.5.1 Involutions of simply connected compact Lie groups

If G is a simply connected compact Lie group and σ is an automorphism of period 2, then the fixed point subgroup K= Gσ is automatically connected. (In fact this is true for any automorphism of G, as shown for inner automorphismsby Steinberg and in general by Borel.) [22]

This can be seen most directly when the involution σ corresponds to a Hermitian symmetric space. In that case σis inner and implemented by an element in a one-parameter subgroup exp tT contained in the center of Gσ. Theinnerness of σ implies that K contains a maximal torus of G, so has maximal rank. On the other hand the centralizerof the subgroup generated by the torus S of elements exp tT is connected, since if x is any element in K there is amaximal torus containing x and S, which lies in the centralizer. On the other hand it contains K since S is central inK and is contained in K since z lies in S. So K is the centralizer of S and hence connected. In particular K containsthe center of G.[23]

For a general involution σ, the connectedness of Gσ can be seen as follows.[24]

The starting point is the Abelian version of the result: if T is a maximal torus of a simply connected group G and σis an involution leaving invariant T and a choice of positive roots (or equivalently a Weyl chamber), then the fixedpoint subgroup Tσ is connected. In fact the kernel of the exponential map from t onto T is a lattice Λ with a Z-basisindexed by simple roots, which σ permutes. Splitting up according to orbits, T can be written as a product of termsT on which σ acts trivially or terms T2 where σ interchanges the factors. The fixed point subgroup just correspondsto taking the diagonal subgroups in the second case, so is connected.Now let x be any element fixed by σ, let S be a maximal torus in CG(x)σ and let T be the identity component of CG(x,S). Then T is a maximal torus in G containing x and S. It is invariant under σ and the identity component of Tσ is S.

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36 CHAPTER 15. COMPLEXIFICATION (LIE GROUP)

In fact since x and S commute, they are contained in a maximal torus which, because it is connected, must lie in T.By construction T is invariant under σ. The identity component of Tσ contains S, lies in CG(x)σ and centralizes S, soit equals S. But S is central in T, to T must be Abelian and hence a maximal torus. For σ acts as multiplication by −1on the Lie algebra t⊖ s , so it and therefore also t are Abelian.The proof is completed by showing that σ preserves a Weyl chamber associated with T. For then Tσ is connected somust equal S. Hence x lies in S. Since x was arbitrary, Gσ must therefore be connected.To produce aWeyl chamber invariant under σ, note that there is no root space gα on which both x and S acted trivially,for this would contradict the fact that CG(x, S) has the same Lie algebra as T. Hence there must be an element s in Ssuch that t = xs acts non-trivially on each root space. In this case t is a regular element of T—the identity componentof its centralizer in G equals T. There is a unique Weyl alcove A in t such that t lies in exp A and 0 lies in the closureof A. Since t is fixed by σ, the alcove is left invariant by σ and hence so also is the Weyl chamber C containing it.

15.5.2 Conjugations on the complexification

Let G be a simply connected compact Lie group with complexification GC. The map c(g) = (g*)−1 defines an au-tomorphism of GC as a real Lie group with G as fixed point subgroup. It is conjugate-linear on gC and satisfies c2= id. Such automorphisms of either GC or gC are called conjugations. Since GC is also simply connected anyconjugation c1 on gC corresponds to a unique automorphism c1 of GC.The classification of conjugations c0 reduces to that of involutions σ ofG because given a c1 there is an automorphismφ of the complex group GC such that

c0 = φ c1 φ−1

commutes with c. The conjugation c0 then leaves G invariant and restricts to an involutive automorphism σ. Bysimple connectivity the same is true at the level of Lie algebras. At the Lie algebra level c0 can be recovered from σby the formula

c0(X + iY ) = σ(X)− iσ(Y )

for X, Y in g .To prove the existence of φ let ψ = c1c an automorphism of the complex groupGC. On the Lie algebra level it definesa self-adjoint operator for the complex inner product

(X,Y ) = −B(X, c(Y )),

where B is the Killing form on gC . Thus ψ2 is a positive operator and an automorphism along with all its real powers.In particular take

φ = (ψ2)1/4

It satisfies

c0c = φc1φ−1c = φcc1φ = (ψ2)1/2ψ−1 = φ−1cc1φ

−1 = cφc1φ−1 = cc0.

15.5.3 Cartan decomposition in a real form

For the complexification GC, the Cartan decomposition is described above. Derived from the polar decompositionin the complex general linear group, it gives a diffeomorphism

GC = G · exp ig = G · P = P ·G.

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15.6. SEE ALSO 37

On GC there is a conjugation operator c corresponding to G as well as an involution σ commuting with c. Let c0 =c σ and let G0 be the fixed point subgroup of c. It is closed in the matrix group GC and therefore a Lie group. Theinvolution σ acts on both G and G0. For the Lie algebra of G there is a decomposition

g = k⊕ p

into the +1 and −1 eigenspaces of σ. The fixed point subgroup K of σ in G is connected since G is simply connected.Its Lie algebra is the +1 eigenspace k . The Lie algebra of G0 is given by

g = k⊕ p

and the fixed point subgroup of σ is again K, so that G ∩ G0 = K. In G0, there is a Cartan decomposition

G0 = K · exp ip = K · P0 = P0 ·K

which is again a diffeomorphism onto the direct and corresponds to the polar decomposition of matrices. It is therestriction of the decomposition on GC. The product gives a diffeomorphism onto a closed subset of G0. To checkthat it is surjective, for g in G0 write g = u ⋅ p with u in G and p in P. Since c0 g = g, uniqueness implies that σu = uand σp = p−1. Hence u lies in K and p in P0.The Cartan decomposition in G0 shows that G0 is connected, simply connected and noncompact, because of thedirect factor P0. Thus G0 is a noncompact real semisimple Lie group.[25]

Moreover given a maximal Abelian subalgebra a in p , A = exp a is a toral subgroup such that σ(a) = a−1 on A; andany two such a 's are conjugate by an element of K. The properties of A can be shown directly. A is closed because theclosure of A is a toral subgroup satisfying σ(a) = a−1, so its Lie algebra lies in m and hence equals a by maximality.A can be generated topologically by a single element exp X, so a is the centralizer of X in m . In the K-orbit of anyelement of m there is an element Y such that (X,Ad k Y) is minimized at k = 1. Setting k = exp tT with T in k , itfollows that (X,[T,Y]) = 0 and hence [X,Y] = 0, so that Y must lie in a . Thus m is the union of the conjugates of a. In particular some conjugate of X lies in any other choice of a , which centralizes that conjugate; so by maximalitythe only possibilities are conjugates of a .[26]

A similar statements hold for the action of K on a0 = ia in p0 . Morevoer, from the Cartan decomposition for G0,if A0 = exp a0 , then

G0 = KA0K.

15.5.4 Iwasawa decomposition in a real form

15.6 See also• Real form (Lie theory)

15.7 Notes[1] See:

• Hochschild 1965• Bourbaki 1981, pp. 212–214

[2] Bourbaki 1981, pp. 210–214

[3] Hochschild 1966

[4] See:

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38 CHAPTER 15. COMPLEXIFICATION (LIE GROUP)

• Hochschild 1965• Chevalley 1946• Bröcker & tom Dieck 1985

[5] See:

• Chevalley 1946• Weyl 1946

[6] Zhelobenko 1973, p. 28

[7] Bump 2001, pp. 202–203

[8] See:

• Bump 2001• Zhelobenko 1973

[9] Zhelobenko 1973

[10] See:

• Gelfand & Naimark 1950, section 18, for SL(n,C)• Bruhat 1956, p. 187 for SO(n,C) and Sp(n,C)• Chevalley 1955 for complexifications of simple compact Lie groups• Helgason 1978, pp. 403–406 for Harish-Chandra's method• Humphreys 1981 for a treatment using algebraic groups• Carter 1972, Chapter 8• Dieudonné 1977, pp. 216–217• Bump 2001, pp. 205–211

[11] Steinberg 1974, p. 73

[12] Chevalley 1955, p. 41

[13] See:

• Steinberg 1974, pp. 73–74• Bourbaki 1981a, pp. 53–54

[14] Sepanski 2007, p. 8

[15] Knapp 2001, p. 117

[16] See:

• Zhelobenko 1973, pp. 288–290• Dieudonné 1977, pp. 197–207• Helgason 1978, pp. 257–262• Bump 2001, pp. 197–204

[17] Bump 2001, pp. 203–204

[18] Zhelobenko 1973, p. 289

[19] Helgason 1978

[20] See:

• Humphreys 1981• Bourbaki 1981a

[21] Dieudonné 1977, pp. 164–173

[22] See:

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15.8. REFERENCES 39

• Helgason 1978, pp. 320–321• Bourbaki 1982, pp. 46–48• Duistermaar & Kolk 2000, pp. 194–195• Dieudonné 1977, p. 151, Exercise 11

[23] Wolf 2010

[24] See: Bourbaki 1982, pp. 46–48

[25] Dieudonné 1977, pp. 166–168

[26] & Helgason 1978, p. 248

15.8 References• Bourbaki, N. (1981), Groupes et Algèbres de Lie (Chapitre 3), Éléments de Mathématique, Hermann, ISBN354033940X

• Bourbaki, N. (1981a), Groupes et Algèbres de Lie (Chapitres 4,5 et 6), Éléments de Mathématique, Masson,ISBN 2225760764

• Bourbaki, N. (1982), Groupes et Algèbres de Lie (Chapitre 9), Éléments de Mathématique, Masson, ISBN354034392X

• Bröcker, T.; tom Dieck, T. (1985), Representations of Compact Lie Groups, Graduate Texts in Mathematics98, Springer, ISBN 3540136789

• Bruhat, F. (1956), “Sur les représentations induites des groupes de Lie”, Bull. Soc. Math. France 84: 97–205

• Bump, Daniel (2004), Lie groups, Graduate Texts in Mathematics 225, Springer, ISBN 0387211543

• Carter, Roger W. (1972), Simple groups of Lie type, Pure and Applied Mathematics 28, Wiley

• Chevalley, C. (1946), Theory of Lie Groups I, Princeton University Press

• Chevalley, C. (1955), “Sur certains groupes simples”, TôhokuMathematical Journal 7: 14–66, doi:10.2748/tmj/1178245104

• Dieudonné, J. (1977), Compact Lie groups and semisimple Lie groups, Chapter XXI, Treatise on analysis 5,Academic Press, ISBN 012215505X

• Duistermaat, J.J.; Kolk, A. (2000), Lie groups, Universitext, Springer, ISBN 3540152938

• Gelfand, I. M.; Naimark, M. A. (1950), “Unitary representations of the classical groups”, Trudy Mat. Inst.Steklov. (in Russian) 36: 3–288

• Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN0821828487

• Helgason, Sigurdur (1994), Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs39 (2nd ed.), American Mathematical Society, ISBN 0821815385

• Hochschild, G. (1965), The structure of Lie groups, Holden-Day

• Hochschild, G. (1966), “Complexification of Real Analytic Groups”, Transactions of the American Mathemat-ical Society 125: 406–413, doi:10.2307/1994572

• Humphreys, James E. (1981), Linear Algebraic Groups, Graduate texts in mathematics 21, Springer, ISBN0387901086

• Humphreys, James E. (1997), Introduction to Lie Algebras and Representation Theory, Graduate texts in math-ematics 9 (2nd ed.), Springer, ISBN 3540900535

• Knapp, Anthony W. (2001), Representation Theory of Semisimple Groups: An Overview Based on Examples,Princeton Mathematical Series 36, Princeton University Press, ISBN 0691090890

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40 CHAPTER 15. COMPLEXIFICATION (LIE GROUP)

• Onishchik, A.L.; Vinberg, E.B. (1994), Lie Groups and Lie Algebras III: Structure of Lie Groups and LieAlgebras, Encyclopaedia of Mathematical Sciences 41, Springer, ISBN 9783540546832

• Sepanski, MarkR. (2007), Compact Lie groups, Graduate Texts inMathematics 235, Springer, ISBN0387302638

• Serre, Jean-Pierre (1954), “Représentations linéaires et espaces homogènes kählériens des groupes de Liecompacts, Exposé no 100”, Séminaire Bourbaki 2

• Steinberg, Robert (1974), Conjugacy classes in algebraic groups, Lecture notes in mathematics 366, Springer

• Weyl, Hermann (1946), The Classical Groups, their Invariants and Representations (2nd ed.), Princeton Uni-versity Press

• Wolf, Joseph A. (2010), Spaces of constant curvature, AMS Chelsea Publishing (6th ed.), American Mathe-matical Society, ISBN 0821852825

• Zhelobenko, D.P. (1973), Compact Lie groups and their representations, Translations of mathematical mono-graphs 40, American Mathematical Society

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Chapter 16

Cuspidal representation

In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely inL2

spaces. The term cuspidal is derived, at a certain distance, from the cusp forms of classical modular form theory. In thecontemporary formulation of automorphic representations, representations take the place of holomorphic functions;these representations may be of adelic algebraic groups.When the group is the general linear group GL2 , the cuspidal representations are directly related to cusp forms andMaass forms. For the case of cusp forms, each Hecke eigenform (newform) corresponds to a cuspidal representation.

16.1 Formulation

Let G be a reductive algebraic group over a number field K and let A denote the adeles of K. Let Z denote the centreof G and let ω be a continuous unitary character from Z(K)\Z(A)× to C×. Fix a Haar measure on G(A) and letL20(G(K)\G(A), ω) denote the Hilbert space of measurable complex-valued functions, f, on G(A) satisfying

1. f(γg) = f(g) for all γ ∈ G(K)

2. f(gz) = f(g)ω(z) for all z ∈ Z(A)

3.∫Z(A)G(K)\G(A) |f(g)|

2 dg <∞

4.∫U(K)\U(A) f(ug) du = 0 for all unipotent radicals, U, of all proper parabolic subgroups of G(A).

This is called the space of cusp forms with central character ω on G(A). A function occurring in such a space iscalled a cuspidal function. This space is a unitary representation of the group G(A) where the action of g ∈ G(A)on a cuspidal function f is given by

(g · f)(x) = f(xg).

The space of cusp forms with central character ω decomposes into a direct sum of Hilbert spaces

L20(G(K)\G(A), ω) =

⊕(π,Vπ)

mπVπ

where the sum is over irreducible subrepresentations of L20(G(K)\G(A), ω) and mπ are positive integers (i.e. eachirreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of G(A) is such a subrep-resentation (π, V) for some ω.The groups for which the multiplicities mπ all equal one are said to have the multiplicity-one property.

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42 CHAPTER 16. CUSPIDAL REPRESENTATION

16.2 References• James W. Cogdell, Henry Hyeongsin Kim, Maruti Ram Murty. Lectures on Automorphic L-functions (2004),Chapter 5.

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Chapter 17

Diagonalizable group

In mathematics, an affine algebraic group is said to be diagonalizable if it is isomorphic to a subgroup of Dn, thegroup of diagonal matrices. A diagonalizable group defined over k is said to split over k or k-split if the isomorphismis defined over k. This coincides with the usual notion of split for an algebraic group. Every diagonalizable groupsplits over the separable closure k of k. Any closed subgroup and image of diagonalizable groups are diagonalizable.The torsion subgroup of a diagonalizable group is dense.The category of diagonalizable groups defined over k is equivalent to the category of finitely generated abelian groupwith Gal(k/k )-equivariant morphisms without p-torsion. This is an analog of Poincaré duality and motivated theterminology.A diagonalizable k-group is said to be anisotropic if it has no nontrivial k-valued character.The so-called "rigidity" states that the identity component of the centralizer of a diagonalizable group coincides withthe identity component of the normalizer of the group. The fact plays a crucial role in the structure theory of solvablegroups.A connected diagonalizable group is called an algebraic torus (which is not necessarily compact, in contrast to acomplex torus). A k-torus is a torus defined over k. The centralizer of a maximal torus is called a Cartan subgroup.

17.1 References• Borel, A. Linear algebraic groups, 2nd ed.

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Chapter 18

Dieudonné module

Inmathematics, aDieudonnémodule introduced byDieudonné (1954, 1957b), is amodule over the non-commutativeDieudonné ring, which is generated over the ring of Witt vectors by two special endomorphisms F and V called theFrobenius and Verschiebung operators. They are used for studying finite flat commutative group schemes.Finite flat commutative group schemes over a perfect field k of positive characteristic p can be studied by trans-ferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the Dieudonné ring D =W(k)F,V/(FV − p), which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vec-tors of k. F and V are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors.Jean Dieudonné and Pierre Cartier constructed an antiequivalence of categories between finite commutative groupschemes over k of order a power of “p” and modules over D with finiteW(k)-length. The Dieudonné module functorin one direction is given by homomorphisms into the abelian sheaf CW of Witt co-vectors. This sheaf is more orless dual to the sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed bytaking a direct limit of finite length Witt vectors under successive Verschiebung maps V :W →W ₊₁, and then com-pleting. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonnémodules, e.g., connected p-group schemes correspond toD-modules for which F is nilpotent, and étale group schemescorrespond to modules for which F is an isomorphism.Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Oda’s 1967 thesisgave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at aboutthe same time, Grothendieck suggested that there should be a crystalline version of the theory that could be usedto analyze p-divisible groups. Galois actions on the group schemes transfer through the equivalences of categories,and the associated deformation theory of Galois representations was used in Wiles's work on the Shimura–Taniyamaconjecture.

18.1 Dieudonné rings

If k is a field of characteristic p, its ring of Witt vectors consists of sequences (w1,w2,w3,...) of elements of k, andhas an endomorphism σ induced by the Frobenius endomorphism of k, so (w1,w2,w3,...)σ = (wp1,wp2,wp3,...). The Dieudonné ring, often denoted by Ek or Dk, is the non-commutative ring over W(k) generated by 2elements F and V subject to the relations

FV = VF = pFw = wσF

wV = Vwσ.

It is a Z-graded ring, where the piece of degree n∈Z is a 1-dimensional free module over W(k), spanned by V−n ifn≤0 and by Fn if n≥0.Some authors define the Dieudonné ring to be the completion of the ring above for the ideal generated by F and V.

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18.2. DIEUDONNÉ MODULES AND GROUPS 45

18.2 Dieudonné modules and groups

Special sorts of modules over the Dieudonné ring correspond to certain algebraic group schemes. For example, finitelength modules over the Dieudonné ring form an abelian category equivalent to the opposite of the category of finitecommutative p-group schemes over k.

18.3 Examples• If X is the constant group scheme Z/pZ over k , then its corresponding Dieudonné module D(X) is k withF = Frobk and V = 0 .

• For the scheme of p-th roots of unity X = µp , then its corresponding Dieudonné module is D(X) = k withF = 0 and V = Frob−1

k .

• For X = αp , defined as the kernel of the Frobenius Ga → Ga , the Dieudonné module is D(X) = k withF = V = 0 .

• If X = E[p] is the p-torsion of an elliptic curve over k (with p-torsion in k), then the Dieudonné moduledepends on whether E is supersingular or not.

18.4 References• Cartier, Pierre (1962), “Groupes algébriques et groupes formels”, Colloq. Théorie des Groupes Algébriques(Bruxelles, 1962), Librairie Universitaire, Louvain, pp. 87–111, MR 0148665

• Dieudonné, Jean (1955), “Lie groups and Lie hyperalgebras over a field of characteristic p>0. IV”, AmericanJournal of Mathematics 77: 429–452, ISSN 0002-9327, JSTOR 2372633, MR 0071718

• Dieudonné, Jean (1957), “Lie groups and Lie hyperalgebras over a field of characteristic p>0. VI”, AmericanJournal of Mathematics 79: 331–388, ISSN 0002-9327, JSTOR 2372686, MR 0094413

• Dieudonné, Jean (1957b), “Groupes de Lie et hyperalgèbres de Lie sur un corps de caractéristique p>0. VII”,Mathematische Annalen 134: 114–133, doi:10.1007/BF01342790, ISSN 0025-5831, MR 0098146

• Dolgachev, I.V. (2001), “D/d031640”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Manin, Ju. I. (1963), “Theory of commutative formal groups over fields of finite characteristic”, AkademiyaNauk SSSR iMoskovskoeMatematicheskoe Obshchestvo. UspekhiMatematicheskikhNauk 18 (6): 3–90, doi:10.1070/RM1963v018n06ABEH001142,ISSN 0042-1316, MR 0157972

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Chapter 19

Differential algebraic group

In mathematics, a differential algebraic group is a differential algebraic variety with a compatible group structure.Differential algebraic groups were introduced by Cassidy (1972).

19.1 References• Cassidy, Phyllis Joan (1972), “Differential algebraic groups”, Amer. J. Math. 94: 891–954, JSTOR 2373764,MR 0360611

• Kolchin, E. R. (1985),Differential algebraic groups, Pure andAppliedMathematics 114, Boston,MA:AcademicPress, ISBN 978-0-12-417640-9, MR 776230

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Chapter 20

Differential Galois theory

In mathematics, differential Galois theory studies the Galois groups of differential equations.

20.1 Overview

Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensionsof differential fields, i.e. fields that are equipped with a derivation, D. Much of the theory of differential Galoistheory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groupsin differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered inalgebraic Galois theory. The problem of finding which integrals of elementary functions can be expressed with otherelementary functions is analogous to the problem of solutions of polynomial equations by radicals in algebraic Galoistheory, and is solved by Picard–Vessiot theory.

20.2 Definitions

For any differential field F, there is a subfield

Con(F) = f in F | Df = 0,

called the constants of F. Given two differential fields F and G, G is called a logarithmic extension of F if G is asimple transcendental extension of F (i.e. G = F(t) for some transcendental t) such that

Dt = Ds/s for some s in F.

This has the form of a logarithmic derivative. Intuitively, one may think of t as the logarithm of some element s ofF, in which case, this condition is analogous to the ordinary chain rule. But it must be remembered that F is notnecessarily equipped with a unique logarithm; one might adjoin many “logarithm-like” extensions to F. Similarly, anexponential extension is a simple transcendental extension which satisfies

Dt = tDs.

With the above caveat in mind, this element may be thought of as an exponential of an element s of F. Finally, Gis called a Liouvillian differential extension of F if there is a finite chain of subfields from F to G where eachextension in the chain is either algebraic, logarithmic, or exponential.

20.3 See also• Liouville’s theorem (differential algebra)• Risch algorithm

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48 CHAPTER 20. DIFFERENTIAL GALOIS THEORY

20.4 References• Bertrand, D. (1996), “Review of “Lectures on differential Galois theory"" (PDF), Bulletin of the AmericanMathematical Society 33 (2), doi:10.1090/s0273-0979-96-00652-0, ISSN 0002-9904

• Beukers, Frits (1992), “8. Differential Galois theory”, in Waldschmidt, Michel; Moussa, Pierre; Luck, Jean-Marc; Itzykson, Claude, From number theory to physics. Lectures of a meeting on number theory and physicsheld at the Centre de Physique, Les Houches (France), March 7–16, 1989, Berlin: Springer-Verlag, pp. 413–439, ISBN 3-540-53342-7, Zbl 0813.12001

• Magid, Andy R. (1994), Lectures on differential Galois theory, University Lecture Series 7, Providence, R.I.:American Mathematical Society, ISBN 978-0-8218-7004-4, MR 1301076

• Magid, Andy R. (1999), “Differential Galois theory” (PDF), Notices of the American Mathematical Society 46(9): 1041–1049, ISSN 0002-9920, MR 1710665

• van der Put, Marius; Singer, Michael F. (2003), Galois theory of linear differential equations, Grundlehren derMathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 328, Berlin, New York:Springer-Verlag, ISBN 978-3-540-44228-8, MR 1960772

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Chapter 21

E6 (mathematics)

In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras e6 , allof which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6. Thedesignation E6 comes from the Cartan–Killing classification of the complex simple Lie algebras (see Élie_Cartan §Work). This classifies Lie algebras into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeledE6, E7, E8, F4, and G2. The E6 algebra is thus one of the five exceptional cases.The fundamental group of the complex form, compact real form, or any algebraic version of E6 is the cyclic groupZ/3Z, and its outer automorphism group is the cyclic group Z/2Z. Its fundamental representation is 27-dimensional(complex), and a basis is given by the 27 lines on a cubic surface. The dual representation, which is inequivalent, isalso 27-dimensional.In particle physics, E6 plays a role in some grand unified theories.

21.1 Real and complex forms

There is a unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78.The complex adjoint Lie group E6 of complex dimension 78 can be considered as a simple real Lie group of realdimension 156. This has fundamental group Z/3Z, has maximal compact subgroup the compact form (see below) ofE6, and has an outer automorphism group non-cyclic of order 4 generated by complex conjugation and by the outerautomorphism which already exists as a complex automorphism.As well as the complex Lie group of type E6, there are five real forms of the Lie algebra, and correspondingly five realforms of the group with trivial center (all of which have an algebraic double cover, and three of which have furthernon-algebraic covers, giving further real forms), all of real dimension 78, as follows:

• The compact form (which is usually the one meant if no other information is given), which has fundamentalgroup Z/3Z and outer automorphism group Z/2Z.

• The split form, EI (or E₆₍₆₎), which has maximal compact subgroup Sp(4)/(±1), fundamental group of order 2and outer automorphism group of order 2.

• The quasi-split form EII (or E₆₍₂₎), which has maximal compact subgroup SU(2) × SU(6)/(center), fundamentalgroup cyclic of order 6 and outer automorphism group of order 2.

• EIII (or E₆₍−₁₄₎), which has maximal compact subgroup SO(2) × Spin(10)/(center), fundamental group Z andtrivial outer automorphism group.

• EIV (or E₆₍−₂₆₎), which has maximal compact subgroup F4, trivial fundamental group cyclic and outer auto-morphism group of order 2.

The EIV form of E6 is the group of collineations (line-preserving transformations) of the octonionic projective planeOP2.[1] It is also the group of determinant-preserving linear transformations of the exceptional Jordan algebra. Theexceptional Jordan algebra is 27-dimensional, which explains why the compact real form of E6 has a 27-dimensionalcomplex representation. The compact real form of E6 is the isometry group of a 32-dimensional Riemannianmanifold

49

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50 CHAPTER 21. E6 (MATHEMATICS)

known as the 'bioctonionic projective plane'; similar constructions for E7 and E8 are known as the Rosenfeld projectiveplanes, and are part of the Freudenthal magic square.

21.2 E6 as an algebraic group

By means of a Chevalley basis for the Lie algebra, one can define E6 as a linear algebraic group over the integers and,consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimesalso known as “untwisted”) adjoint form of E6. Over an algebraically closed field, this and its triple cover are theonly forms; however, over other fields, there are often many other forms, or “twists” of E6, which are classified inthe general framework of Galois cohomology (over a perfect field k) by the set H1(k, Aut(E6)) which, because theDynkin diagram of E6 (see below) has automorphism group Z/2Z, maps to H1(k, Z/2Z) = Hom (Gal(k), Z/2Z) withkernel H1(k, E₆,ₐ ).[2]

Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E6 coincidewith the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all adjointforms of E6 have fundamental group Z/3Z in the sense of algebraic geometry, with Galois action as on the thirdroots of unity; this means that they admit exactly one triple cover (which may be trivial on the real points); thefurther non-compact real Lie group forms of E6 are therefore not algebraic and admit no faithful finite-dimensionalrepresentations. The compact real form of E6 as well as the noncompact forms EI=E₆₍₆₎ and EIV=E₆₍−₂₆₎ are said tobe inner or of type 1E6 meaning that their class lies in H1(k, E₆,ₐ ) or that complex conjugation induces the trivialautomorphism on the Dynkin diagram, whereas the other two real forms are said to be outer or of type 2E6.Over finite fields, the Lang–Steinberg theorem implies that H1(k, E6) = 0, meaning that E6 has exactly one twistedform, known as 2E6: see below.

21.3 Algebra

21.3.1 Dynkin diagram

TheDynkin diagram for E6 is given by , whichmay also be drawn as or .

21.3.2 Roots of E6

Although they span a six-dimensional space, it is much more symmetrical to consider them as vectors in a six-dimensional subspace of a nine-dimensional space.

(1,−1,0;0,0,0;0,0,0), (−1,1,0;0,0,0;0,0,0),(−1,0,1;0,0,0;0,0,0), (1,0,−1;0,0,0;0,0,0),(0,1,−1;0,0,0;0,0,0), (0,−1,1;0,0,0;0,0,0),(0,0,0;1,−1,0;0,0,0), (0,0,0;−1,1,0;0,0,0),(0,0,0;−1,0,1;0,0,0), (0,0,0;1,0,−1;0,0,0),(0,0,0;0,1,−1;0,0,0), (0,0,0;0,−1,1;0,0,0),(0,0,0;0,0,0;1,−1,0), (0,0,0;0,0,0;−1,1,0),(0,0,0;0,0,0;−1,0,1), (0,0,0;0,0,0;1,0,−1),(0,0,0;0,0,0;0,1,−1), (0,0,0;0,0,0;0,−1,1),

All 27 combinations of (3; 3; 3) where 3 is one of(23 ,−

13 ,−

13

),(−1

3 ,23 ,−

13

),(−1

3 ,−13 ,

23

)All 27 combinations of (3; 3; 3) where 3 is one of

(−2

3 ,13 ,

13

),(13 ,−

23 ,

13

),(13 ,

13 ,−

23

)Simple roots

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21.3. ALGEBRA 51

The 72 vertices of the 122 polytope represent the root vectors of the E6, as shown in this Coxeter plane projection.Coxeter-Dynkin diagram:

(0,0,0;0,0,0;0,1,−1)

(0,0,0;0,0,0;1,−1,0)

(0,0,0;0,1,−1;0,0,0)

(0,0,0;1,−1,0;0,0,0)

(0,1,−1;0,0,0;0,0,0)(1

3,−2

3,1

3;−2

3,1

3,1

3;−2

3,1

3,1

3

)

An alternative description

An alternative (6-dimensional) description of the root system, which is useful in considering E6 × SU(3) as a subgroupof E8, is the following:

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52 CHAPTER 21. E6 (MATHEMATICS)

Graph of E6 as a subgroup of E8 projected into the Coxeter plane

All 4×(52

)permutations of

(±1,±1, 0, 0, 0, 0)

and all of the following roots with an odd number of plus signs

(±1

2,±1

2,±1

2,±1

2,±1

2,±

√3

2

).

Thus the 78 generators consist of the following subalgebras:

A 45-dimensional SO(10) subalgebra, including the above 4 ×(52

)generators plus the five Cartan

generators corresponding to the first five entries.Two 16-dimensional subalgebras that transform as a Weyl spinor of spin(10) and its complex conjugate.These have a non-zero last entry.1 generator which is their chirality generator, and is the sixth Cartan generator.

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21.3. ALGEBRA 53

2

3

1 3

1 4 6

2 4 6

2 562 4

3 5

3

43

4 6 61 5 41 2 62 5

6 54 1 35 16 2

5 3 1 51 36

3 5 21 4

4 2 51

2 4

3

6

(1, 0, 0, 0, 0, 0)

(1, 1, 0, 0, 0, 0)

(0, 1, 0, 0, 0, 0)

(0, 1, 1, 0, 0, 0)

(0, 0, 1, 0, 0, 0)

(0, 0, 1, 1, 0, 0) (0, 0, 1, 0, 0, 1)

(0, 0, 0, 1, 0, 0)

(0, 0, 0, 1, 1, 0)

(0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 1)

(1, 1, 1, 0, 0, 0) (0, 1, 1, 1, 0, 0) (0, 1, 1, 0, 0, 1) (0, 0, 1, 1, 1, 0)(0, 0, 1, 1, 0, 1)

(1, 1, 1, 1, 0, 0) (1, 1, 1, 0, 0, 1) (0, 1, 1, 1, 0, 1) (0, 1, 1, 1, 1, 0) (0, 0, 1, 1, 1, 1)

(1, 1, 1, 1, 0, 1) (0, 1, 2, 1, 0, 1)(0, 1, 1, 1, 1, 1) (1, 1, 1, 1, 1, 0)

(1, 1, 1, 1, 1, 1) (1, 1, 2, 1, 0, 1) (0, 1, 2, 1, 1, 1)

(1, 1, 2, 1, 1, 1) (1, 2, 2, 1, 0, 1)(0, 1, 2, 2, 1, 1)

(1, 1, 2, 2, 1, 1) (1, 2, 2, 1, 1, 1)

(1, 2, 2, 2, 1, 1)

(1, 2, 3, 2, 1, 1)

(1, 2, 3, 2, 1, 2)

Hasse diagram of E6 root poset with edge labels identifying added simple root position

One choice of simple roots for E6 is given by the rows of the followingmatrix, indexed in the order 1 2 3 4 5

6

:

1 −1 0 0 0 00 1 −1 0 0 00 0 1 −1 0 00 0 0 1 1 0

− 12 − 1

2 − 12 − 1

2 − 12

√3

20 0 0 1 −1 0

21.3.3 Weyl group

The Weyl group of E6 is of order 51840: it is the automorphism group of the unique simple group of order 25920(which can be described as any of: PSU4(2), PSΩ6

−(2), PSp4(3) or PSΩ5(3)).[3]

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54 CHAPTER 21. E6 (MATHEMATICS)

21.3.4 Cartan matrix 2 −1 0 0 0 0−1 2 −1 0 0 00 −1 2 −1 0 −10 0 −1 2 −1 00 0 0 −1 2 00 0 −1 0 0 2

21.4 Important subalgebras and representations

The Lie algebra E6 has an F4 subalgebra, which is the fixed subalgebra of an outer automorphism, and an SU(3) ×SU(3) × SU(3) subalgebra. Other maximal subalgebras which have an importance in physics (see below) and can beread off the Dynkin diagram, are the algebras of SO(10) × U(1) and SU(6) × SU(2).In addition to the 78-dimensional adjoint representation, there are two dual 27-dimensional “vector” representations.The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all givenby the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A121737in OEIS):

1, 27 (twice), 78, 351 (four times), 650, 1728 (twice), 2430, 2925, 3003 (twice), 5824 (twice), 7371(twice), 7722 (twice), 17550 (twice), 19305 (four times), 34398 (twice), 34749, 43758, 46332 (twice),51975 (twice), 54054 (twice), 61425 (twice), 70070, 78975 (twice), 85293, 100386 (twice), 105600,112320 (twice), 146432 (twice), 252252 (twice), 314496 (twice), 359424 (four times), 371800 (twice),386100 (twice), 393822 (twice), 412776 (twice), 442442 (twice)…

The underlined terms in the sequence above are the dimensions of those irreducible representations possessed by theadjoint form of E6 (equivalently, those whose weights belong to the root lattice of E6), whereas the full sequencegives the dimensions of the irreducible representations of the simply connected form of E6.The symmetry of the Dynkin diagram of E6 explains why many dimensions occur twice, the corresponding represen-tations being related by the non-trivial outer automorphism; however, there are sometimes even more representationsthan this, such as four of dimension 351, two of which are fundamental and two of which are not.The fundamental representations have dimensions 27, 351, 2925, 351, 27 and 78 (corresponding to the six nodes inthe Dynkin diagram in the order chosen for the Cartan matrix above, i.e., the nodes are read in the five-node chainfirst, with the last node being connected to the middle one).

21.5 E6 polytope

TheE6 polytope is the convex hull of the roots of E6. It therefore exists in 6 dimensions; its symmetry group containsthe Coxeter group for E6 as an index 2 subgroup.

21.6 Chevalley and Steinberg groups of type E6 and 2E6

Main article: ²E₆

The groups of type E6 over arbitrary fields (in particular finite fields) were introduced by Dickson (1901, 1908).The points over a finite field with q elements of the (split) algebraic group E6 (see above), whether of the adjoint(centerless) or simply connected form (its algebraic universal cover), give a finite Chevalley group. This is closelyconnected to the group written E6(q), however there is ambiguity in this notation, which can stand for several things:

• the finite group consisting of the points over Fq of the simply connected form of E6 (for clarity, this can bewritten E₆, (q) or more rarely E6(q) and is known as the “universal” Chevalley group of type E6 over Fq),

• (rarely) the finite group consisting of the points over Fq of the adjoint form of E6 (for clarity, this can be writtenE₆,ₐ (q), and is known as the “adjoint” Chevalley group of type E6 over Fq), or

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21.7. IMPORTANCE IN PHYSICS 55

• the finite group which is the image of the natural map from the former to the latter: this is what will be denotedby E6(q) in the following, as is most common in texts dealing with finite groups.

From the finite group perspective, the relation between these three groups, which is quite analogous to that betweenSL(n,q), PGL(n,q) and PSL(n,q), can be summarized as follows: E6(q) is simple for any q, E₆, (q) is its Schurcover, and E₆,ₐ (q) lies in its automorphism group; furthermore, when q−1 is not divisible by 3, all three coincide, andotherwise (when q is congruent to 1mod 3), the Schurmultiplier of E6(q) is 3 and E6(q) is of index 3 in E₆,ₐ (q), whichexplains why E₆, (q) and E₆,ₐ (q) are often written as 3·E6(q) and E6(q)·3. From the algebraic group perspective, itis less common for E6(q) to refer to the finite simple group, because the latter is not in a natural way the set of pointsof an algebraic group over Fq unlike E₆, (q) and E₆,ₐ (q).Beyond this “split” (or “untwisted”) form of E6, there is also one other form of E6 over the finite field Fq, knownas 2E6, which is obtained by twisting by the non-trivial automorphism of the Dynkin diagram of E6. Concretely,2E6(q), which is known as a Steinberg group, can be seen as the subgroup of E6(q2) fixed by the composition ofthe non-trivial diagram automorphism and the non-trivial field automorphism of Fq2. Twisting does not change thefact that the algebraic fundamental group of 2E₆,ₐ is Z/3Z, but it does change those q for which the covering of2E₆,ₐ by 2E₆, is non-trivial on the Fq-points. Precisely: 2E₆, (q) is a covering of 2E6(q), and 2E₆,ₐ (q) lies in itsautomorphism group; when q+1 is not divisible by 3, all three coincide, and otherwise (when q is congruent to 2 mod3), the degree of 2E₆, (q) over 2E6(q) is 3 and 2E6(q) is of index 3 in 2E₆,ₐ (q), which explains why 2E₆, (q) and2E₆,ₐ (q) are often written as 3·2E6(q) and 2E6(q)·3.Two notational issues should be raised concerning the groups 2E6(q). One is that this is sometimes written 2E6(q2),a notation which has the advantage of transposing more easily to the Suzuki and Ree groups, but the disadvantage ofdeviating from the notation for the Fq-points of an algebraic group. Another is that whereas 2E₆, (q) and 2E₆,ₐ (q)are the Fq-points of an algebraic group, the group in question also depends on q (e.g., the points over Fq2 of the samegroup are the untwisted E₆, (q2) and E₆,ₐ (q2)).The groups E6(q) and 2E6(q) are simple for any q,[4][5] and constitute two of the infinite families in the classificationof finite simple groups. Their order is given by the following formula (sequence A008872 in OEIS):

|E6(q)| =1

gcd(3, q − 1)q36(q12 − 1)(q9 − 1)(q8 − 1)(q6 − 1)(q5 − 1)(q2 − 1)

|2E6(q)| =1

gcd(3, q + 1)q36(q12 − 1)(q9 + 1)(q8 − 1)(q6 − 1)(q5 + 1)(q2 − 1)

(sequence A008916 in OEIS). The order of E₆, (q) or E₆,ₐ (q) (both are equal) can be obtained by removing thedividing factor gcd(3,q−1) from the first formula (sequence A008871 in OEIS), and the order of 2E₆, (q) or 2E₆,ₐ (q)(both are equal) can be obtained by removing the dividing factor gcd(3,q+1) from the second (sequence A008915 inOEIS).The Schur multiplier of E6(q) is always gcd(3,q−1) (i.e., E₆, (q) is its Schur cover). The Schur multiplier of 2E6(q)is gcd(3,q+1) (i.e., 2E₆, (q) is its Schur cover) outside of the exceptional case q=2 where it is 22·3 (i.e., there isan additional 22-fold cover). The outer automorphism group of E6(q) is the product of the diagonal automorphismgroup Z/gcd(3,q−1)Z (given by the action of E₆,ₐ (q)), the group Z/2Z of diagram automorphisms, and the group offield automorphisms (i.e., cyclic of order f if q=pf where p is prime). The outer automorphism group of 2E6(q) isthe product of the diagonal automorphism group Z/gcd(3,q+1)Z (given by the action of 2E₆,ₐ (q)) and the group offield automorphisms (i.e., cyclic of order f if q=pf where p is prime).

21.7 Importance in physics

N=8 supergravity in five dimensions, which is a dimensional reduction from 11 dimensional supergravity, admits anE6 bosonic global symmetry and an Sp(8) bosonic local symmetry. The fermions are in representations of Sp(8), thegauge fields are in a representation of E6, and the scalars are in a representation of both (Gravitons are singlets withrespect to both). Physical states are in representations of the coset E6/Sp(8).In grand unification theories, E6 appears as a possible gauge group which, after its breaking, gives rise to the SU(3) ×SU(2) × U(1) gauge group of the standard model (also see Importance in physics of E8). One way of achieving thisis through breaking to SO(10) × U(1). The adjoint 78 representation breaks, as explained above, into an adjoint 45,spinor 16 and 16 as well as a singlet of the SO(10) subalgebra. Including the U(1) charge we have

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56 CHAPTER 21. E6 (MATHEMATICS)

WW'

B

g3g8

The pattern of weak isospin, W, weaker isospin, W', strong g3 and g8, and baryon minus lepton, B, charges for particles in theSO(10) Grand Unified Theory, rotated to show the embedding in E6.

78 → 450 ⊕ 16−3 ⊕ 163 + 10.

Where the subscript denotes the U(1) charge.

21.8 See also

• En (Lie algebra)

• ADE classification

• Freudenthal magic square

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21.9. REFERENCES 57

21.9 References• Adams, J. Frank (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University ofChicago Press, ISBN 978-0-226-00526-3, MR 1428422

• Baez, John (2002). “TheOctonions, Section 4.4: E6". Bull. Amer. Math. Soc. 39 (2): 145–205. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979.. Online HTML version at .

• Cremmer, E.; J. Scherk; J. H. Schwarz (1979). “Spontaneously Broken N=8 Supergravity”. Phys. Lett. B 84(1): 83–86. doi:10.1016/0370-2693(79)90654-3.. Online scanned version at .

• Dickson, Leonard Eugene (1901), “A class of groups in an arbitrary realm connected with the configuration ofthe 27 lines on a cubic surface”, The quarterly journal of pure and applied mathematics 33: 145–173, Reprintedin volume 5 of his collected works

• Dickson, Leonard Eugene (1908), “A class of groups in an arbitrary realm connected with the configurationof the 27 lines on a cubic surface (second paper)", The quarterly journal of pure and applied mathematics 39:205–209, Reprinted in volume VI of his collected works

• Ichiro, Yokota. “Exceptional Lie groups”. arXiv.org. Retrieved June 28, 2015.

[1] Rosenfeld, Boris (1997), Geometry of Lie Groups (theorem 7.4 on page 335, and following paragraph).

[2] Платонов, Владимир П.; Рапинчук, Андрей С. (1991). Алгебраические группы и теория чисел. Наука. ISBN 5-02-014191-7. (English translation: Platonov, Vladimir P.; Rapinchuk, Andrei S. (1994). Algebraic groups and number theory.Academic Press. ISBN 0-12-558180-7.), §2.2.4

[3] Conway, John Horton; Curtis, Robert Turner; Norton, Simon Phillips; Parker, Richard A; Wilson, Robert Arnott (1985).Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford University Press. p. 26.ISBN 0-19-853199-0.

[4] Carter, Roger W. (1989). Simple Groups of Lie Type. Wiley Classics Library. John Wiley & Sons. ISBN 0-471-50683-4.

[5] Wilson, Robert A. (2009). The Finite Simple Groups. Graduate Texts in Mathematics 251. Springer-Verlag. ISBN 1-84800-987-9.

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Chapter 22

E7 (mathematics)

In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7,all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7.The designation E7 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall intofour infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E7 algebrais thus one of the five exceptional cases.The fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E7 is the cyclicgroup Z/2Z, and its outer automorphism group is the trivial group. The dimension of its fundamental representationis 56.

22.1 Real and complex forms

There is a unique complex Lie algebra of type E7, corresponding to a complex group of complex dimension 133.The complex adjoint Lie group E7 of complex dimension 133 can be considered as a simple real Lie group of realdimension 266. This has fundamental group Z/2Z, has maximal compact subgroup the compact form (see below) ofE7, and has an outer automorphism group of order 2 generated by complex conjugation.As well as the complex Lie group of type E7, there are four real forms of the Lie algebra, and correspondingly fourreal forms of the group with trivial center (all of which have an algebraic double cover, and three of which havefurther non-algebraic covers, giving further real forms), all of real dimension 133, as follows:

• The compact form (which is usually the one meant if no other information is given), which has fundamentalgroup Z/2Z and has trivial outer automorphism group.

• The split form, EV (or E₇₍₇₎), which has maximal compact subgroup SU(8)/±1, fundamental group cyclic oforder 4 and outer automorphism group of order 2.

• EVI (or E₇₍−₅₎), which has maximal compact subgroup SU(2)·SO(12)/(center), fundamental group non-cyclicof order 4 and trivial outer automorphism group.

• EVII (or E₇₍−₂₅₎), which has maximal compact subgroup SO(2)·E6/(center), infinite cyclic findamental groupand outer automorphism group of order 2.

For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups.The compact real form of E7 is the isometry group of the 64-dimensional exceptional compact Riemannian symmetricspace EVI (in Cartan’s classification). It is known informally as the "quateroctonionic projective plane" because itcan be built using an algebra that is the tensor product of the quaternions and the octonions, and is also knownas a Rosenfeld projective plane, though it does not obey the usual axioms of a projective plane. This can be seensystematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits.The Tits–Koecher construction produces forms of the E7 Lie algebra from Albert algebras, 27-dimensional excep-tional Jordan algebras.

58

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22.2. E7 AS AN ALGEBRAIC GROUP 59

22.2 E7 as an algebraic group

By means of a Chevalley basis for the Lie algebra, one can define E7 as a linear algebraic group over the integers and,consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimesalso known as “untwisted”) adjoint form of E7. Over an algebraically closed field, this and its double cover are theonly forms; however, over other fields, there are often many other forms, or “twists” of E7, which are classified inthe general framework of Galois cohomology (over a perfect field k) by the set H1(k, Aut(E7)) which, because theDynkin diagram of E7 (see below) has no automorphisms, coincides with H1(k, E₇, ₐ ).[1]

Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E7 coincidewith the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all adjointforms of E7 have fundamental group Z/2Z in the sense of algebraic geometry, meaning that they admit exactly onedouble cover; the further non-compact real Lie group forms of E7 are therefore not algebraic and admit no faithfulfinite-dimensional representations.Over finite fields, the Lang–Steinberg theorem implies that H1(k, E7) = 0, meaning that E7 has no twisted forms: seebelow.

22.3 Algebra

22.3.1 Dynkin diagram

The Dynkin diagram for E7 is given by .

22.3.2 Root system

Even though the roots span a 7-dimensional space, it is more symmetric and convenient to represent them as vectorslying in a 7-dimensional subspace of an 8-dimensional vector space.

The roots are all the 8×7 permutations of (1,−1,0,0,0,0,0,0) and all the(84

)permutations of (½,½,½,½,−½,−½,−½,−½)

Note that the 7-dimensional subspace is the subspace where the sum of all the eight coordinates is zero. There are126 roots.The simple roots are

(0,−1,1,0,0,0,0,0)(0,0,−1,1,0,0,0,0)(0,0,0,−1,1,0,0,0)(0,0,0,0,−1,1,0,0)(0,0,0,0,0,−1,1,0)(0,0,0,0,0,0,−1,1)(½,½,½,½,−½,−½,−½,−½)

We have ordered them so that their corresponding nodes in the Dynkin diagram are ordered from left to right (in thediagram depicted above) with the side node last.

An alternative description

An alternative (7-dimensional) description of the root system, which is useful in considering E7 × SU(2) as a subgroupof E8, is the following:

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60 CHAPTER 22. E7 (MATHEMATICS)

The 126 vertices of the 231 polytope represent the root vectors of E7, as shown in this Coxeter plane projectionCoxeter–Dynkin diagram:

All 4 ×(62

)permutations of (±1,±1,0,0,0,0,0) preserving the zero at the last entry, all of the following roots with

an even number of +½

(±1

2,±1

2,±1

2,±1

2,±1

2,±1

2,± 1√

2

)and the two following roots

(0, 0, 0, 0, 0, 0,±

√2).

Thus the generators consist of a 66-dimensional so(12) subalgebra as well as 65 generators that transform as twoself-conjugate Weyl spinors of spin(12) of opposite chirality and their chirality generator, and two other generatorsof chiralities ±

√2 .

Given the E7 Cartan matrix (below) and a Dynkin diagram node ordering of: 1 2 3 4 5 6

7

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22.3. ALGEBRA 61

one choice of simple roots is given by the rows of the following matrix:

1 −1 0 0 0 0 00 1 −1 0 0 0 00 0 1 −1 0 0 00 0 0 1 −1 0 00 0 0 0 1 1 0

−12 − 1

2 −12 −1

2 − 12 −1

2

√22

0 0 0 0 1 −1 0

.

22.3.3 Weyl group

The Weyl group of E7 is of order 2903040: it is the direct product of the cyclic group of order 2 and the uniquesimple group of order 1451520 (which can be described as PSp6(2) or PSΩ7(2)).[2]

22.3.4 Cartan matrix

2

3

13

14

2 4

2 5 7

3 5 7

3 6735

4 6

4

54

415 7 72 65

2 3 73 6

5 77 1651

2 46 27 3

6 1 4 2 62 4675

7 1

1 46 4 6 3172

5

4 1 533

6 16 2

3 5 3151 26

65 2 3 1 4

54 21 7

2 7 4 1

3 7 2

7 3

4

5

6

(1, 0, 0, 0, 0, 0, 0)

(1, 1, 0, 0, 0, 0, 0)

(0, 1, 0, 0, 0, 0, 0)

(0, 1, 1, 0, 0, 0, 0)

(0, 0, 1, 0, 0, 0, 0)

(0, 0, 1, 1, 0, 0, 0)

(0, 0, 0, 1, 0, 0, 0)

(0, 0, 0, 1, 1, 0, 0) (0, 0, 0, 1, 0, 0, 1)

(0, 0, 0, 0, 1, 0, 0)

(0, 0, 0, 0, 1, 1, 0)

(0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 0, 1)

(1, 1, 1, 0, 0, 0, 0)(0, 1, 1, 1, 0, 0, 0) (0, 0, 1, 1, 1, 0, 0) (0, 0, 1, 1, 0, 0, 1) (0, 0, 0, 1, 1, 1, 0)(0, 0, 0, 1, 1, 0, 1)

(1, 1, 1, 1, 0, 0, 0)(0, 1, 1, 1, 1, 0, 0) (0, 1, 1, 1, 0, 0, 1) (0, 0, 1, 1, 1, 0, 1) (0, 0, 1, 1, 1, 1, 0) (0, 0, 0, 1, 1, 1, 1)

(0, 1, 1, 1, 1, 0, 1) (0, 0, 1, 2, 1, 0, 1)(0, 0, 1, 1, 1, 1, 1) (1, 1, 1, 1, 1, 0, 0) (1, 1, 1, 1, 0, 0, 1)(0, 1, 1, 1, 1, 1, 0)

(0, 1, 1, 1, 1, 1, 1) (1, 1, 1, 1, 1, 0, 1) (0, 1, 1, 2, 1, 0, 1)(1, 1, 1, 1, 1, 1, 0) (0, 0, 1, 2, 1, 1, 1)

(1, 1, 1, 1, 1, 1, 1) (0, 1, 1, 2, 1, 1, 1) (1, 1, 1, 2, 1, 0, 1) (0, 1, 2, 2, 1, 0, 1) (0, 0, 1, 2, 2, 1, 1)

(1, 1, 1, 2, 1, 1, 1) (0, 1, 1, 2, 2, 1, 1)(0, 1, 2, 2, 1, 1, 1) (1, 1, 2, 2, 1, 0, 1)

(1, 2, 2, 2, 1, 0, 1)(1, 1, 2, 2, 1, 1, 1) (1, 1, 1, 2, 2, 1, 1) (0, 1, 2, 2, 2, 1, 1)

(1, 2, 2, 2, 1, 1, 1)(1, 1, 2, 2, 2, 1, 1) (0, 1, 2, 3, 2, 1, 1)

(1, 1, 2, 3, 2, 1, 1) (1, 2, 2, 2, 2, 1, 1) (0, 1, 2, 3, 2, 1, 2)

(1, 2, 2, 3, 2, 1, 1) (1, 1, 2, 3, 2, 1, 2)

(1, 2, 3, 3, 2, 1, 1) (1, 2, 2, 3, 2, 1, 2)

(1, 2, 3, 3, 2, 1, 2)

(1, 2, 3, 4, 2, 1, 2)

(1, 2, 3, 4, 3, 1, 2)

(1, 2, 3, 4, 3, 2, 2)

Hasse diagram of E7 root poset with edge labels identifying added simple root position

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62 CHAPTER 22. E7 (MATHEMATICS)

2 −1 0 0 0 0 0−1 2 −1 0 0 0 00 −1 2 −1 0 0 00 0 −1 2 −1 0 −10 0 0 −1 2 −1 00 0 0 0 −1 2 00 0 0 −1 0 0 2

.

22.4 Important subalgebras and representations

E7 has an SU(8) subalgebra, as is evident by noting that in the 8-dimensional description of the root system, the firstgroup of roots are identical to the roots of SU(8) (with the same Cartan subalgebra as in the E7).In addition to the 133-dimensional adjoint representation, there is a 56-dimensional “vector” representation, to befound in the E8 adjoint representation.The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all givenby the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A121736in OEIS):

1, 56, 133, 912, 1463, 1539, 6480, 7371, 8645, 24320, 27664, 40755, 51072, 86184, 150822, 152152,238602, 253935, 293930, 320112, 362880, 365750, 573440, 617253, 861840, 885248, 915705, 980343,2273920, 2282280, 2785552, 3424256, 3635840...

The underlined terms in the sequence above are the dimensions of those irreducible representations possessed by theadjoint form of E7 (equivalently, those whose weights belong to the root lattice of E7), whereas the full sequence givesthe dimensions of the irreducible representations of the simply connected form of E7. There exist non-isomorphicirreducible representation of dimensions 1903725824, 16349520330, etc.The fundamental representations are those with dimensions 133, 8645, 365750, 27664, 1539, 56 and 912 (corre-sponding to the seven nodes in the Dynkin diagram in the order chosen for the Cartan matrix above, i.e., the nodesare read in the six-node chain first, with the last node being connected to the third).

22.4.1 E7 Polynomial Invariants

E7 is the automorphism group of the following pair of polynomials in 56 non-commutative variables. We divide thevariables into two groups of 28, (p, P) and (q, Q) where p and q are real variables and P and Q are 3x3 octonionhermitian matrices. Then the first invariant is the symplectic invariant of Sp(56, R):

C1 = pq − qp+ Tr[PQ]− Tr[QP ]

The second more complicated invariant is a symmetric quartic polynomial:

C2 = (pq + Tr[P Q])2 + pTr[Q Q] + qTr[P P ] + Tr[P Q]

Where P ≡ det(P )P−1 and the binary circle operator is defined by A B = (AB +BA)/2 .An alternative quartic polynomial invariant constructed by Cartan uses two anti-symmetric 8x8 matrices each with28 components.

C2 = Tr[(XY )2]− 1

4Tr[XY ]2 +

1

96ϵijklmnop

(XijXklXmnXop + Y ijY klY mnY op

)

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22.5. CHEVALLEY GROUPS OF TYPE E7 63

22.5 Chevalley groups of type E7

The points over a finite field with q elements of the (split) algebraic group E7 (see above), whether of the adjoint(centerless) or simply connected form (its algebraic universal cover), give a finite Chevalley group. This is closelyconnected to the group written E7(q), however there is ambiguity in this notation, which can stand for several things:

• the finite group consisting of the points over Fq of the simply connected form of E7 (for clarity, this can bewritten E₇, (q) and is known as the “universal” Chevalley group of type E7 over Fq),

• (rarely) the finite group consisting of the points over Fq of the adjoint form of E7 (for clarity, this can be writtenE₇,ₐ (q), and is known as the “adjoint” Chevalley group of type E7 over Fq), or

• the finite group which is the image of the natural map from the former to the latter: this is what will be denotedby E7(q) in the following, as is most common in texts dealing with finite groups.

From the finite group perspective, the relation between these three groups, which is quite analogous to that betweenSL(n, q), PGL(n, q) and PSL(n, q), can be summarized as follows: E7(q) is simple for any q, E₇, (q) is its Schurcover, and the E₇,ₐ (q) lies in its automorphism group; furthermore, when q is a power of 2, all three coincide, andotherwise (when q is odd), the Schur multiplier of E7(q) is 2 and E7(q) is of index 2 in E₇,ₐ (q), which explains whyE₇, (q) and E₇,ₐ (q) are often written as 2·E7(q) and E7(q)·2. From the algebraic group perspective, it is less commonfor E7(q) to refer to the finite simple group, because the latter is not in a natural way the set of points of an algebraicgroup over Fq unlike E₇, (q) and E₇,ₐ (q).As mentioned above, E7(q) is simple for any q,[3][4] and it constitutes one of the infinite families addressed by theclassification of finite simple groups. Its number of elements is given by the formula (sequence A008870 in OEIS):

1

gcd(2, q − 1)q63(q18 − 1)(q14 − 1)(q12 − 1)(q10 − 1)(q8 − 1)(q6 − 1)(q2 − 1)

The order of E₇, (q) or E₇,ₐ (q) (both are equal) can be obtained by removing the dividing factor gcd(2, q−1) (se-quence A008869 in OEIS). The Schur multiplier of E7(q) is gcd(2, q−1), and its outer automorphism group is theproduct of the diagonal automorphism group Z/gcd(2, q−1)Z (given by the action of E₇,ₐ (q)) and the group of fieldautomorphisms (i.e., cyclic of order f if q = pf where p is prime).

22.6 Importance in physics

N = 8 supergravity in four dimensions, which is a dimensional reduction from 11 dimensional supergravity, admit anE7 bosonic global symmetry and an SU(8) bosonic local symmetry. The fermions are in representations of SU(8),the gauge fields are in a representation of E7, and the scalars are in a representation of both (Gravitons are singletswith respect to both). Physical states are in representations of the coset E7 / SU(8).In string theory, E7 appears as a part of the gauge group of one the (unstable and non-supersymmetric) versions ofthe heterotic string. It can also appear in the unbroken gauge group E8 × E7 in six-dimensional compactifications ofheterotic string theory, for instance on the four-dimensional surface K3.

22.7 See also

• En (Lie algebra)

• ADE classification

• List of simple Lie groups

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64 CHAPTER 22. E7 (MATHEMATICS)

22.8 Notes[1] Platonov, Vladimir; Rapinchuk, Andrei (1994) [1991],Algebraic groups and number theory, Pure andAppliedMathematics

139, Boston, MA: Academic Press, ISBN 978-0-12-558180-6, MR 1278263 (original version: Платонов, Владимир П.;Рапинчук, Андрей С. (1991). Алгебраические группы и теория чисел. Наука. ISBN 5-02-014191-7.), §2.2.4

[2] Conway, John Horton; Curtis, Robert Turner; Norton, Simon Phillips; Parker, Richard A; Wilson, Robert Arnott (1985).Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford University Press. p. 46.ISBN 0-19-853199-0.

[3] Carter, Roger W. (1989). Simple Groups of Lie Type. Wiley Classics Library. John Wiley & Sons. ISBN 0-471-50683-4.

[4] Wilson, Robert A. (2009). The Finite Simple Groups. Graduate Texts in Mathematics 251. Springer-Verlag. ISBN 1-84800-987-9.

22.9 References• Adams, J. Frank (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University ofChicago Press, ISBN 978-0-226-00526-3, MR 1428422

• John Baez, The Octonions, Section 4.5: E7, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTMLversion at http://math.ucr.edu/home/baez/octonions/node18.html.

• E. Cremmer and B. Julia, The N = 8 Supergravity Theory. 1. The Lagrangian, Phys.Lett.B80:48,1978.Online scanned version at http://ac.els-cdn.com/0370269378903039/1-s2.0-0370269378903039-main.pdf?_tid=79273f80-539d-11e4-a133-00000aab0f6c&acdnat=1413289833_5f3539a6365149b108ddcec889200964.

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Chapter 23

E8 (mathematics)

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Liealgebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. Thedesignation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into fourinfinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E8 algebra isthe largest and most complicated of these exceptional cases.Wilhelm Killing (1888a, 1888b, 1889, 1890) discovered the complex Lie algebra E8 during his classification ofsimple compact Lie algebras, though he did not prove its existence, which was first shown by Élie Cartan. Cartandetermined that a complex simple Lie algebra of type E8 admits three real forms. Each of them gives rise to a simpleLie group of dimension 248, exactly one of which is compact. Chevalley (1955) introduced algebraic groups and Liealgebras of type E8 over other fields: for example, in the case of finite fields they lead to an infinite family of finitesimple groups of Lie type.

23.1 Basic description

The Lie group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is 8. Therefore thevectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article.The Weyl group of E8, which is the group of symmetries of the maximal torus which are induced by conjugations inthe whole group, has order 214 3 5 5 2 7 = 696729600.The compact group E8 is unique among simple compact Lie groups in that its non-trivial representation of smallestdimension is the adjoint representation (of dimension 248) acting on the Lie algebra E8 itself; it is also the uniqueone which has the following four properties: trivial center, compact, simply connected, and simply laced (all rootshave the same length).There is a Lie algebra En for every integer n ≥ 3, which is infinite dimensional if n is greater than 8.

23.2 Real and complex forms

There is a unique complex Lie algebra of type E8, corresponding to a complex group of complex dimension 248.The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of real dimension496. This is simply connected, has maximal compact subgroup the compact form (see below) of E8, and has an outerautomorphism group of order 2 generated by complex conjugation.As well as the complex Lie group of type E8, there are three real forms of the Lie algebra, three real forms of thegroup with trivial center (two of which have non-algebraic double covers, giving two further real forms), all of realdimension 248, as follows:

• The compact form (which is usually the one meant if no other information is given), which is simply connectedand has trivial outer automorphism group.

• The split form, EVIII (or E₈₍₈₎), which has maximal compact subgroup Spin(16)/(Z/2Z), fundamental group

65

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66 CHAPTER 23. E8 (MATHEMATICS)

of order 2 (implying that it has a double cover, which is a simply connected Lie real group but is not algebraic,see below) and has trivial outer automorphism group.

• EIX (or E₈₍−₂₄₎), which has maximal compact subgroup E7×SU(2)/(−1,−1), fundamental group of order 2(again implying a double cover, which is not algebraic) and has trivial outer automorphism group.

For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups.

23.3 E8 as an algebraic group

By means of a Chevalley basis for the Lie algebra, one can define E8 as a linear algebraic group over the integers and,consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimesalso known as “untwisted”) form of E8. Over an algebraically closed field, this is the only form; however, over otherfields, there are often many other forms, or “twists” of E8, which are classified in the general framework of Galoiscohomology (over a perfect field k) by the set H1(k,Aut(E8)) which, because the Dynkin diagram of E8 (see below)has no automorphisms, coincides with H1(k,E8).[1]

Over R, the real connected component of the identity of these algebraically twisted forms of E8 coincide with thethree real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all forms of E8 aresimply connected in the sense of algebraic geometry, meaning that they admit no non-trivial algebraic coverings;the non-compact and simply connected real Lie group forms of E8 are therefore not algebraic and admit no faithfulfinite-dimensional representations.Over finite fields, the Lang–Steinberg theorem implies that H1(k,E8)=0, meaning that E8 has no twisted forms: seebelow.

23.4 Representation theory

The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all givenby the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A121732in OEIS):

1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000,26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500,820260000, 1094951000, 2172667860, 2275896000, 2642777280, 2903770000, 3929713760, 4076399250,4825673125, 6899079264, 8634368000 (twice), 12692520960…

The 248-dimensional representation is the adjoint representation. There are two non-isomorphic irreducible represen-tations of dimension 8634368000 (it is not unique; however, the next integer with this property is 175898504162692612600853299200000(sequenceA181746 inOEIS)). The fundamental representations are thosewith dimensions 3875, 6696000, 6899079264,146325270, 2450240, 30380, 248 and 147250 (corresponding to the eight nodes in the Dynkin diagram in the orderchosen for the Cartan matrix below, i.e., the nodes are read in the seven-node chain first, with the last node beingconnected to the third).The coefficients of the character formulas for infinite dimensional irreducible representations of E8 depend on somelarge square matrices consisting of polynomials, the Lusztig–Vogan polynomials, an analogue of Kazhdan–Lusztigpolynomials introduced for reductive groups in general by George Lusztig and David Kazhdan (1983). The values at1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations (whosecharacters are easy to describe) with the irreducible representations.These matrices were computed after four years of collaboration by a group of 18 mathematicians and computerscientists, led by Jeffrey Adams, with much of the programming done by Fokko du Cloux. The most difficult case (forexceptional groups) is the split real form of E8 (see above), where the largest matrix is of size 453060×453060. TheLusztig–Vogan polynomials for all other exceptional simple groups have been known for some time; the calculationfor the split form of E8 is far longer than any other case. The announcement of the result in March 2007 receivedextraordinary attention from the media (see the external links), to the surprise of the mathematicians working on it.The representations of the E8 groups over finite fields are given by Deligne–Lusztig theory.

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23.5. CONSTRUCTIONS 67

23.5 Constructions

One can construct the (compact form of the) E8 group as the automorphism group of the corresponding e8 Liealgebra. This algebra has a 120-dimensional subalgebra so(16) generated by Jij as well as 128 new generators Qathat transform as a Weyl–Majorana spinor of spin(16). These statements determine the commutators

[Jij , Jkℓ] = δjkJiℓ − δjℓJik − δikJjℓ + δiℓJjk

as well as

[Jij , Qa] =1

4(γiγj − γjγi)abQb,

while the remaining commutator (not anticommutator!) is defined as

[Qa, Qb] = γ[iacγj]cbJij .

It is then possible to check that the Jacobi identity is satisfied.

23.6 Geometry

The compact real form of E8 is the isometry group of the 128-dimensional exceptional compact Riemannian symmet-ric space EVIII (in Cartan’s classification). It is known informally as the "octooctonionic projective plane" becauseit can be built using an algebra that is the tensor product of the octonions with themselves, and is also known as aRosenfeld projective plane, though it does not obey the usual axioms of a projective plane. This can be seen sys-tematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits (Landsberg &Manivel 2001).

23.7 E8 root system

A root system of rank r is a particular finite configuration of vectors, called roots, which span an r-dimensionalEuclidean space and satisfy certain geometrical properties. In particular, the root system must be invariant underreflection through the hyperplane perpendicular to any root.TheE8 root system is a rank 8 root system containing 240 root vectors spanningR8. It is irreducible in the sense thatit cannot be built from root systems of smaller rank. All the root vectors in E8 have the same length. It is convenientfor a number of purposes to normalize them to have length √2. These 240 vectors are the vertices of a semi-regularpolytope discovered by Thorold Gosset in 1900, sometimes known as the 421 polytope.

23.7.1 Construction

In the so-called even coordinate system, E8 is given as the set of all vectors in R8 with length squared equal to 2 suchthat coordinates are either all integers or all half-integers and the sum of the coordinates is even.Explicitly, there are 112 roots with integer entries obtained from

(±1,±1, 0, 0, 0, 0, 0, 0)

by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots with half-integer entries obtained from

(±1

2 ,±12 ,±

12 ,±

12 ,±

12 ,±

12 ,±

12 ,±

12

)

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68 CHAPTER 23. E8 (MATHEMATICS)

Zome model of the E8 root system, projected into three-space, and represented by the vertices of the 421 polytope,

by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even).There are 240 roots in all.The 112 roots with integer entries form a D8 root system. The E8 root system also contains a copy of A8 (which has72 roots) as well as E6 and E7 (in fact, the latter two are usually defined as subsets of E8).In the odd coordinate system, E8 is given by taking the roots in the even coordinate system and changing the sign ofany one coordinate. The roots with integer entries are the same while those with half-integer entries have an oddnumber of minus signs rather than an even number.

23.7.2 Dynkin diagram

The Dynkin diagram for E8 is given by .This diagram gives a concise visual summary of the root structure. Each node of this diagram represents a simpleroot. A line joining two simple roots indicates that they are at an angle of 120° to each other. Two simple rootswhich are not joined by a line are orthogonal.

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23.7. E8 ROOT SYSTEM 69

E8 with thread made by hand

23.7.3 Cartan matrix

The Cartan matrix of a rank r root system is an r × r matrix whose entries are derived from the simple roots.Specifically, the entries of the Cartan matrix are given by

Aij = 2(αi, αj)

(αi, αi)

where (−,−) is the Euclidean inner product and αi are the simple roots. The entries are independent of the choice ofsimple roots (up to ordering).The Cartan matrix for E8 is given by

2 −1 0 0 0 0 0 0−1 2 −1 0 0 0 0 00 −1 2 −1 0 0 0 00 0 −1 2 −1 0 0 00 0 0 −1 2 −1 0 −10 0 0 0 −1 2 −1 00 0 0 0 0 −1 2 00 0 0 0 −1 0 0 2

.

The determinant of this matrix is equal to 1.

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70 CHAPTER 23. E8 (MATHEMATICS)

2

3

13

14

24

25

3 5

3 6 8

4 68

4 784 6

5 7

5

65

41526 8 83 763 4 84 7

516 88 2 76 23 5 7 38 4

72 5 3 73 56 81 78 16 8 2

2 5715 742 78 6 8 1 3 6

1 5 2 6485 714 72 7 3

5 41 6 42 621 374 7

37 64 171 6 314 2 5

2736 165 31 2 84

35 76 21 3 815 2

63 84 81 3 125

54 2 8 381 4 1

1 52 84 8 4 2

52 1 63 8 4

8 3 5 2 6 1 7

5 63 2 7 1

46 3 7 2

74 36

7 5 4

5 87

68 7

6 8

5

4

3

2

1

(1,0,0,0,0,0,0,0)

(1,1,0,0,0,0,0,0)

(0,1,0,0,0,0,0,0)

(0,1,1,0,0,0,0,0)

(0,0,1,0,0,0,0,0)

(0,0,1,1,0,0,0,0)

(0,0,0,1,0,0,0,0)

(0,0,0,1,1,0,0,0)

(0,0,0,0,1,0,0,0)

(0,0,0,0,1,1,0,0) (0,0,0,0,1,0,0,1)

(0,0,0,0,0,1,0,0)

(0,0,0,0,0,1,1,0)

(0,0,0,0,0,0,1,0)(0,0,0,0,0,0,0,1)

(1,1,1,0,0,0,0,0)(0,1,1,1,0,0,0,0)(0,0,1,1,1,0,0,0) (0,0,0,1,1,1,0,0) (0,0,0,1,1,0,0,1) (0,0,0,0,1,1,1,0)(0,0,0,0,1,1,0,1)

(1,1,1,1,0,0,0,0)(0,1,1,1,1,0,0,0)(0,0,1,1,1,1,0,0) (0,0,1,1,1,0,0,1) (0,0,0,1,1,1,0,1) (0,0,0,1,1,1,1,0) (0,0,0,0,1,1,1,1)

(0,0,1,1,1,1,0,1) (0,0,0,1,2,1,0,1) (0,0,0,1,1,1,1,1)(1,1,1,1,1,0,0,0)(0,1,1,1,1,1,0,0) (0,1,1,1,1,0,0,1) (0,0,1,1,1,1,1,0)

(0,0,1,1,1,1,1,1)(0,1,1,1,1,1,0,1) (0,0,1,1,2,1,0,1) (1,1,1,1,1,1,0,0) (1,1,1,1,1,0,0,1) (0,1,1,1,1,1,1,0) (0,0,0,1,2,1,1,1)

(0,1,1,1,1,1,1,1) (0,0,1,1,2,1,1,1)(1,1,1,1,1,1,1,0)(1,1,1,1,1,1,0,1)(0,1,1,1,2,1,0,1) (0,0,1,2,2,1,0,1) (0,0,0,1,2,2,1,1)

(1,1,1,1,1,1,1,1) (0,1,1,1,2,1,1,1) (0,0,1,1,2,2,1,1)(0,0,1,2,2,1,1,1)(0,1,1,2,2,1,0,1) (1,1,1,1,2,1,0,1)

(1,1,1,2,2,1,0,1) (1,1,1,1,2,1,1,1) (0,1,2,2,2,1,0,1)(0,1,1,2,2,1,1,1) (0,1,1,1,2,2,1,1) (0,0,1,2,2,2,1,1)

(1,1,2,2,2,1,0,1)(1,1,1,2,2,1,1,1) (0,1,2,2,2,1,1,1)(0,1,1,2,2,2,1,1) (0,0,1,2,3,2,1,1)(1,1,1,1,2,2,1,1)

(1,1,1,2,2,2,1,1) (1,2,2,2,2,1,0,1)(1,1,2,2,2,1,1,1)(0,1,1,2,3,2,1,1) (0,1,2,2,2,2,1,1) (0,0,1,2,3,2,1,2)

(1,2,2,2,2,1,1,1)(1,1,1,2,3,2,1,1) (0,1,2,2,3,2,1,1) (0,1,1,2,3,2,1,2)(1,1,2,2,2,2,1,1)

(1,2,2,2,2,2,1,1)(1,1,2,2,3,2,1,1) (1,1,1,2,3,2,1,2)(0,1,2,3,3,2,1,1) (0,1,2,2,3,2,1,2)

(0,1,2,3,3,2,1,2)(1,1,2,3,3,2,1,1) (1,2,2,2,3,2,1,1) (1,1,2,2,3,2,1,2)

(1,1,2,3,3,2,1,2) (0,1,2,3,4,2,1,2)(1,2,2,3,3,2,1,1) (1,2,2,2,3,2,1,2)

(1,2,3,3,3,2,1,1) (1,2,2,3,3,2,1,2) (1,1,2,3,4,2,1,2) (0,1,2,3,4,3,1,2)

(1,2,3,3,3,2,1,2) (1,2,2,3,4,2,1,2) (1,1,2,3,4,3,1,2) (0,1,2,3,4,3,2,2)

(1,2,3,3,4,2,1,2) (1,2,2,3,4,3,1,2) (1,1,2,3,4,3,2,2)

(1,2,3,3,4,3,1,2) (1,2,2,3,4,3,2,2)(1,2,3,4,4,2,1,2)

(1,2,3,4,4,3,1,2) (1,2,3,3,4,3,2,2)

(1,2,3,4,4,3,2,2) (1,2,3,4,5,3,1,2)

(1,2,3,4,5,3,2,2) (1,2,3,4,5,3,1,3)

(1,2,3,4,5,3,2,3) (1,2,3,4,5,4,2,2)

(1,2,3,4,5,4,2,3)

(1,2,3,4,6,4,2,3)

(1,2,3,5,6,4,2,3)

(1,2,4,5,6,4,2,3)

(1,3,4,5,6,4,2,3)

(2,3,4,5,6,4,2,3)

Hasse diagram of E8 root poset with edge labels identifying added simple root position.

23.7.4 Simple roots

A set of simple roots for a root system Φ is a set of roots that form a basis for the Euclidean space spanned by Φwith the special property that each root has components with respect to this basis that are either all nonnegative or allnonpositive.

Given the E8 Cartan matrix (above) and a Dynkin diagram node ordering of: 1 2 3 4 5 6 7

8

One choice of simple roots is given by the rows of the following matrix:

1 −1 0 0 0 0 0 00 1 −1 0 0 0 0 00 0 1 −1 0 0 0 00 0 0 1 −1 0 0 00 0 0 0 1 −1 0 00 0 0 0 0 1 1 0

− 12 − 1

2 − 12 − 1

2 − 12 − 1

2 − 12 − 1

20 0 0 0 0 1 −1 0

.

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23.8. CHEVALLEY GROUPS OF TYPE E8 71

23.7.5 Weyl group

The Weyl group of E8 is of order 696729600, and can be described as O+8(2): it is of the form 2.G.2 (that is, a stem extension by the cyclic group of order 2 of an extension of the cyclicgroup of order 2 by a group G) where G is the unique simple group of order 174182400 (which can be described asPSΩ8

+(2)).[2]

23.7.6 E8 root lattice

Main article: E8 lattice

The integral span of the E8 root system forms a lattice in R8 naturally called the E8 root lattice. This lattice is ratherremarkable in that it is the only (nontrivial) even, unimodular lattice with rank less than 16.

23.7.7 Simple subalgebras of E8

The Lie algebra E8 contains as subalgebras all the exceptional Lie algebras as well as many other important Liealgebras in mathematics and physics. The height of the Lie algebra on the diagram approximately corresponds to therank of the algebra. A line from an algebra down to a lower algebra indicates that the lower algebra is a subalgebraof the higher algebra.

23.8 Chevalley groups of type E8

Chevalley (1955) showed that the points of the (split) algebraic group E8 (see above) over a finite field with q elementsform a finite Chevalley group, generally written E8(q), which is simple for any q,[3][4] and constitutes one of theinfinite families addressed by the classification of finite simple groups. Its number of elements is given by the formula(sequence A008868 in OEIS):

q120(q30 − 1)(q24 − 1)(q20 − 1)(q18 − 1)(q14 − 1)(q12 − 1)(q8 − 1)(q2 − 1)

The first term in this sequence, the order of E8(2), namely 337804753143634806261388190614085595079991692242467651576160959909068800000≈ 3.38×1074, is already larger than the size of the Monster group. This group E8(2) is the last one described (butwithout its character table) in the ATLAS of Finite Groups.[5]

The Schur multiplier of E8(q) is trivial, and its outer automorphism group is that of field automorphisms (i.e., cyclicof order f if q=pf where p is prime).Lusztig (1979) described the unipotent representations of finite groups of type E8.

23.9 Subgroups

The smaller exceptional groups E7 and E6 sit inside E8. In the compact group, both E7×SU(2)/(−1,−1) and E6×SU(3)/(Z/3Z)are maximal subgroups of E8.The 248-dimensional adjoint representation of E8 may be considered in terms of its restricted representation to thefirst of these subgroups. It transforms under E7×SU(2) as a sum of tensor product representations, which may belabelled as a pair of dimensions as (3,1) + (1,133) + (2,56) (since there is a quotient in the product, these notationsmay strictly be taken as indicating the infinitesimal (Lie algebra) representations). Since the adjoint representationcan be described by the roots together with the generators in the Cartan subalgebra, we may see that decompositionby looking at these. In this description,

• (3,1) consists of the roots (0,0,0,0,0,0,1,−1), (0,0,0,0,0,0,−1,1) and the Cartan generator corresponding to thelast dimension;

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72 CHAPTER 23. E8 (MATHEMATICS)

E8

E7

E6

O(16)

O(12)O(13)

O(14)O(15)

O(11)O(10)

O(9)O(8)

F4

O(7) O(6)~SU(4)

SU(3)

SU(2)~O(3)~USp(2)

U(1)~O(2)

G2 O(5)~USp(4)O(4)

Subgroup Tree of E8

SU(6)

SU(7)

SU(8)

SU(5)

USp(6)

USp(8)

An incomplete simple subgroup tree of E8

• (1,133) consists of all roots with (1,1), (−1,−1), (0,0), (−½,−½) or (½,½) in the last two dimensions, togetherwith the Cartan generators corresponding to the first seven dimensions;

• (2,56) consists of all roots with permutations of (1,0), (−1,0) or (½,−½) in the last two dimensions.

The 248-dimensional adjoint representation of E8, when similarly restricted, transforms under E6×SU(3) as: (8,1) +(1,78) + (3,27) + (3,27). We may again see the decomposition by looking at the roots together with the generatorsin the Cartan subalgebra. In this description,

• (8,1) consists of the roots with permutations of (1,−1,0) in the last three dimensions, together with the Cartangenerator corresponding to the last two dimensions;

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23.10. APPLICATIONS 73

• (1,78) consists of all roots with (0,0,0), (−½,−½,−½) or (½,½,½) in the last three dimensions, together withthe Cartan generators corresponding to the first six dimensions;

• (3,27) consists of all roots with permutations of (1,0,0), (1,1,0) or (−½,½,½) in the last three dimensions.

• (3,27) consists of all roots with permutations of (−1,0,0), (−1,−1,0) or (½,−½,−½) in the last three dimensions.

The finite quasisimple groups that can embed in (the compact form of) E8 were found by Griess & Ryba (1999).The Dempwolff group is a subgroup of (the compact form of) E8. It is contained in the Thompson sporadic group,which acts on the underlying vector space of the Lie group E8 but does not preserve the Lie bracket. The Thompsongroup fixes a lattice and does preserve the Lie bracket of this lattice mod 3, giving an embedding of the Thompsongroup into E8(F3).

23.10 Applications

The E8 Lie group has applications in theoretical physics and especially in string theory and supergravity. E8×E8 isthe gauge group of one of the two types of heterotic string and is one of two anomaly-free gauge groups that can becoupled to the N = 1 supergravity in ten dimensions. E8 is the U-duality group of supergravity on an eight-torus (inits split form).One way to incorporate the standard model of particle physics into heterotic string theory is the symmetry breakingof E8 to its maximal subalgebra SU(3)×E6.In 1982, Michael Freedman used the E8 lattice to construct an example of a topological 4-manifold, the E8 manifold,which has no smooth structure.Antony Garrett Lisi's incomplete "An Exceptionally Simple Theory of Everything" attempts to describe all knownfundamental interactions in physics as part of the E8 Lie algebra.[6][7]

R. Coldea, D. A. Tennant, and E. M. Wheeler et al. (2010) reported an experiment where the electron spins of acobalt-niobium crystal exhibited, under certain conditions, two of the eight peaks related to E8 that were predictedby Zamolodchikov (1989).[8][9]

23.11 Notes

[1] Платонов, Владимир П.; Рапинчук, Андрей С. (1991), Алгебраические группы и теория чисел, Наука, ISBN 5-02-014191-7 (English translation: Platonov, Vladimir P.; Rapinchuk, Andrei S. (1994), Algebraic groups and number theory,Academic Press, ISBN 0-12-558180-7), §2.2.4

[2] Conway, John Horton; Curtis, Robert Turner; Norton, Simon Phillips; Parker, Richard A; Wilson, Robert Arnott (1985),Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Oxford University Press, p. 85,ISBN 0-19-853199-0

[3] Carter, Roger W. (1989), Simple Groups of Lie Type, Wiley Classics Library, John Wiley & Sons, ISBN 0-471-50683-4

[4] Wilson, Robert A. (2009), The Finite Simple Groups, Graduate Texts inMathematics 251, Springer-Verlag, ISBN 1-84800-987-9

[5] Conway &al, op. cit., p. 235.

[6] A.G. Lisi; J. O.Weatherall (2010). “AGeometric Theory of Everything”. Scientific American 303 (6): 54–61. doi:10.1038/scientificamerican1210-54. PMID 21141358.

[7] Greg Boustead (2008-11-17). “Garrett Lisi’s Exceptional Approach to Everything”. SEED Magazine.

[8] Most beautiful math structure appears in lab for first time, New Scientist, January 2010 (retrieved January 8, 2010).

[9] Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?, Notices of the American Mathematical Society, Septem-ber 2011.

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74 CHAPTER 23. E8 (MATHEMATICS)

23.12 References

• Adams, J. Frank (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University ofChicago Press, ISBN 978-0-226-00526-3, MR 1428422

• Baez, John C. (2002), “The octonions”, American Mathematical Society. Bulletin. New Series 39 (2): 145–205,doi:10.1090/S0273-0979-01-00934-X, MR 1886087

• Chevalley, Claude (1955), “Sur certains groupes simples”, The Tohoku Mathematical Journal. Second Series7: 14–66, doi:10.2748/tmj/1178245104, ISSN 0040-8735, MR 0073602

• Coldea, R.; Tennant, D. A.; Wheeler, E. M.; Wawrzynska, E.; Prabhakaran, D.; Telling, M.; Habicht, K.;Smeibidl, P.; Kiefer, K. (2010), “Quantum Criticality in an Ising Chain: Experimental Evidence for EmergentE8 Symmetry”, Science 327 (5962): 177–180, doi:10.1126/science.1180085

• Griess, Robert L.; Ryba, A. J. E. (1999), “Finite simple groups which projectively embed in an exceptional Liegroup are classified!", AmericanMathematical Society. Bulletin. New Series 36 (1): 75–93, doi:10.1090/S0273-0979-99-00771-5, MR 1653177

• Killing, Wilhelm (1888a), “Die Zusammensetzung der stetigen endlichen Transformationsgruppen”, Mathe-matische Annalen 31 (2): 252–290, doi:10.1007/BF01211904

• Killing, Wilhelm (1888b), “Die Zusammensetzung der stetigen endlichen Transformationsgruppen”, Mathe-matische Annalen 33 (1): 1–48, doi:10.1007/BF01444109

• Killing, Wilhelm (1889), “Die Zusammensetzung der stetigen endlichen Transformationsgruppen”,Mathema-tische Annalen 34 (1): 57–122, doi:10.1007/BF01446792

• Killing, Wilhelm (1890), “Die Zusammensetzung der stetigen endlichen Transformationsgruppen”,Mathema-tische Annalen 36 (2): 161–189, doi:10.1007/BF01207837

• J.M. Landsberg and L. Manivel (2001), The projective geometry of Freudenthal’s magic square, Journal ofAlgebra, Volume 239, Issue 2, pages 477–512, doi:10.1006/jabr.2000.8697, arXiv:math/9908039v1.

• Lusztig, George (1979), “Unipotent representations of a finite Chevalley group of type E8”, The QuarterlyJournal of Mathematics. Oxford. Second Series 30 (3): 315–338, doi:10.1093/qmath/30.3.301, ISSN 0033-5606, MR R545068

• Lusztig, George; Vogan, David (1983), “Singularities of closures of K-orbits on flag manifolds”, InventionesMathematicae (Springer-Verlag) 71 (2): 365–379, doi:10.1007/BF01389103

• Zamolodchikov, A. B. (1989), “Integrals of motion and S-matrix of the (scaled) T=T Ising model with mag-netic field”, International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology. NuclearPhysics 4 (16): 4235–4248, doi:10.1142/S0217751X8900176X, MR 1017357

23.13 External links

Lusztig–Vogan polynomial calculation

• Atlas of Lie groups

• Kazhdan–Lusztig–Vogan Polynomials for E8

• Narrative of the Project to compute Kazhdan–Lusztig Polynomials for E8

• Slides for The Character Table for E8, or How We Wrote Down a 453,060 × 453,060 Matrix and FoundHappiness by D. Vogan.

• American Institute of Mathematics (March 2007), Mathematicians Map E8

• The n-Category Café, a University of Texas blog posting by John Baez on E8.

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23.13. EXTERNAL LINKS 75

Other links

• Graphic representation of E8 root system.

• The list of dimensions of irreducible representations of the complex form of E8 is sequence A121732 in theOEIS.

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Chapter 24

F4 (mathematics)

In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Liegroups. F4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism groupis the trivial group. Its fundamental representation is 26-dimensional.The compact real form of F4 is the isometry group of a 16-dimensional Riemannian manifold known as the octonionicprojective plane OP2. This can be seen systematically using a construction known as the magic square, due to HansFreudenthal and Jacques Tits.There are 3 real forms: a compact one, a split one, and a third one. They are the isometry groups of the three realAlbert algebras.The F4 Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36-dimensional Liealgebra so(9), in analogy with the construction of E8.In older books and papers, F4 is sometimes denoted by E4.

24.1 Algebra

24.1.1 Dynkin diagram

The Dynkin diagram for F4 is .

24.1.2 Weyl/Coxeter group

Its Weyl/Coxeter group is the symmetry group of the 24-cell: it is a solvable group of order 1152.

24.1.3 Cartan matrix2 −1 0 0

−1 2 −2 00 −1 2 −10 0 −1 2

24.1.4 F4 lattice

The F4 lattice is a four-dimensional body-centered cubic lattice (i.e. the union of two hypercubic lattices, each lyingin the center of the other). They form a ring called the Hurwitz quaternion ring. The 24 Hurwitz quaternions of norm1 form the vertices of a 24-cell centered at the origin.

76

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24.1. ALGEBRA 77

24.1.5 Roots of F4

The 24 vertices of 24-cell (red) and 24 vertices of its dual (yellow) represent the 48 root vectors of F4 in this Coxeter plane projection

The 48 root vectors of F4 can be found as the vertices of the 24-cell in two dual configurations:

24-cell vertices:

• 24 roots by (±1,±1,0,0), permuting coordinate positions

Dual 24-cell vertices:

• 8 roots by (±1, 0, 0, 0), permuting coordinate positions

• 16 roots by (±½, ±½, ±½, ±½).

Simple roots

One choice of simple roots for F4, , is given by the rows of the following matrix:

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78 CHAPTER 24. F4 (MATHEMATICS)

0 1 −1 00 0 1 −10 0 0 112 −1

2 −12 − 1

2

2

3

1 3

1 3 4

2 4

2

3

3 41 41 3

2 4 3 1 4

4 42 1

34 2

3 4

3

2

1

(1, 0, 0, 0)

(1, 1, 0, 0)

(0, 1, 0, 0)

(0, 1, 1, 0)

(0, 0, 1, 0)

(0, 0, 1, 1)

(0, 0, 0, 1)

(1, 1, 1, 0) (0, 1, 2, 0) (0, 1, 1, 1)

(1, 1, 2, 0) (1, 1, 1, 1) (0, 1, 2, 1)

(1, 2, 2, 0) (1, 1, 2, 1) (0, 1, 2, 2)

(1, 2, 2, 1) (1, 1, 2, 2)

(1, 2, 2, 2) (1, 2, 3, 1)

(1, 2, 3, 2)

(1, 2, 4, 2)

(1, 3, 4, 2)

(2, 3, 4, 2)

Hasse diagram of F4 root poset with edge labels identifying added simple root position

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24.2. REPRESENTATIONS 79

24.1.6 F4 polynomial invariant

Just as O(n) is the group of automorphisms which keep the quadratic polynomials x2 + y2 + ... invariant, F4 is thegroup of automorphisms of the following set of 3 polynomials in 27 variables. (The first can easily be substituted intoother two making 26 variables).

C1 = x+ y + z

C2 = x2 + y2 + z2 + 2XX + 2Y Y + 2ZZ

C3 = xyz − xXX − yY Y − zZZ +XY Z +XY Z

Where x, y, z are real valued and X, Y, Z are octonion valued. Another way of writing these invariants is as (combi-nations of) Tr(M), Tr(M2) and Tr(M3) of the hermitian octonion matrix:

M =

x Z YZ y XY X z

24.2 Representations

The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all givenby the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A121738in OEIS):

1, 26, 52, 273, 324, 1053 (twice), 1274, 2652, 4096, 8424, 10829, 12376, 16302, 17901, 19278,19448, 29172, 34749, 76076, 81081, 100776, 106496, 107406, 119119, 160056 (twice), 184756,205751, 212992, 226746, 340119, 342056, 379848, 412776, 420147, 627912…

The 52-dimensional representation is the adjoint representation, and the 26-dimensional one is the trace-free part ofthe action of F4 on the exceptional Albert algebra of dimension 27.There are two non-isomorphic irreducible representations of dimensions 1053, 160056, 4313088, etc. The fundamentalrepresentations are those with dimensions 52, 1274, 273, 26 (corresponding to the four nodes in the Dynkin diagramin the order such that the double arrow points from the second to the third).

24.3 See also• Albert algebra

• Cayley plane

• Dynkin diagram

• Fundamental representation

• Simple Lie group

24.4 References• Adams, J. Frank (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University ofChicago Press, ISBN 978-0-226-00526-3, MR 1428422

• John Baez, The Octonions, Section 4.2: F4, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTMLversion at http://math.ucr.edu/home/baez/octonions/node15.html.

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80 CHAPTER 24. F4 (MATHEMATICS)

• Chevalley C, Schafer RD (February 1950). “The Exceptional Simple Lie Algebras F(4) and E(6)". Proc. Natl.Acad. Sci. U.S.A. 36 (2): 137–41. doi:10.1073/pnas.36.2.137. PMC 1063148. PMID 16588959.

• Jacobson, Nathan (1971-06-01). Exceptional Lie Algebras (1 ed.). CRC Press. ISBN 0-8247-1326-5.

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Chapter 25

Fixed-point subgroup

In algebra, the fixed-point subgroup Gf of an automorphism f of a group G is the subgroup of G:

Gf = g ∈ G | f(g) = g.

For example, take G to be the group of invertible n-by-n real matrices and f(g) = (gT )−1 . Then Gf is the groupO(n) of n-by-n orthogonal matrices.The same definition applies to rings as well. Let R be a ring and f an automorphism of R. Then the subring fixed byf is the subring of R:

Rf = r ∈ R | f(r) = r.

Slightly more generally, if G is a subgroup of the automorphism group Aut(R) of R, then RG , the intersection ofRg, g ∈ G is a subring called the subring fixed by H or, more commonly, the ring of invariants. A basic exampleappears in Galois theory; see Fundamental theorem of Galois theory.

81

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Chapter 26

Formal group

In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the productof a Lie group. They were introduced by S. Bochner (1946). The term formal group sometimes means the same asformal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Liegroups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology.

26.1 Definitions

A one-dimensional formal group law over a commutative ring R is a power series F(x,y) with coefficients in R,such that

1. F(x,y) = x + y + terms of higher degree

2. F(x, F(y,z)) = F(F(x,y), z) (associativity).

The simplest example is the additive formal group law F(x, y) = x + y. The idea of the definition is that F shouldbe something like the formal power series expansion of the product of a Lie group, where we choose coordinates sothat the identity of the Lie group is the origin.More generally, an n-dimensional formal group law is a collection of n power series Fi(x1, x2, ..., xn, y1, y2, ...,yn) in 2n variables, such that

1. F(x,y) = x + y + terms of higher degree

2. F(x, F(y,z)) = F(F(x,y), z)

where we write F for (F1, ..., Fn), x for (x1,..., xn), and so on.The formal group law is called commutative if F(x,y) = F(y,x).

Prop. If R is Z -torsion free then any one-dimensional formal group law over R is commutative.

Proof. The torsion freeness gives us the exponential and logarithm which allows us to write F as F(x,y)= exp(log(x) + log(y)).

There is no need for an axiom analogous to the existence of an inverse for groups, as this turns out to follow auto-matically from the definition of a formal group law. In other words we can always find a (unique) power series Gsuch that F(x,G(x)) = 0.A homomorphism from a formal group law F of dimensionm to a formal group lawG of dimension n is a collectionf of n power series in m variables, such that

G(f(x), f(y)) = f(F(x, y)).

82

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26.2. EXAMPLES 83

A homomorphism with an inverse is called an isomorphism, and is called a strict isomorphism if in addition f(x)=x + terms of higher degree. Two formal group laws with an isomorphism between them are essentially the same; theydiffer only by a “change of coordinates”.

26.2 Examples• The additive formal group law is given by

F (x, y) = x+ y.

• The multiplicative formal group law is given by

F (x, y) = x+ y + xy.

This rule can be understood as follows. The product G in the (multiplicative group of the) ring R is given by G(a,b)= ab. If we “change coordinates” to make 0 the identity by putting a = 1 + x, b = 1 + y, and G = 1 + F, then we findthat F(x, y) = x + y + xy. Over the rational numbers, there is an isomorphism from the additive formal group law tothe multiplicative one, given by exp(x) − 1. Over general commutative rings R there is no such homomorphism asdefining it requires non-integral rational numbers, and the additive and multiplicative formal groups are usually notisomorphic.

• More generally, we can construct a formal group law of dimension n from any algebraic group or Lie groupof dimension n, by taking coordinates at the identity and writing down the formal power series expansion ofthe product map. The additive and multiplicative formal group laws are obtained in this way from the additiveand multiplicative algebraic groups. Another important special case of this is the formal group (law) of anelliptic curve (or abelian variety).

• F(x,y) = (x + y)/(1 + xy) is a formal group law coming from the addition formula for the hyperbolic tangentfunction: tanh(x + y) = F(tanh(x), tanh(y)), and is also the formula for addition of velocities in special relativity(with the speed of light equal to 1).

• F (x, y) = (x√1− y4 + y

√1− x4)/(1 + x2y2) is a formal group law over Z[1/2] found by Euler, in the

form of the addition formula for an elliptic integral:

∫ x

0

dt√1− t4

+

∫ y

0

dt√1− t4

=

∫ F (x,y)

0

dt√1− t4

.

26.3 Lie algebras

Any n-dimensional formal group law gives an n dimensional Lie algebra over the ring R, defined in terms of thequadratic part F2 of the formal group law.

[x,y] = F2(x,y) − F2(y,x)

The natural functor from Lie groups or algebraic groups to Lie algebras can be factorized into a functor from Liegroups to formal group laws, followed by taking the Lie algebra of the formal group:

Lie groups → Formal group laws → Lie algebras

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84 CHAPTER 26. FORMAL GROUP

Over fields of characteristic 0, formal group laws are essentially the same as finite-dimensional Lie algebras: moreprecisely, the functor from finite-dimensional formal group laws to finite-dimensional Lie algebras is an equivalenceof categories. Over fields of non-zero characteristic, formal group laws are not equivalent to Lie algebras. In fact,in this case it is well known that passing from an algebraic group to its Lie algebra often throws away too muchinformation, but passing instead to the formal group law often keeps enough information. So in some sense formalgroup laws are the “right” substitute for Lie algebras in characteristic p > 0.

26.4 The logarithm of a commutative formal group law

If F is a commutative n-dimensional formal group law over a commutativeQ-algebra R, then it is strictly isomorphicto the additive formal group law. In other words, there is a strict isomorphism f from the additive formal group to F,called the logarithm of F, so that

f(F(x,y)) = f(x) + f(y)

Examples:

• The logarithm of F(x, y) = x + y is f(x) = x.

• The logarithm of F(x, y) = x + y + xy is f(x) = log(1 + x), because log(1 + x + y + xy) = log(1 + x) + log(1 + y).

If R does not contain the rationals, a map f can be constructed by extension of scalars to R⊗Q, but this will sendeverything to zero if R has positive characteristic. Formal group laws over a ring R are often constructed by writingdown their logarithm as a power series with coefficients in R⊗Q, and then proving that the coefficients of the corre-sponding formal group over R⊗Q actually lie in R. When working in positive characteristic, one typically replaces Rwith a mixed characteristic ring that has a surjection to R, such as the ringW(R) of Witt vectors, and reduces to R atthe end.

26.5 The formal group ring of a formal group law

The formal group ring of a formal group law is a cocommutative Hopf algebra analogous to the group ring of a groupand to the universal enveloping algebra of a Lie algebra, both of which are also cocommutative Hopf algebras. Ingeneral cocommutative Hopf algebras behave very much like groups.For simplicity we describe the 1-dimensional case; the higher-dimensional case is similar except that notation becomesmessier.Suppose that F is a (1-dimensional) formal group law over R. Its formal group ring (also called its hyperalgebraor its covariant bialgebra) is a cocommutative Hopf algebra H constructed as follows.

• As an R-module, H is free with a basis 1 = D(0), D(1), D(2), ...

• The coproduct Δ is given by ΔD(n) = ∑D(i) ⊗ D(n−i) (so the dual of this coalgebra is just the ring of formalpower series).

• The counit η is given by the coefficient of D(0).

• The identity is 1 = D(0).

• The antipode S takes D(n) to (−1)nD(n).

• The coefficient of D(1) in the product D(i)D(j) is the coefficient of xiyj in F(x, y).

Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law Ffrom it. So 1-dimensional formal group laws are essentially the same as Hopf algebras whose coalgebra structure isgiven above.

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26.6. FORMAL GROUP LAWS AS FUNCTORS 85

26.6 Formal group laws as functors

Given an n-dimensional formal group law F over R and a commutative R-algebra S, we can form a group F(S) whoseunderlying set is Nn where N is the set of nilpotent elements of S. The product is given by using F to multiply elementsof Nn; the point is that all the formal power series now converge because they are being applied to nilpotent elements,so there are only a finite number of nonzero terms. This makes F into a functor from commutative R-algebras S togroups.We can extend the definition of F(S) to some topological R-algebras. In particular, if S is an inverse limit of discreteR algebras, we can define F(S) to be the inverse limit of the corresponding groups. For example, this allows us todefine F(Zp) with values in the p-adic numbers.The group-valued functor of F can also be described using the formal group ring H of F. For simplicity we willassume that F is 1-dimensional; the general case is similar. For any cocommutative Hopf algebra, an element g iscalled group-like if Δg = g ⊗ g and εg = 1, and the group-like elements form a group under multiplication. In thecase of the Hopf algebra of a formal group law over a ring, the group like elements are exactly those of the form

D(0) + D(1)x + D(2)x2 + ...

for nilpotent elements x. In particular we can identify the group-like elements of H⊗S with the nilpotent elements ofS, and the group structure on the group-like elements of H⊗S is then identified with the group structure on F(S).

26.7 The height of a formal group law

Suppose that f is a homomorphism between one dimensional formal group laws over a field of characteristic p > 0.Then f is either zero, or the first nonzero term in its power series expansion is axph for some non-negative integer h,called the height of the homomorphism f. The height of the zero homomorphism is defined to be ∞.The height of a one dimensional formal group law over a field of characteristic p > 0 is defined to be the height of itsmultiplication by p map.Two one dimensional formal group laws over an algebraically closed field of characteristic p > 0 are isomorphic ifand only if they have the same height, and the height can be any positive integer or ∞.Examples:

• The additive formal group law F(x, y) = x + y has height ∞, as its pth power map is 0.

• The multiplicative formal group law F(x, y) = x + y + xy has height 1, as its pth power map is (1 + x)p − 1 = xp.

• The formal group law of an elliptic curve has height either one or two, depending on whether the curve isordinary or supersingular. Supersingularity can be detected by the vanishing of the Eisenstein series Ep−1 .

26.8 Lazard ring

Main article: Lazard’s universal ring

There is a universal commutative one-dimensional formal group law over a universal commutative ring defined asfollows. We let

F(x, y)

be

x + y + Σci,j xiyj

for indeterminates

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86 CHAPTER 26. FORMAL GROUP

ci,j,

and we define the universal ring R to be the commutative ring generated by the elements ci,j, with the relations thatare forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring Rhas the following universal property:

For any commutative ring S, one-dimensional formal group laws over S correspond to ring homomor-phisms from R to S.

The commutative ring R constructed above is known as Lazard’s universal ring. At first sight it seems to beincredibly complicated: the relations between its generators are very messy. However Lazard proved that it has avery simple structure: it is just a polynomial ring (over the integers) on generators of degrees 2, 4, 6, ... (where ci,jhas degree 2(i + j − 1)). Daniel Quillen proved that the coefficient ring of complex cobordism is naturally isomorphicas a graded ring to Lazard’s universal ring, explaining the unusual grading.

26.9 Formal groups

A formal group is a group object in the category of formal schemes.

• If G is a functor from Artinian algebras to groups which is left exact, then it representable (G is the functor ofpoints of a formal group. (left exactness of a functor is equivalent to commuting with finite projective limits).

• IfG is a group scheme then G , the formal completion of G at the identity has the structure of a formal group.

• A smooth group scheme is isomorphic to Spf(R[[T1, . . . , Tn]]) . Some people call a formal group schemesmooth if the converse holds.

• formal smoothness asserts the existence of lifts of deformations and can apply to formal schemes that are largerthan points. A smooth formal group scheme is a special case of a formal group scheme.

• Given a smooth formal group, one can construct a formal group law and a field by choosing a uniformizing setof sections.

• The (non-strict) isomorphisms between formal group laws induced by change of parameters make up the ele-ments of the group of coordinate changes on the formal group.

Formal groups and formal group laws can also be defined over arbitrary schemes, rather than just over commutativerings or fields, and families can be classified by maps from the base to a parametrizing object.The moduli space of formal group laws is a disjoint union of infinite-dimensional affine spaces, whose componentsare parametrized by dimension, and whose points are parametrized by admissible coefficients of the power seriesF. The corresponding moduli stack of smooth formal groups is a quotient of this space by a canonical action of theinfinite-dimensional groupoid of coordinate changes.Over an algebraically closed field, the substack of one dimensional formal groups is either a point (in characteristiczero) or an infinite chain of stacky points parametrizing heights. In characteristic zero, the closure of each pointcontains all points of greater height. This difference gives formal groups a rich geometric theory in positive andmixed characteristic, with connections to the Steenrod algebra, p-divisible groups, Dieudonné theory, and Galoisrepresentations. For example, the Serre-Tate theorem implies that the deformations of a group scheme are stronglycontrolled by those of its formal group, especially in the case of supersingular abelian varieties. For supersingularelliptic curves, this control is complete, and this is quite different from the characteristic zero situation where theformal group has no deformations.A formal group is sometimes defined as a cocommutative Hopf algebra (usually with some extra conditions added,such as being pointed or connected).[1] This is more or less dual to the notion above. In the smooth case, choosingcoordinates is equivalent to taking a distinguished basis of the formal group ring.Some authors use the term formal group to mean formal group law.

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26.10. LUBIN–TATE FORMAL GROUP LAWS 87

26.10 Lubin–Tate formal group laws

Main article: Lubin–Tate formal group law

We let Zp be the ring of p-adic integers. The Lubin–Tate formal group law is the unique (1-dimensional) formalgroup law F such that e(x) = px + xp is an endomorphism of F, in other words

e(F (x, y)) = F (e(x), e(y)).

More generally we can allow e to be any power series such that e(x) = px + higher-degree terms and e(x) = xp mod p.All the group laws for different choices of e satisfying these conditions are strictly isomorphic.[2]

For each element a in Zp there is a unique endomorphism f of the Lubin–Tate formal group law such that f(x) = ax+ higher-degree terms. This gives an action of the ring Zp on the Lubin–Tate formal group law.There is a similar construction with Zp replaced by any complete discrete valuation ring with finite residue classfield.[3]

This construction was introduced by Lubin & Tate (1965), in a successful effort to isolate the local field part of theclassical theory of complex multiplication of elliptic functions. It is also a major ingredient in some approaches tolocal class field theory.[4]

26.11 See also• Witt vector

• Artin–Hasse exponential

26.12 References[1] Underwood, Robert G. (2011). An introduction to Hopf algebras. Berlin: Springer-Verlag. p. 121. ISBN 978-0-387-

72765-3. Zbl 1234.16022.

[2] Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of MathematicalSciences 49 (Second ed.). p. 168. ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002.

[3] Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). Springer-Verlag. pp.62–63. ISBN 3-540-63003-1. Zbl 0819.11044.

[4] e.g. Serre, Jean-Pierre (1967). “Local class field theory”. In Cassels, J.W.S.; Fröhlich, Albrecht. Algebraic Number Theory.Academic Press. pp. 128–161. Zbl 0153.07403.Hazewinkel, Michiel (1975). “Local class field theory is easy”. Advancesin Math. 18 (2): 148–181. doi:10.1016/0001-8708(75)90156-5. Zbl 0312.12022.Iwasawa, Kenkichi (1986). Local classfield theory. OxfordMathematical Monographs. The Clarendon Press Oxford University Press. ISBN 978-0-19-504030-2.MR 863740. Zbl 0604.12014.

• Adams, J. Frank (1974), Stable homotopy and generalised homology, University of Chicago Press, ISBN 978-0-226-00524-9

• Bochner, Salomon (1946), “Formal Lie groups”, Annals of Mathematics. Second Series 47: 192–201, ISSN0003-486X, JSTOR 1969242, MR 0015397

• M. Demazure, Lectures on p-divisible groups Lecture Notes in Mathematics, 1972. ISBN 0-387-06092-8

• Fröhlich, A. (1968), Formal groups, Lecture Notes in Mathematics 74, Berlin, New York: Springer-Verlag,doi:10.1007/BFb0074373, MR 0242837

• P. Gabriel, Étude infinitésimale des schémas en groupes SGA 3 Exp. VIIB

• Formal Groups and Applications (Pure and Applied Math 78) Michiel Hazewinkel Publisher: Academic Pr(June 1978) ISBN 0-12-335150-2

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88 CHAPTER 26. FORMAL GROUP

• Lazard, Michel (1975), Commutative formal groups, Lecture Notes in Mathematics 443, Berlin, New York:Springer-Verlag, doi:10.1007/BFb0070554, ISBN 978-3-540-07145-7, MR 0393050

• Lubin, Jonathan; Tate, John (1965), “Formal complex multiplication in local fields”, Annals of Mathematics.Second Series 81: 380–387, ISSN 0003-486X, JSTOR 1970622, MR 0172878, Zbl 0128.26501

• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322,Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859

• Strickland, N. “Formal groups” (PDF).

• Zarkhin, Yu.G. (2001), “F/f040820”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

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Chapter 27

Fundamental lemma (Langlands program)

In the mathematical theory of automorphic forms, the fundamental lemma relates orbital integrals on a reductivegroup over a local field to stable orbital integrals on its endoscopic groups. It was conjectured by Langlands (1983)in the course of developing the Langlands program. The fundamental lemma was proved by Gérard Laumon andNgô Bảo Châu in the case of unitary groups and then by Ngô for general reductive groups, building on a series ofimportant reductions made by Jean-LoupWaldspurger to the case of Lie algebras. Timemagazine placed Ngô's proofon the list of the “Top 10 scientific discoveries of 2009”.[1] In 2010 Ngô was awarded the Fields medal for this proof.

27.1 Motivation and history

Robert Langlands outlined a strategy for proving local and global Langlands conjectures using the Arthur–Selbergtrace formula, but in order for this approach to work, the geometric sides of the trace formula for different groupsmust be related in a particular way. This relationship takes the form of identities between orbital integrals on reductivegroups G and H over a nonarchimedean local field F, where the group H, called an endoscopic group of G, is con-structed from G and some additional data.The first case considered was G = SL2 (Labesse & Langlands 1979). Langlands and Shelstad (1987) then developedthe general framework for the theory of endoscopic transfer and formulated specific conjectures. However, duringthe next two decades only partial progress was made towards proving the fundamental lemma.[2][3] Harris called ita “bottleneck limiting progress on a host of arithmetic questions”.[4] Langlands himself, writing on the origins ofendoscopy, commented:

27.2 Statement

The fundamental lemma states that an orbital integral O for a group G is equal to a stable orbital integral SO for anendoscopic group H, up to a transfer factor Δ (Nadler 2012):

SOγH (1KH ) = ∆(γH , γG)OκγG

(1KG)

where

• F is a local field

• G is an unramified group defined over F, in other words a quasi-split reductive group defined over F that splitsover an unramified extension of F

• H is an unramified endoscopic group of G associated to κ

• KG and KH are hyperspecial maximal compact subgroups of G and H, which means roughly that they are thesubgroups of points with coefficients in the ring of integers of F.

89

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90 CHAPTER 27. FUNDAMENTAL LEMMA (LANGLANDS PROGRAM)

• 1KG and 1KH are the characteristic functions of KG and KH.

• Δ(γH,γG) is a transfer factor, a certain elementary expression depending on γH and γG

• γH and γG are elements of G and H representing stable conjugacy classes, such that the stable conjugacy classof G is the transfer of the stable conjugacy class of H.

• κ is a character of the group of conjugacy classes in the stable conjugacy class of γG

• SO and O are stable orbital integrals and orbital integrals depending on their parameters.

27.3 Approaches

Shelstad (1982) proved the fundamental lemma for Archimedean fields.Waldspurger (1991) verified the fundamental lemma for general linear groups.Kottwitz (1992) and Blasius & Rogawski (1992) verified some cases of the fundamental lemma for 3-dimensionalunitary groups.Hales (1997) andWeissauer (2009) verified the fundamental lemma for the symplectic and general symplectic groupsSp4, GSp4.A paper of George Lusztig and David Kazhdan pointed out that orbital integrals could be interpreted as countingpoints on certain algebraic varieties over finite fields. Further, the integrals in question can be computed in a way thatdepends only on the residue field of F; and the issue can be reduced to the Lie algebra version of the orbital integrals.Then the problem was restated in terms of the Springer fiber of algebraic groups.[6] The circle of ideas was connectedto a purity conjecture; Laumon gave a conditional proof based on such a conjecture, for unitary groups. Laumonand Ngô (2008) then proved the fundamental lemma for unitary groups, using Hitchin fibration introduced by Ngô(2006), which is an abstract geometric analogue of the Hitchin system of complex algebraic geometry. Waldspurger(2006) showed for Lie algebras that the function field case implies the fundamental lemma over all local fields, andWaldspurger (2008) showed that the fundamental lemma for Lie algebras implies the fundamental lemma for groups.

27.4 Notes[1] Top 10 Scientific Discoveries of 2009, Time

[2] Kottwitz and Rogawski for U3, Wadspurger for SLn, Hales and Weissauer for Sp4.

[3] Fundamental Lemma and Hitchin Fibration, Gérard Laumon, May 13, 2009

[4] INTRODUCTION TO “THE STABLE TRACE FORMULA, SHIMURA VARIETIES, AND ARITHMETIC APPLI-CATIONS”, p. 1., Michael Harris

[5] publications.ias.edu

[6] The Fundamental Lemma for Unitary Groups, at p. 12., Gérard Laumon

27.5 References• Blasius, Don; Rogawski, Jonathan D. (1992), “Fundamental lemmas for U(3) and related groups”, in Lang-lands, Robert P.; Ramakrishnan, Dinakar, The zeta functions of Picard modular surfaces, Montreal, QC: Univ.Montréal, pp. 363–394, ISBN 978-2-921120-08-1, MR 1155234

• Casselman, W. (2009), Langlands’ Fundamental Lemma for SL(2)

• Dat, Jean-François (November 2004), Lemme fondamental et endoscopie, une approche géométrique, d'aprèsGérard Laumon et Ngô Bao Châu, Séminaire Bourbaki, no 940

• Hales, Thomas C. (1997), “The fundamental lemma for Sp(4)", Proceedings of the American MathematicalSociety 125 (1): 301–308, doi:10.1090/S0002-9939-97-03546-6, ISSN 0002-9939, MR 1346977

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27.6. EXTERNAL LINKS 91

• Harris, M. (ed.), Stabilisation de la formule des traces, variétés de Shimura, et applications arithmétiques

• Kazhdan, David; Lusztig, George (1988), “Fixed point varieties on affine flag manifolds”, Israel Journal ofMathematics 62 (2): 129–168, doi:10.1007/BF02787119, ISSN 0021-2172, MR 947819

• Kottwitz, Robert E. (1992), “Calculation of some orbital integrals”, in Langlands, Robert P.; Ramakrishnan,Dinakar, The zeta functions of Picard modular surfaces, Montreal, QC: Univ. Montréal, pp. 349–362, ISBN978-2-921120-08-1, MR 1155233

• Labesse, Jean-Pierre; Langlands, R. P. (1979), “L-indistinguishability for SL(2)", Canadian Journal of Math-ematics 31 (4): 726–785, doi:10.4153/CJM-1979-070-3, ISSN 0008-414X, MR 540902

• Langlands, Robert P. (1983), Les débuts d'une formule des traces stable, Publications Mathématiques del'Université Paris VII [Mathematical Publications of the University of Paris VII] 13, Paris: Université deParis VII U.E.R. de Mathématiques, MR 697567

• Langlands, Robert P.; Shelstad, Diana (1987), “On the definition of transfer factors”, Mathematische Annalen278 (1): 219–271, doi:10.1007/BF01458070, ISSN 0025-5831, MR 909227

• Laumon, Gérard (2006), “Aspects géométriques du LemmeFondamental de Langlands-Shelstad”, InternationalCongress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, pp. 401–419, MR 2275603

• Laumon, Gérard; Ngô, Bao Châu (2008), “Le lemme fondamental pour les groupes unitaires”, Annals of Math-ematics. Second Series 168 (2): 477–573, doi:10.4007/annals.2008.168.477, ISSN 0003-486X, MR 2434884

• Nadler, David (2012), “The geometric nature of the fundamental lemma”, Bulletin of the American Mathemat-ical Society 49: 1–50, doi:10.1090/S0273-0979-2011-01342-8, ISSN 0002-9904

• Ngô, Bao Châu (2006), “Fibration de Hitchin et endoscopie”, Inventiones Mathematicae 164 (2): 399–453,doi:10.1007/s00222-005-0483-7, ISSN 0020-9910, MR 2218781

• Ngô, Bao Châu (2010), “Le lemme fondamental pour les algèbres de Lie”, Institut des Hautes Études Scien-tifiques. Publications Mathématiques 111: 1–169, doi:10.1007/s10240-010-0026-7, ISSN 0073-8301, MR2653248

• Shelstad, Diana (1982), “L-indistinguishability for real groups”, Mathematische Annalen 259 (3): 385–430,doi:10.1007/BF01456950, ISSN 0025-5831, MR 661206

• Waldspurger, Jean-Loup (1991), “Sur les intégrales orbitales tordues pour les groupes linéaires: un lemmefondamental”, Canadian Journal ofMathematics 43 (4): 852–896, doi:10.4153/CJM-1991-049-5, ISSN 0008-414X, MR 1127034

• Waldspurger, Jean-Loup (2006), “Endoscopie et changement de caractéristique”, Journal of the Institute ofMathematics of Jussieu. JIMJ. Journal de l'Institut deMathématiques de Jussieu 5 (3): 423–525, doi:10.1017/S1474748006000041,ISSN 1474-7480, MR 2241929

• Waldspurger, Jean-Loup (2008), “L'endoscopie tordue n'est pas si tordue”,Memoirs of the AmericanMathemat-ical Society (Providence, R.I.: American Mathematical Society) 194 (908): 261, ISBN 978-0-8218-4469-4,ISSN 0065-9266, MR 2418405

• Weissauer, Rainer (2009), Endoscopy for GSp(4) and the cohomology of Siegel modular threefolds, LectureNotes inMathematics 1968, Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-540-89306-6, ISBN 978-3-540-89305-9, MR 2498783

27.6 External links• Gerard Laumon lecture on the fundamental lemma for unitary groups

• Basken, Paul (September 12, 2010). “Understanding the Langlands Fundamental Lemma”. The Chronicle ofHigher Education.

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Chapter 28

G2 (mathematics)

In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split realform), their Lie algebras g2 , as well as some algebraic groups. They are the smallest of the five exceptional simpleLie groups. G2 has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.The compact form of G2 can be described as the automorphism group of the octonion algebra or, equivalently, as thesubgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional real spinor representation. RobertBryant introduced the definition of G2 as the subgroup of GL(R7) that preserves the non-degenerate 3-form

dx124 + dx235 + dx346 + dx450 + dx561 + dx602 + dx013,

(invariant under the cyclic permutation (0123456)) with dxijk denoting dxi ∧ dxj ∧ dxk.In older books and papers, G2 is sometimes denoted by E2.

28.1 Real forms

There are 3 simple real Lie algebras associated with this root system:

• The underlying real Lie algebra of the complex Lie algebra G2 has dimension 28. It has complex conjugationas an outer automorphism and is simply connected. The maximal compact subgroup of its associated group isthe compact form of G2.

• The Lie algebra of the compact form is 14-dimensional. The associated Lie group has no outer automorphisms,no center, and is simply connected and compact.

• The Lie algebra of the non-compact (split) form has dimension 14. The associated simple Lie group hasfundamental group of order 2 and its outer automorphism group is the trivial group. Its maximal compactsubgroup is SU(2) × SU(2)/(−1,−1). It has a non-algebraic double cover that is simply connected.

28.2 Algebra

28.2.1 Dynkin diagram and Cartan matrix

The Dynkin diagram for G2 is given by .Its Cartan matrix is:

[2 −1

−3 2

]92

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28.3. POLYNOMIAL INVARIANT 93

28.2.2 Roots of G2

Although they span a 2-dimensional space, as drawn, it is much more symmetric to consider them as vectors in a2-dimensional subspace of a three-dimensional space.

One set of simple roots, for is:

(0,1,−1), (1,−2,1)

28.2.3 Weyl/Coxeter group

Its Weyl/Coxeter group is the dihedral group, D6 of order 12.

28.2.4 Special holonomy

G2 is one of the possible special groups that can appear as the holonomy group of a Riemannianmetric. Themanifoldsof G2 holonomy are also called G2-manifolds.

28.3 Polynomial Invariant

G2 is the automorphism group of the following two polynomials in 7 non-commutative variables.

C1 = t2 + u2 + v2 + w2 + x2 + y2 + z2

C2 = tuv + wtx+ ywu+ zyt+ vzw + xvy + uxz

which comes from the octonion algebra. The variables must be non-commutative otherwise the second polynomialwould be identically zero.

28.4 Generators

Adding a representation of the 14 generators with coefficients A..N gives the matrix:

Aλ1 + ...+Nλ14 =

0 C −B E −D −G −F +M−C 0 A F −G+N D −K E + LB −A 0 −N M L K−E −F N 0 −A+H −B + I −C + JD G−N −M A−H 0 J −IG K −D −L B − I −J 0 H

F −M −E − L −K C − J I −H 0

28.5 Representations

The characters of finite-dimensional representations of the real and complex Lie algebras and Lie groups are all givenby the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A104599in OEIS):

1, 7, 14, 27, 64, 77 (twice), 182, 189, 273, 286, 378, 448, 714, 729, 748, 896, 924, 1254, 1547, 1728,1729, 2079 (twice), 2261, 2926, 3003, 3289, 3542, 4096, 4914, 4928 (twice), 5005, 5103, 6630, 7293,7371, 7722, 8372, 9177, 9660, 10206, 10556, 11571, 11648, 12096, 13090….

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94 CHAPTER 28. G2 (MATHEMATICS)

The 14-dimensional representation is the adjoint representation, and the 7-dimensional one is action of G2 on theimaginary octonions.There are two non-isomorphic irreducible representations of dimensions 77, 2079, 4928, 28652, etc. The fundamentalrepresentations are those with dimensions 14 and 7 (corresponding to the two nodes in the Dynkin diagram in theorder such that the triple arrow points from the first to the second).Vogan (1994) described the (infinite-dimensional) unitary irreducible representations of the split real form of G2.

28.6 Finite groups

The group G2(q) is the points of the algebraic group G2 over the finite field Fq. These finite groups were firstintroduced by Leonard Eugene Dickson in Dickson (1901) for odd q and Dickson (1905) for even q. The order ofG2(q) is q6(q6 − 1)(q2 − 1). When q ≠ 2, the group is simple, and when q = 2, it has a simple subgroup of index 2isomorphic to 2A2(32), and is the automorphism group of a maximal order of the octonions. The Janko group J1 wasfirst constructed as a subgroup of G2(11). Ree (1960) introduced twisted Ree groups 2G2(q) of order q3(q3 + 1)(q− 1) for q = 32n+1, an odd power of 3.

28.7 See also• Cartan matrix

• Dynkin diagram

• Exceptional Jordan algebra

• Fundamental representation

• G2-structure

• Lie group

• Seven-dimensional cross product

• Simple Lie group

28.8 References• Adams, J. Frank (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University ofChicago Press, ISBN 978-0-226-00526-3, MR 1428422

• Agricola, Ilka (2008), Old and New on the Exceptional Group G2 (PDF) 55 (8)

• Baez, John (2002), “The Octonions”, Bull. Amer. Math. Soc. 39 (2): 145–205, doi:10.1090/S0273-0979-01-00934-X.

See section 4.1: G2; an online HTML version of which is available at http://math.ucr.edu/home/baez/octonions/node14.html.

• Bryant, Robert (1987), “Metrics with Exceptional Holonomy”, Annals of Mathematics, 2 126 (3): 525–576,doi:10.2307/1971360

• Dickson, Leonard Eugene (1901), “Theory of Linear Groups in AnArbitrary Field”, Transactions of the Ameri-canMathematical Society (Providence, R.I.: AmericanMathematical Society) 2 (4): 363–394, doi:10.1090/S0002-9947-1901-1500573-3, ISSN 0002-9947, JSTOR 1986251, Reprinted in volume II of his collected papersLeonard E. Dickson reported groups of type G2 in fields of odd characteristic.

• Dickson, L. E. (1905), “A new system of simple groups”,Math. Ann. 60: 137–150, doi:10.1007/BF01447497Leonard E. Dickson reported groups of type G2 in fields of even characteristic.

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Chapter 29

Geometric invariant theory

In mathematics Geometric invariant theory (or GIT) is a method for constructing quotients by group actions inalgebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas fromthe paper (Hilbert 1893) in classical invariant theory.Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and provides tech-niques for forming the 'quotient' of X by G as a scheme with reasonable properties. One motivation was to constructmoduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980sthe theory developed interactions with symplectic geometry and equivariant topology, and was used to constructmoduli spaces of objects in differential geometry, such as instantons and monopoles.

29.1 Background

Main article: Invariant theory

Invariant theory is concerned with a group action of a group G on an algebraic variety (or a scheme) X. Classicalinvariant theory addresses the situation when X = V is a vector space and G is either a finite group, or one of theclassical Lie groups that acts linearly on V. This action induces a linear action of G on the space of polynomialfunctions R(V) on V by the formula

g · f(v) = f(g−1v), g ∈ G, v ∈ V.

The polynomial invariants of the G-action on V are those polynomial functions f on V which are fixed under the'change of variables’ due to the action of the group, so that g·f = f for all g in G. They form a commutative algebraA = R(V)G, and this algebra is interpreted as the algebra of functions on the 'invariant theory quotient' V //G. In thelanguage of modern algebraic geometry,

V //G = SpecA = SpecR(V )G.

Several difficulties emerge from this description. The first one, successfully tackled by Hilbert in the case of a generallinear group, is to prove that the algebra A is finitely generated. This is necessary if one wanted the quotient to be anaffine algebraic variety. Whether a similar fact holds for arbitrary groups G was the subject of Hilbert’s fourteenthproblem, and Nagata demonstrated that the answer was negative in general. On the other hand, in the course ofdevelopment of representation theory in the first half of the twentieth century, a large class of groups for which theanswer is positive was identified; these are called reductive groups and include all finite groups and all classical groups.The finite generation of the algebra A is but the first step towards the complete description of A, and the progressin resolving this more delicate question was rather modest. The invariants had classically been described only in arestricted range of situations, and the complexity of this description beyond the first few cases held out little hope forfull understanding of the algebras of invariants in general. Furthermore, it may happen that all polynomial invariantsf take the same value on a given pair of points u and v in V, yet these points are in different orbits of the G-action. A

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29.2. MUMFORD’S BOOK 97

simple example is provided by themultiplicative groupC* of non-zero complex numbers that acts on an n-dimensionalcomplex vector space Cn by scalar multiplication. In this case, every polynomial invariant is a constant, but there aremany different orbits of the action. The zero vector forms an orbit by itself, and the non-zeromultiples of any non-zerovector form an orbit, so that non-zero orbits are paramatrized by the points of the complex projective spaceCPn−1. Ifthis happens, one says that “invariants do not separate the orbits”, and the algebra A reflects the topological quotientspace X /G rather imperfectly. Indeed, the latter space is frequently non-separated. In 1893 Hilbert formulated andproved a criterion for determining those orbits which are not separated from the zero orbit by invariant polynomials.Rather remarkably, unlike his earlier work in invariant theory, which led to the rapid development of abstract algebra,this result of Hilbert remained little known and little used for the next 70 years. Much of the development of invarianttheory in the first half of the twentieth century concerned explicit computations with invariants, and at any rate,followed the logic of algebra rather than geometry.

29.2 Mumford’s book

Geometric invariant theory was founded and developed by Mumford in a monograph, first published in 1965, thatapplied ideas of nineteenth century invariant theory, including some results of Hilbert, to modern algebraic geometryquestions. (The book was greatly expanded in two later editions, with extra appendices by Fogarty and Mumford,and a chapter on symplectic quotients by Kirwan.) The book uses both scheme theory and computational techniquesavailable in examples. The abstract setting used is that of a group action on a scheme X. The simple-minded idea ofan orbit space

G\X,

i.e. the quotient space of X by the group action, runs into difficulties in algebraic geometry, for reasons that areexplicable in abstract terms. There is in fact no general reason why equivalence relations should interact well with the(rather rigid) regular functions (polynomial functions), which are at the heart of algebraic geometry. The functionson the orbit space G\X that should be considered are those on X that are invariant under the action of G. The directapproach can be made, by means of the function field of a variety (i.e. rational functions): take the G-invariantrational functions on it, as the function field of the quotient variety. Unfortunately this — the point of view ofbirational geometry — can only give a first approximation to the answer. As Mumford put it in the Preface to thebook:

The problem is, within the set of all models of the resulting birational class, there is one model whosegeometric points classify the set of orbits in some action, or the set of algebraic objects in some moduliproblem.

In Chapter 5 he isolates further the specific technical problem addressed, in a moduli problem of quite classicaltype — classify the big 'set' of all algebraic varieties subject only to being non-singular (and a requisite condition onpolarization). The moduli are supposed to describe the parameter space. For example for algebraic curves it has beenknown from the time of Riemann that there should be connected components of dimensions

0, 1, 3, 6, 9, …

according to the genus g =0, 1, 2, 3, 4, …, and the moduli are functions on each component. In the coarse moduliproblem Mumford considers the obstructions to be:

• non-separated topology on the moduli space (i.e. not enough parameters in good standing)

• infinitely many irreducible components (which isn't avoidable, but local finiteness may hold)

• failure of components to be representable as schemes, although respectable topologically.

It is the third point that motivated the whole theory. As Mumford puts it, if the first two difficulties are resolved

[the third question] becomes essentially equivalent to the question of whether an orbit space of some locallyclosed subset of the Hilbert or Chow schemes by the projective group exists.

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98 CHAPTER 29. GEOMETRIC INVARIANT THEORY

To deal with this he introduced a notion (in fact three) of stability. This enabled him to open up the previouslytreacherous area — much had been written, in particular by Francesco Severi, but the methods of the literaturehad limitations. The birational point of view can afford to be careless about subsets of codimension 1. To havea moduli space as a scheme is on one side a question about characterising schemes as representable functors (asthe Grothendieck school would see it); but geometrically it is more like a compactification question, as the stabilitycriteria revealed. The restriction to non-singular varieties will not lead to a compact space in any sense as modulispace: varieties can degenerate to having singularities. On the other hand the points that would correspond to highlysingular varieties are definitely too 'bad' to include in the answer. The correct middle ground, of points stable enoughto be admitted, was isolated by Mumford’s work. The concept was not entirely new, since certain aspects of it wereto be found in David Hilbert's final ideas on invariant theory, before he moved on to other fields.The book’s Preface also enunciated the Mumford conjecture, later proved by William Haboush.

29.3 Stability

“Stable point” redirects here. It is not to be confused with Stable fixed point.

If a reductive group G acts linearly on a vector space V, then a non-zero point of V is called

• unstable if 0 is in the closure of its orbit,

• semi-stable if 0 is not in the closure of its orbit,

• stable if its orbit is closed, and its stabilizer is finite.

There are equivalent ways to state these (this criterion is known as the Hilbert–Mumford criterion):

• A non-zero point x is unstable if and only if there is a 1-parameter subgroup of G all of whose weights withrespect to x are positive.

• A non-zero point x is unstable if and only if every invariant polynomial has the same value on 0 and x.

• A non-zero point x is semistable if and only if there is no 1-parameter subgroup of G all of whose weights withrespect to x are positive.

• A non-zero point x is semistable if and only if some invariant polynomial has different values on 0 and x.

• A non-zero point x is stable if and only if every 1-parameter subgroup of G has positive (and negative) weightswith respect to x.

• A non-zero point x is stable if and only if for every y not in the orbit of x there is some invariant polynomial thathas different values on y and x, and the ring of invariant polynomials has transcendence degree dim(V)−dim(G).

A point of the corresponding projective space of V is called unstable, semi-stable, or stable if it is the image of apoint in V with the same property. “Unstable” is the opposite of “semistable” (not “stable”). The unstable pointsform a Zariski closed set of projective space, while the semistable and stable points both form Zariski open sets(possibly empty). These definitions are from (Mumford 1977) and are not equivalent to the ones in the first editionof Mumford’s book.Many moduli spaces can be constructed as the quotients of the space of stable points of some subset of projectivespace by some group action. These spaces can often compactified by adding certain equivalence classes of semistablepoints. Different stable orbits correspond to different points in the quotient, but two different semistable orbits maycorrespond to the same point in the quotient if their closures intersect.Example: (Deligne & Mumford 1969) A stable curve is a reduced connected curve of genus ≥2 such that its onlysingularities are ordinary double points and every non-singular rational component meets the other components inat least 3 points. The moduli space of stable curves of genus g is the quotient of a subset of the Hilbert scheme ofcurves in P5g−6 with Hilbert polynomial (6n−1)(g−1) by the group PGL₅g₋₅.Example: A vector bundle W over an algebraic curve (or over a Riemann surface) is a stable vector bundle if andonly if

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29.4. SEE ALSO 99

deg(V )

rank(V )<

deg(W )

rank(W )

for all proper non-zero subbundles V ofW and is semistable if this condition holds with < replaced by ≤.

29.4 See also• Geometric complexity theory

• Geometric quotient

• Categorical quotient

• Quantization commutes with reduction

29.5 References• Deligne, Pierre; Mumford, David (1969), “The irreducibility of the space of curves of given genus”, PublicationsMathématiques de l'IHÉS 36 (36): 75–109, doi:10.1007/BF02684599, MR 0262240

• Hilbert, D. (1893), "Über die vollen Invariantensysteme”,Math. Annalen 42 (3): 313, doi:10.1007/BF01444162

• Kirwan, Frances, Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes, 31.Princeton University Press, Princeton, NJ, 1984. i+211 pp. MR 0766741 ISBN 0-691-08370-3

• Kraft, Hanspeter, Geometrische Methoden in der Invariantentheorie. (German) (Geometrical methods in in-variant theory) Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984. x+308 pp. MR0768181 ISBN 3-528-08525-8

• Mumford, David (1977), “Stability of projective varieties”, L'Enseignement Mathématique. Revue Interna-tionale. IIe Série 23 (1): 39–110, ISSN 0013-8584, MR 0450272

• Mumford, David; Fogarty, J.; Kirwan, F. (1994), Geometric invariant theory, Ergebnisse der Mathematikund ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)] 34 (3rd ed.), Berlin, New York:Springer-Verlag, ISBN978-3-540-56963-3,MR0214602( 1st ed 1965) [[Mathematical Reviews|MR]]&nbsp;[http://www.ams.org/mathscinet-getitem?mr=0719371 0719371] (2nd ed) [[Mathematical Reviews|MR]]&nbsp;[http://www.ams.org/mathscinet-getitem?mr=1304906 1304906](3rd ed.)

• E.B. Vinberg, V.L. Popov, Invariant theory, in Algebraic geometry. IV. Encyclopaedia of Mathematical Sci-ences, 55 (translated from 1989 Russian edition) Springer-Verlag, Berlin, 1994. vi+284 pp. ISBN 3-540-54682-0

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Chapter 30

Glossary of algebraic groups

There are a number of mathematical notions to study and classify algebraic groups.In the sequel, G denotes an algebraic group over a field k.

30.1 References[1] These two are the only connected one-dimensional linear groups, Springer 1998, Theorem 3.4.9

• Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics 126 (2nd ed.), Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012

• Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics 9 (2nd ed.), Boston, MA:Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713

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Chapter 31

Good filtration

In mathematical representation theory, a good filtration is a filtration of a representation of a reductive algebraicgroup G such that the subquotients are isomorphic to the spaces of sections F(λ) of line bundles λ over G/B for aBorel subgroup B. In characteristic 0 this is automatically true as the irreducible modules are all of the form F(λ),but this is not usually true in positive characteristic. Mathieu (1990) showed that the tensor product of two modulesF(λ)⊗F(μ) has a good filtration, completing the results of Donkin (1985) who proved it in most cases and Wang(1982) who proved it in large characteristic. Littelmann (1992) showed that the existence of good filtrations for thesetensor products also follows from standard monomial theory.

31.1 References• Donkin, Stephen (1985), Rational representations of algebraic groups, Lecture Notes in Mathematics 1140,Berlin, New York: Springer-Verlag, doi:10.1007/BFb0074637, ISBN 978-3-540-15668-0, MR 804233

• Littelmann, Peter (1992), “Good filtrations and decomposition rules for representations with standard mono-mial theory”, Journal für die reine und angewandte Mathematik 433: 161–180, doi:10.1515/crll.1992.433.161,ISSN 0075-4102, MR 1191604

• Mathieu, Olivier (1990), “Filtrations of G-modules”, Annales Scientifiques de l'École Normale Supérieure. Qua-trième Série 23 (4): 625–644, ISSN 0012-9593, MR 1072820

• Wang, Jian Pan (1982), “Sheaf cohomology on G/B and tensor products of Weyl modules”, Journal of Algebra77 (1): 162–185, doi:10.1016/0021-8693(82)90284-8, ISSN 0021-8693, MR 665171

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Chapter 32

Grosshans subgroup

Inmathematics, in the representation theory of algebraic groups, aGrosshans subgroup, named after FrankGrosshans,is an algebraic subgroup of an algebraic group that is an observable subgroup for which the ring of functions on thequotient variety is finitely generated.[1]

32.1 References[1] Dolgachev, Igor (2003), Lectures on Invariant Theory, London Mathematical Society Lecture Note Series 296, Cambridge

University Press, p. 50, ISBN 9780521525480.

32.2 External links• Invariants of Unipotent subgroups

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Chapter 33

Group of Lie type

In mathematics, a group of Lie type is a group closely related to the group G(k) of rational points of a reductivelinear algebraic group G with values in the field k. Finite groups of Lie type give the bulk of nonabelian finite simplegroups. Special cases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki–Ree groups.Dieudonné (1971) and Carter (1989) are standard references for groups of Lie type.

33.1 Classical groups

Main article: Classical group

An initial approach to this question was the definition and detailed study of the so-called classical groups over finiteand other fields by Jordan (1870). These groups were studied by L. E. Dickson and Jean Dieudonné. Emil Artininvestigated the orders of such groups, with a view to classifying cases of coincidence.A classical group is, roughly speaking, a special linear, orthogonal, symplectic, or unitary group. There are severalminor variations of these, given by taking derived subgroups or central quotients, the latter yielding projective lineargroups. They can be constructed over finite fields (or any other field) in much the same way that they are constructedover the real numbers. They correspond to the series An, Bn, Cn, Dn,2An, 2Dn of Chevalley and Steinberg groups.

33.2 Chevalley groups

The theory was clarified by the theory of algebraic groups, and the work of Chevalley (1955) on Lie algebras, bymeans of which the Chevalley group concept was isolated. Chevalley constructed a Chevalley basis (a sort of integralform) for all the complex simple Lie algebras (or rather of their universal enveloping algebras), which can be used todefine the corresponding algebraic groups over the integers. In particular, he could take their points with values in anyfinite field. For the Lie algebras An, Bn, Cn, Dn this gave well known classical groups, but his construction also gavegroups associated to the exceptional Lie algebras E6, E7, E8, F4, and G2. The ones of type G2 (sometimes calledDickson groups) had already been constructed by Dickson (1905), and the ones of type E6 by Dickson (1901).

33.3 Steinberg groups

Main article: ³D₄Main article: ²E₆

Chevalley’s construction did not give all of the known classical groups: it omitted the unitary groups and the non-splitorthogonal groups. Steinberg (1959) found a modification of Chevalley’s construction that gave these groups and two

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104 CHAPTER 33. GROUP OF LIE TYPE

new families 3D4, 2E6, the second of which was discovered at about the same time from a different point of view byTits (1958). This construction generalizes the usual construction of the unitary group from the general linear group.The unitary group arises as follows: the general linear group over the complex numbers has a diagram automorphismgiven by reversing the Dynkin diagram An (which corresponds to taking the transpose inverse), and a field automor-phism given by taking complex conjugation, which commute. The unitary group is the group of fixed points of theproduct of these two automorphisms.In the same way, many Chevalley groups have diagram automorphisms induced by automorphisms of their Dynkindiagrams, and field automorphisms induced by automorphisms of a finite field. Analogously to the unitary case,Steinberg constructed families of groups by taking fixed points of a product of a diagram and a field automorphism.These gave:

• the unitary groups 2An, from the order 2 automorphism of An;• further orthogonal groups 2Dn, from the order 2 automorphism of Dn;• the new series 2E6, from the order 2 automorphism of E6;• the new series 3D4, from the order 3 automorphism of D4.

The groups of type 3D4 have no analogue over the reals, as the complex numbers have no automorphism of order 3.The symmetries of the D4 diagram also give rise to triality.

33.4 Suzuki–Ree groups

Main articles: Suzuki groups and Ree group

Suzuki (1960) found a new infinite series of groups that at first sight seemed unrelated to the known algebraic groups.Ree (1960, 1961) knew that the algebraic group B2 had an “extra” automorphism in characteristic 2 whose squarewas the Frobenius automorphism. He found that if a finite field of characteristic 2 also has an automorphism whosesquare was the Frobenius map, then an analogue of Steinberg’s construction gave the Suzuki groups. The fields withsuch an automorphism are those of order 22n+1, and the corresponding groups are the Suzuki groups

2B2(22n+1) = Suz(22n+1).

(Strictly speaking, the group Suz(2) is not counted as a Suzuki group as it is not simple: it is the Frobenius group oforder 20.) Ree was able to find two new similar families

2F4(22n+1)

and2G2(32n+1)

of simple groups by using the fact that F4 and G2 have extra automorphisms in characteristic 2 and 3. (Roughlyspeaking, in characteristic p one is allowed to ignore the arrow on bonds of multiplicity p in the Dynkin diagramwhen taking diagram automorphisms.) The smallest group 2F4(2) of type 2F4 is not simple, but it has a simplesubgroup of index 2, called the Tits group (named after the mathematician Jacques Tits). The smallest group 2G2(3)of type 2G2 is not simple, but it has a simple normal subgroup of index 3, isomorphic to A1(8). In the classificationof finite simple groups, the Ree groups

2G2(32n+1)

are the ones whose structure is hardest to pin down explicitly. These groups also played a role in the discovery ofthe first modern sporadic group. They have involution centralizers of the form Z/2Z × PSL(2, q) for q = 3n, and byinvestigating groups with an involution centralizer of the similar form Z/2Z × PSL(2, 5) Janko found the sporadicgroup J1.The Suzuki groups are the only finite non-abelian simple groups with order not divisible by 3. They have order22(2n+1)(22(2n+1) + 1)(2(2n+1) −1).

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33.5. RELATIONS WITH FINITE SIMPLE GROUPS 105

33.5 Relations with finite simple groups

Finite groups of Lie type were among the first groups to be considered in mathematics, after cyclic, symmetric andalternating groups, with the projective special linear groups over prime finite fields, PSL(2, p) being constructed byÉvariste Galois in the 1830s. The systematic exploration of finite groups of Lie type started with Camille Jordan'stheorem that the projective special linear group PSL(2, q) is simple for q ≠ 2, 3. This theorem generalizes to projectivegroups of higher dimensions and gives an important infinite family PSL(n, q) of finite simple groups. Other classicalgroups were studied by Leonard Dickson in the beginning of 20th century. In the 1950s Claude Chevalley realizedthat after an appropriate reformulation, many theorems about semisimple Lie groups admit analogues for algebraicgroups over an arbitrary field k, leading to construction of what are now called Chevalley groups. Moreover, as inthe case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups(Tits simplicity theorem). Although it was known since 19th century that other finite simple groups exist (for example,Mathieu groups), gradually a belief formed that nearly all finite simple groups can be accounted for by appropriateextensions of Chevalley’s construction, together with cyclic and alternating groups. Moreover, the exceptions, thesporadic groups, share many properties with the finite groups of Lie type, and in particular, can be constructed andcharacterized based on their geometry in the sense of Tits.The belief has now become a theorem – the classification of finite simple groups. Inspection of the list of finite simplegroups shows that groups of Lie type over a finite field include all the finite simple groups other than the cyclic groups,the alternating groups, the Tits group, and the 26 sporadic simple groups.

33.6 Small groups of Lie type

In general the finite group associated to an endomorphism of a simply connected simple algebraic group is the universalcentral extension of a simple group, so is perfect and has trivial Schur multiplier. However some of the smallest groupsin the families above are either not perfect or have a Schur multiplier larger than “expected”.Cases where the group is not perfect include

• A1(2) = SL(2, 2) Solvable of order 6 (the symmetric group on 3 points)

• A1(3) = SL(2, 3) Solvable of order 24 (a double cover of the alternating group on 4 points)

• 2A2(4) Solvable

• B2(2) Not perfect, but is isomorphic to the symmetric group on 6 points so its derived subgroup has index 2and is simple of order 360.

• 2B2(2) = Suz(2) Solvable of order 20 (a Frobenius group)

• 2F4(2) Not perfect, but the derived group has index 2 and is the simple Tits group.

• G2(2) Not perfect, but the derived group has index 2 and is simple of order 6048.

• 2G2(3) Not perfect, but the derived group has index 3 and is the simple group of order 504.

Some cases where the group is perfect but has a Schur multiplier that is larger than expected include:

• A1(4) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 insteadof 1.

• A1(9) The Schur multiplier has an extra Z/3Z, so the Schur multiplier of the simple group has order 6 insteadof 2.

• A2(2) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 insteadof 1.

• A2(4) The Schur multiplier has an extra Z/4Z × Z/4Z, so the Schur multiplier of the simple group has order48 instead of 3.

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106 CHAPTER 33. GROUP OF LIE TYPE

• A3(2) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 insteadof 1.

• B3(2) = C3(2) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order2 instead of 1.

• B3(3) The Schur multiplier has an extra Z/3Z, so the Schur multiplier of the simple group has order 6 insteadof 2.

• D4(2) The Schur multiplier has an extra Z/2Z × Z/2Z, so the Schur multiplier of the simple group has order 4instead of 1.

• F4(2) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 insteadof 1.

• G2(3) The Schur multiplier has an extra Z/3Z, so the Schur multiplier of the simple group has order 3 insteadof 1.

• G2(4) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 insteadof 1.

• 2A3(4) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 insteadof 1.

• 2A3(9) The Schur multiplier has an extra Z/3Z × Z/3Z, so the Schur multiplier of the simple group has order36 instead of 4.

• 2A5(4) The Schur multiplier has an extra Z/2Z × Z/2Z, so the Schur multiplier of the simple group has order12 instead of 3.

• 2E6(4) The Schur multiplier has an extra Z/2Z × Z/2Z, so the Schur multiplier of the simple group has order12 instead of 3.

• 2B2(8) The Schur multiplier has an extra Z/2Z × Z/2Z, so the Schur multiplier of the simple group has order4 instead of 1.

There is a bewildering number of “accidental” isomorphisms between various small groups of Lie type (and alternatinggroups). For example, the groups SL(2, 4), PSL(2, 5), and the alternating group on 5 points are all isomorphic.For a complete list of these exceptions see the list of finite simple groups. Many of these special properties are relatedto certain sporadic simple groups.Alternating groups sometimes behave as if they were groups of Lie type over the field with one element. Some of thesmall alternating groups also have exceptional properties. The alternating groups usually have an outer automorphismgroup of order 2, but the alternating group on 6 points has an outer automorphism group of order 4. Alternatinggroups usually have a Schur multiplier of order 2, but the ones on 6 or 7 points have a Schur multiplier of order 6.

33.7 Notation issues

Unfortunately there is no standard notation for the finite groups of Lie type, and the literature contains dozens ofincompatible and confusing systems of notation for them.

• The simple group PSL(n, q) is not usually the same as the group PSL(n, Fq) of Fq-valued points of thealgebraic group PSL(n). The problem is that a surjective map of algebraic groups such as SL(n) → PSL(n)does not necessarily induce a surjective map of the corresponding groups with values in some (non algebraicallyclosed) field. There are similar problems with the points of other algebraic groups with values in finite fields.

• The groups of type An₋₁ are sometimes denoted by PSL(n, q) (the projective special linear group) or by L(n,q).

• The groups of type Cn are sometimes denoted by Sp(2n, q) (the symplectic group) or (confusingly) by Sp(n,q).

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33.8. SEE ALSO 107

• The notation for groups of type Dn (“orthogonal” groups) is particularly confusing. Some symbols used areO(n, q), O−(n, q), PSO(n, q), Ωn(q), but there are so many conventions that it is not possible to say exactly whatgroups these correspond to without it being specified explicitly. The source of the problem is that the simplegroup is not the orthogonal group O, nor the projective special orthogonal group PSO, but rather a subgroupof PSO,[1] which accordingly does not have a classical notation. A particularly nasty trap is that some authors,such as the ATLAS, use O(n, q) for a group that is not the orthogonal group, but the corresponding simplegroup. The notation Ω, PΩ was introduced by Jean Dieudonné, though his definition is not simple for n ≤ 4and thus the same notation may be used for a slightly different group, which agrees in n ≥ 5 but not in lowerdimension.[1]

• For the Steinberg groups, some authors write 2An(q2) (and so on) for the group that other authors denote by2An(q). The problem is that there are two fields involved, one of order q2, and its fixed field of order q, andpeople have different ideas on which should be included in the notation. The "2An(q2)" convention is morelogical and consistent, but the "2An(q)" convention is far more common and is closer to the convention foralgebraic groups.

• Authors differ on whether groups such as An(q) are the groups of points with values in the simple or the simplyconnected algebraic group. For example, An(q) may mean either the special linear group SL(n+1, q) or theprojective special linear group PSL(n+1, q). So 2A2(4) may be any one of 4 different groups, depending onthe author.

33.8 See also

• Deligne–Lusztig theory

• Modular Lie algebra

33.9 Notes

[1] ATLAS, p. xi

33.10 References

• Carter, Roger W. (1989) [1972], Simple groups of Lie type, Wiley Classics Library, New York: John Wiley &Sons, ISBN 978-0-471-50683-6, MR 0407163

• Chevalley, Claude (1955), “Sur certains groupes simples”, The Tohoku Mathematical Journal. Second Series7: 14–66, doi:10.2748/tmj/1178245104, ISSN 0040-8735, MR 0073602

• Dickson, Leonard Eugene (1901b), “Theory of Linear Groups in An Arbitrary Field”, Transactions of theAmericanMathematical Society (Providence, R.I.: AmericanMathematical Society) 2 (4): 363–394, doi:10.1090/S0002-9947-1901-1500573-3, ISSN 0002-9947, JSTOR 1986251, Reprinted in volume II of his collected papers

• Dickson, Leonard Eugene (1901), “A class of groups in an arbitrary realm connected with the configuration ofthe 27 lines on a cubic surface”, The quarterly journal of pure and applied mathematics 33: 145–173, Reprintedin volume 5 of his collected works

• Dickson, L. E. (1905), “A new system of simple groups”,Math. Ann. 60: 137–150, doi:10.1007/BF01447497Leonard E. Dickson reported groups of type G2

• Dieudonné, Jean A. (1971) [1955], La géométrie des groupes classiques (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-05391-2, MR 0310083

• Jordan, Camille (1870), Traité des substitutions et des équations algébriques, Paris: Gauthier-Villars

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108 CHAPTER 33. GROUP OF LIE TYPE

• Ree, Rimhak (1960), “A family of simple groups associated with the simple Lie algebra of type (G2)", Bulletinof the American Mathematical Society 66: 508–510, doi:10.1090/S0002-9904-1960-10523-X, ISSN 0002-9904, MR 0125155

• Ree, Rimhak (1961), “A family of simple groups associated with the simple Lie algebra of type (F4)", Bulletinof the American Mathematical Society 67: 115–116, doi:10.1090/S0002-9904-1961-10527-2, ISSN 0002-9904, MR 0125155

• Steinberg, Robert (1959), “Variations on a theme of Chevalley”, Pacific Journal of Mathematics 9: 875–891,doi:10.2140/pjm.1959.9.875, ISSN 0030-8730, MR 0109191

• Steinberg, Robert (1968), Lectures on Chevalley groups, Yale University, New Haven, Conn., MR 0466335

• Suzuki, Michio (1960), “A new type of simple groups of finite order”, Proceedings of the National Academy ofSciences of the United States of America 46: 868–870, doi:10.1073/pnas.46.6.868, ISSN 0027-8424, JSTOR70960, MR 0120283

• Tits, Jacques (1958), Les “formes réelles” des groupes de type E6, Séminaire Bourbaki; 10e année: 1957/1958.Textes des conférences; Exposés 152 à 168; 2e èd. corrigée, Exposé 162 15, Paris: Secrétariat math'ematique,MR 0106247

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Chapter 34

Group scheme

In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Groupschemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraicgroups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over afield. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand andanswer questions of arithmetic significance. The category of group schemes is somewhat better behaved than thatof group varieties, since all homomorphisms have kernels, and there is a well-behaved deformation theory. Groupschemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since theycome up in contexts of Galois representations and moduli problems. The initial development of the theory of groupschemes was due to Alexander Grothendieck, Michel Raynaud and Michel Demazure in the early 1960s.

34.1 Definition

A group scheme is a group object in a category of schemes that has fiber products and some final object S. That is, itis an S-scheme G equipped with one of the equivalent sets of data

• a triple of morphisms μ: G ×S G → G, e: S → G, and ι: G → G, satisfying the usual compatibilities of groups(namely associativity of μ, identity, and inverse axioms)

• a functor from schemes over S to the category of groups, such that composition with the forgetful functor tosets is equivalent to the presheaf corresponding to G under the Yoneda embedding.

A homomorphism of group schemes is a map of schemes that respects multiplication. This can be precisely phrasedeither by saying that a map f satisfies the equation fμ = μ(f × f), or by saying that f is a natural transformation offunctors from schemes to groups (rather than just sets).A left action of a group schemeG on a schemeX is a morphismG ×SX→X that induces a left action of the groupG(T)on the set X(T) for any S-scheme T. Right actions are defined similarly. Any group scheme admits natural left andright actions on its underlying scheme by multiplication and conjugation. Conjugation is an action by automorphisms,i.e., it commutes with the group structure, and this induces linear actions on naturally derived objects, such as its Liealgebra, and the algebra of left-invariant differential operators.An S-group scheme G is commutative if the group G(T) is an abelian group for all S-schemes T. There are severalother equivalent conditions, such as conjugation inducing a trivial action, or inversion map ι being a group schemeautomorphism.

34.2 Constructions• Given a group G, one can form the constant group scheme GS. As a scheme, it is a disjoint union of copies ofS, and by choosing an identification of these copies with elements of G, one can define the multiplication, unit,and inverse maps by transport of structure. As a functor, it takes any S-scheme T to a product of copies of thegroup G, where the number of copies is equal to the number of connected components of T. GS is affine over

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S if and only if G is a finite group. However, one can take a projective limit of finite constant group schemesto get profinite group schemes, which appear in the study of fundamental groups and Galois representations orin the theory of the fundamental group scheme, and these are affine of infinite type. More generally, by takinga locally constant sheaf of groups on S, one obtains a locally constant group scheme, for which monodromy onthe base can induce non-trivial automorphisms on the fibers.

• The existence of fiber products of schemes allows one to make several constructions. Finite direct products ofgroup schemes have a canonical group scheme structure. Given an action of one group scheme on another byautomorphisms, one can form semidirect products by following the usual set-theoretic construction. Kernelsof group scheme homomorphisms are group schemes, by taking a fiber product over the unit map from thebase. Base change sends group schemes to group schemes.

• Group schemes can be formed from smaller group schemes by taking restriction of scalars with respect to somemorphism of base schemes, although one needs finiteness conditions to be satisfied to ensure representabilityof the resulting functor. When this morphism is along a finite extension of fields, it is known asWeil restriction.

• For any abelian group A, one can form the corresponding diagonalizable group D(A), defined as a functor bysetting D(A)(T) to be the set of abelian group homomorphisms from A to invertible global sections of OT foreach S-scheme T. If S is affine, D(A) can be formed as the spectrum of a group ring. More generally, one canform groups of multiplicative type by letting A be a non-constant sheaf of abelian groups on S.

• For a subgroup schemeH of a group schemeG, the functor that takes an S-scheme T toG(T)/H(T) is in generalnot a sheaf, and even its sheafification is in general not representable as a scheme. However, if H is finite, flat,and closed in G, then the quotient is representable, and admits a canonical left G-action by translation. If therestriction of this action to H is trivial, then H is said to be normal, and the quotient scheme admits a naturalgroup law. Representability holds in many other cases, such as when H is closed in G and both are affine.[1]

34.3 Examples• The multiplicative groupG has the punctured affine line as its underlying scheme, and as a functor, it sends anS-scheme T to the multiplicative group of invertible global sections of the structure sheaf. It can be describedas the diagonalizable group D(Z) associated to the integers. Over an affine base such as Spec A, it is thespectrum of the ring A[x,y]/(xy − 1), which is also written A[x, x−1]. The unit map is given by sending x to one,multiplication is given by sending x to x ⊗ x, and the inverse is given by sending x to x−1. Algebraic tori form animportant class of commutative group schemes, defined either by the property of being locally on S a productof copies of G , or as groups of multiplicative type associated to finitely generated free abelian groups.

• The general linear group GLn is an affine algebraic variety that can be viewed as the multiplicative group of then by nmatrix ring variety. As a functor, it sends an S-scheme T to the group of invertible n by nmatrices whoseentries are global sections of T. Over an affine base, one can construct it as a quotient of a polynomial ring inn2 + 1 variables by an ideal encoding the invertibility of the determinant. Alternatively, it can be constructedusing 2n2 variables, with relations describing an ordered pair of mutually inverse matrices.

• For any positive integer n, the group μ is the kernel of the nth power map from G to itself. As a functor,it sends any S-scheme T to the group of global sections f of T such that fn = 1. Over an affine base such asSpec A, it is the spectrum of A[x]/(xn−1). If n is not invertible in the base, then this scheme is not smooth. Inparticular, over a field of characteristic p, μ is not smooth.

• The additive group Gₐ has the affine line A1 as its underlying scheme. As a functor, it sends any S-scheme Tto the underlying additive group of global sections of the structure sheaf. Over an affine base such as Spec A,it is the spectrum of the polynomial ring A[x]. The unit map is given by sending x to zero, the multiplication isgiven by sending x to 1 ⊗ x + x ⊗ 1, and the inverse is given by sending x to −x.

• If p = 0 in S for some prime number p, then the taking of pth powers induces an endomorphism ofGₐ, and thekernel is the group scheme α . As a scheme, it is isomorphic to μ , but the group structures are different. Overan affine base such as Spec A, it is the spectrum of A[x]/(xp).

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34.4. BASIC PROPERTIES 111

• The automorphism group of the affine line is isomorphic to the semidirect product of Gₐ by G , where theadditive group acts by translations, and the multiplicative group acts by dilations. The subgroup fixing a chosenbasepoint is isomorphic to the multiplicative group, and taking the basepoint to be the identity of an additivegroup structure identifies G with the automorphism group of Gₐ.

• A smooth genus one curve with a marked point (i.e., an elliptic curve) has a unique group scheme structurewith that point as the identity. Unlike the previous positive-dimensional examples, elliptic curves are projective(in particular proper).

34.4 Basic properties

Suppose that G is a group scheme of finite type over a field k. Let G0 be the connected component of the identity,i.e., the maximal connected subgroup scheme. Then G is an extension of a finite étale group scheme by G0. G has aunique maximal reduced subscheme Gᵣₑ , and if k is perfect, then Gᵣₑ is a smooth group variety that is a subgroupscheme of G. The quotient scheme is the spectrum of a local ring of finite rank.Any affine group scheme is the spectrum of a commutative Hopf algebra (over a base S, this is given by the relativespectrum of an OS-algebra). The multiplication, unit, and inverse maps of the group scheme are given by the comul-tiplication, counit, and antipode structures in the Hopf algebra. The unit and multiplication structures in the Hopfalgebra are intrinsic to the underlying scheme. For an arbitrary group scheme G, the ring of global sections also hasa commutative Hopf algebra structure, and by taking its spectrum, one obtains the maximal affine quotient group.Affine group varieties are known as linear algebraic groups, since they can be embedded as subgroups of generallinear groups.Complete connected group schemes are in some sense opposite to affine group schemes, since the completenessimplies all global sections are exactly those pulled back from the base, and in particular, they have no nontrivial mapsto affine schemes. Any complete group variety (variety here meaning reduced and geometrically irreducible separatedscheme of finite type over a field) is automatically commutative, by an argument involving the action of conjugationon jet spaces of the identity. Complete group varieties are called abelian varieties. This generalizes to the notionof abelian scheme; a group scheme G over a base S is abelian if the structural morphism from G to S is proper andsmooth with geometrically connected fibers They are automatically projective, and they have many applications, e.g.,in geometric class field theory and throughout algebraic geometry. A complete group scheme over a field need notbe commutative, however; for example, any finite group scheme is complete.

34.5 Finite flat group schemes

A group scheme G over a noetherian scheme S is finite and flat if and only if OG is a locally free OS-module of finiterank. The rank is a locally constant function on S, and is called the order of G. The order of a constant group schemeis equal to the order of the corresponding group, and in general, order behaves well with respect to base change andfinite flat restriction of scalars.Among the finite flat group schemes, the constants (cf. example above) form a special class, and over an algebraicallyclosed field of characteristic zero, the category of finite groups is equivalent to the category of constant finite groupschemes. Over bases with positive characteristic or more arithmetic structure, additional isomorphism types exist.For example, if 2 is invertible over the base, all group schemes of order 2 are constant, but over the 2-adic integers,μ2 is non-constant, because the special fiber isn't smooth. There exist sequences of highly ramified 2-adic rings overwhich the number of isomorphism types of group schemes of order 2 grows arbitrarily large. More detailed analysisof commutative finite flat group schemes over p-adic rings can be found in Raynaud’s work on prolongations.Commutative finite flat group schemes often occur in nature as subgroup schemes of abelian and semi-abelian vari-eties, and in positive or mixed characteristic, they can capture a lot of information about the ambient variety. Forexample, the p-torsion of an elliptic curve in characteristic zero is locally isomorphic to the constant elementaryabelian group scheme of order p2, but over F , it is a finite flat group scheme of order p2 that has either p connectedcomponents (if the curve is ordinary) or one connected component (if the curve is supersingular). If we consider afamily of elliptic curves, the p-torsion forms a finite flat group scheme over the parametrizing space, and the supersin-gular locus is where the fibers are connected. This merging of connected components can be studied in fine detail bypassing from a modular scheme to a rigid analytic space, where supersingular points are replaced by discs of positiveradius.

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34.6 Cartier duality

Cartier duality is a scheme-theoretic analogue of Pontryagin duality. Given any finite flat commutative group schemeG over S, its Cartier dual is the group of characters, defined as the functor that takes any S-scheme T to the abeliangroup of group scheme homomorphisms from the base changeGT toG ,T and anymap of S-schemes to the canonicalmap of character groups. This functor is representable by a finite flat S-group scheme, and Cartier duality forms anadditive involutive antiequivalence from the category of finite flat commutative S-group schemes to itself. If G is aconstant commutative group scheme, then its Cartier dual is the diagonalizable group D(G), and vice versa. If S isaffine, then the duality functor is given by the duality of the Hopf algebras of functions.The definition of Cartier dual extends usefully to much more general situations where the resulting functor on schemesis no longer represented as a group scheme. Common cases include fppf sheaves of commutative groups over S, andcomplexes thereof. These more general geometric objects can be useful when one wants to work with categories thathave good limit behavior. There are cases of intermediate abstraction, such as commutative algebraic groups over afield, where Cartier duality gives an antiequivalence with commutative affine formal groups, so if G is the additivegroupGₐ, then its Cartier dual is the multiplicative formal group Gm , and if G is a torus, then its Cartier dual is étaleand torsion-free. For loop groups of tori, Cartier duality defines the tame symbol in local geometric class field theory.Laumon introduced a sheaf-theoretic Fourier transform for quasi-coherent modules over 1-motives that specializesto many of these equivalences.Example: The Cartier dual of the cyclic group Z/n of order n is the n-th roots of unity µn .

34.7 Dieudonné modules

Main article: Dieudonné module

Finite flat commutative group schemes over a perfect field k of positive characteristic p can be studied by trans-ferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the Dieudonné ring D =W(k)F,V/(FV − p), which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vec-tors of k. F and V are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors.Dieudonne and Cartier constructed an antiequivalence of categories between finite commutative group schemes overk of order a power of “p” and modules over D with finite W(k)-length. The Dieudonné module functor in one di-rection is given by homomorphisms into the abelian sheaf CW of Witt co-vectors. This sheaf is more or less dual tothe sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed by taking a directlimit of finite length Witt vectors under successive Verschiebung maps V:W →W ₊₁, and then completing. Manyproperties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g.,connected p-group schemes correspond to D-modules for which F is nilpotent, and étale group schemes correspondto modules for which F is an isomorphism.Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Oda’s 1967 thesisgave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at aboutthe same time, Grothendieck suggested that there should be a crystalline version of the theory that could be usedto analyze p-divisible groups. Galois actions on the group schemes transfer through the equivalences of categories,and the associated deformation theory of Galois representations was used in Wiles's work on the Shimura–Taniyamaconjecture.

34.8 References[1] Raynaud, Michel (1967), Passage au quotient par une relation d'équivalence plate, Berlin, New York: Springer-Verlag, MR

0232781

• Demazure, Michel; Alexandre Grothendieck, eds. (1970). Séminaire de Géométrie Algébrique du Bois Marie– 1962–64 – Schémas en groupes – (SGA 3) – vol. 1 (Lecture notes in mathematics 151) (in French). Berlin;New York: Springer-Verlag. pp. xv+564.

• Demazure, Michel; Alexandre Grothendieck, eds. (1970). Séminaire de Géométrie Algébrique du Bois Marie– 1962–64 – Schémas en groupes – (SGA 3) – vol. 2 (Lecture notes in mathematics 152) (in French). Berlin;

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34.8. REFERENCES 113

New York: Springer-Verlag. pp. ix+654.

• Demazure, Michel; Alexandre Grothendieck, eds. (1970). Séminaire de Géométrie Algébrique du Bois Marie– 1962–64 – Schémas en groupes – (SGA 3) – vol. 3 (Lecture notes in mathematics 153) (in French). Berlin;New York: Springer-Verlag. vii+529.

• Gabriel, Peter; Demazure, Michel (1980). Introduction to algebraic geometry and algebraic groups. Amster-dam: North-Holland Pub. Co. ISBN 0-444-85443-6.

• Berthelot, Breen, Messing Théorie de Dieudonné Crystalline II

• Laumon, Transformation de Fourier généralisée

• Shatz, Stephen S. (1986), “Group schemes, formal groups, and p-divisible groups”, in Cornell, Gary; Silverman,Joseph H., Arithmetic geometry (Storrs, Conn., 1984), Berlin, New York: Springer-Verlag, pp. 29–78, ISBN978-0-387-96311-2, MR 861972

• Serre, Jean-Pierre (1984), Groupes algébriques et corps de classes, Publications de l'Institut Mathématique del'Université de Nancago [Publications of the Mathematical Institute of the University of Nancago], 7, Paris:Hermann, ISBN 978-2-7056-1264-1, MR 907288

• John Tate, Finite flat group schemes, from Modular Forms and Fermat’s Last Theorem

• Waterhouse, William (1979), Introduction to affine group schemes, Graduate Texts in Mathematics 66, Berlin,New York: Springer-Verlag, doi:10.1007/978-1-4612-6217-6, ISBN 978-0-387-90421-4, MR 0547117

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Chapter 35

Haboush’s theorem

InmathematicsHaboush’s theorem, often still referred to as theMumford conjecture, states that for any semisimplealgebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, given v ≠ 0 in V thatis fixed by the action of G, there is a G-invariant polynomial F on V, without constant term, such that

F(v) ≠ 0.

The polynomial can be taken to be homogeneous, in other words an element of a symmetric power of the dual of V,and if the characteristic is p>0 the degree of the polynomial can be taken to be a power of p. WhenK has characteristic0 this was well known; in fact Weyl’s theorem on the complete reducibility of the representations of G implies thatF can even be taken to be linear. Mumford’s conjecture about the extension to prime characteristic p was proved byW. J. Haboush (1975), about a decade after the problem had been posed by David Mumford, in the introduction tothe first edition of his book Geometric Invariant Theory.

35.1 Applications

Haboush’s theorem can be used to generalize results of geometric invariant theory from characteristic 0, where theywere already known, to characteristic p>0. In particular Nagata’s earlier results together with Haboush’s theoremshow that if a reductive group (over an algebraically closed field) acts on a finitely generated algebra then the fixedsubalgebra is also finitely generated.Haboush’s theorem implies that if G is a reductive algebraic group acting regularly on an affine algebraic variety, thendisjoint closed invariant sets X and Y can be separated by an invariant function f (this means that f is 0 on X and 1on Y).C.S. Seshadri (1977) extended Haboush’s theorem to reductive groups over schemes.It follows from the work of Nagata (1963), Haboush, and Popov that the following conditions are equivalent for anaffine algebraic group G over a field K:

• G is reductive (its unipotent radical is trivial).

• For any non-zero invariant vector in a rational representation of G, there is an invariant homogeneous polyno-mial that does not vanish on it.

• For any finitely generatedK algebra on whichG act rationally, the algebra of fixed elements is finitely generated.

35.2 Proof

The theorem is proved in several steps as follows:

• We can assume that the group is defined over an algebraically closed field K of characteristic p>0.

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• Finite groups are easy to deal with as one can just take a product over all elements, so one can reduce to the caseof connected reductive groups (as the connected component has finite index). By taking a central extensionwhich is harmless one can also assume the group G is simply connected.

• Let A(G) be the coordinate ring of G. This is a representation of G with G acting by left translations. Pick anelement v′ of the dual of V that has value 1 on the invariant vector v. The map V to A(G) by sending w∈V tothe element a∈A(G) with a(g) = v′(g(w)). This sends v to 1∈A(G), so we can assume that V⊂A(G) and v=1.

• The structure of the representation A(G) is given as follows. Pick a maximal torus T of G, and let it act onA(G) by right translations (so that it commutes with the action of G). Then A(G) splits as a sum over charactersλ of T of the subrepresentations A(G)λ of elements transforming according to λ. So we can assume that V iscontained in the T-invariant subspace A(G)λ of A(G).

• The representation A(G)λ is an increasing union of subrepresentations of the form Eλ₊nᵨ⊗Enᵨ, where ρ is theWeyl vector for a choice of simple roots of T, n is a positive integer, and Eμ is the space of sections of the linebundle over G/B corresponding to a character μ of T, where B is a Borel subgroup containing T.

• If n is sufficiently large then Enᵨ has dimension (n+1)N whereN is the number of positive roots. This is becausein characteristic 0 the corresponding module has this dimension by the Weyl character formula, and for n largeenough that the line bundle over G/B is very ample, Enᵨ has the same dimension as in characteristic 0.

• If q=pr for a positive integer r, and n=q−1, thenEnᵨ contains the Steinberg representation ofG(Fq) of dimensionqN . (Here Fq ⊂ K is the finite field of order q.) The Steinberg representation is an irreducible representation ofG(Fq) and therefore of G(K), and for r large enough it has the same dimension as Enᵨ, so there are infinitelymany values of n such that Enᵨ is irreducible.

• If Enᵨ is irreducible it is isomorphic to its dual, so Enᵨ⊗Enᵨ is isomorphic to End(Enᵨ). Therefore the T-invariant subspace A(G)λ of A(G) is an increasing union of subrepresentations of the form End(E) for repre-sentations E (of the form E₍q₋₁₎ᵨ)). However for representations of the form End(E) an invariant polynomialthat separates 0 and 1 is given by the determinant. This completes the sketch of the proof of Haboush’s theorem.

35.3 References• Demazure, Michel (1976), “Démonstration de la conjecture de Mumford (d'après W. Haboush)", SéminaireBourbaki (1974/1975: Exposés Nos. 453-−470), Lecture Notes in Math. 514, Berlin: Springer, pp. 138–144,doi:10.1007/BFb0080063, ISBN 978-3-540-07686-5, MR 0444786

• Haboush, W. J. (1975), “Reductive groups are geometrically reductive”, Ann. Of Math. (The Annals ofMathematics, Vol. 102, No. 1) 102 (1): 67–83, doi:10.2307/1970974, JSTOR 1970974

• Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematikund ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994.xiv+292 pp. MR 1304906 ISBN 3-540-56963-4

• Nagata, Masayoshi (1963), “Invariants of a group in an affine ring”, Journal of Mathematics of Kyoto University3: 369–377, ISSN 0023-608X, MR 0179268

• M. Nagata, T. Miyata, “Note on semi-reductive groups” J. Math. Kyoto Univ., 3 (1964) pp. 379–382

• Popov, V.L. (2001), “Mumford hypothesis”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• C.S. Seshadri, “Geometric reductivity over arbitrary base” Adv. Math., 26 (1977) pp. 225–274

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Chapter 36

Hochschild–Mostow group

In mathematics, the Hochschild–Mostow group, introduced by Hochschild and Mostow (1957), is the universalpro-affine algebraic group generated by a group.

36.1 References• Hochschild, Gerhard; Mostow, GeorgeD. (1957), “Representations and representative functions of Lie groups”,Annals of Mathematics. Second Series 66: 495–542, ISSN 0003-486X, JSTOR 1969906, MR 0098796

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Chapter 37

Hyperspecial subgroup

In the theory of reductive groups over local fields, a hyperspecial subgroup of a reductive group G is a certain typeof compact subgroup of G.In particular, let F be a nonarchimedean local field, O its ring of integers, k its residue field and G a reductive groupover F. A subgroup K of G(F) is called hyperspecial if there exists a smooth group scheme Γ over O such that

• ΓF=G,

• Γ is a connected reductive group, and

• Γ(O)=K.

The original definition of a hyperspecial subgroup (appearing in section 1.10.2 of [1]) was in terms of hyperspecialpoints in the Bruhat-Tits Building of G. The equivalent definition above is given in the same paper of Tits, section3.8.1.Hyperspecial subgroups of G(F) exist if, and only if, G is unramified over F.[2]

An interesting property of hyperspecial subgroups, is that among all compact subgroups of G(F), the hyperspecialsubgroups have maximum measure.

37.1 References[1] Tits, Jacques, Reductive Groups over Local Fields in Automorphic forms, representations and L-functions, Part 1, Proc.

Sympos. Pure Math. XXXIII, 1979, pp. 29-69.

[2] Milne, James, The points on a Shimura variety modulo a prime of good reduction in The zeta functions of Picardmodular surfaces, Publications du CRM, 1992, pp. 151-253.

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Chapter 38

Inner form

In mathematics, an inner form of an algebraic group G over a field K is another algebraic group associated to anelement of H1(Gal(K/K), Inn(G)) where Inn(G) is the group of inner automorphisms of G.

38.1 References• Tits, Jacques (1966), “Classification of algebraic semisimple groups”, in Borel, Armand; Mostow, George D.,Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Providence,R.I.: American Mathematical Society, pp. 33–62, ISBN 978-0-8218-1409-3, MR 0224710

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Chapter 39

Iwahori subgroup

In algebra, an Iwahori subgroup is a subgroup of a reductive algebraic group over a local field that is analogous toa Borel subgroup of an algebraic group. A parahoric subgroup is a subgroup that is a finite union of double cosetsof an Iwahori subgroup, so is analogous to a parabolic subgroup of an algebraic group. Iwahori subgroups are namedafter Nagayoshi Iwahori, and “parahoric” is a portmanteau of “parabolic” and “Iwahori”. Iwahori & Matsumoto(1965) studied Iwahori subgroups for Chevalley groups over p-adic fields, and Bruhat & Tits (1972) extended theirwork to more general groups.Roughly speaking, an Iwahori subgroup of an algebraic group G(K), for a local field K with integers O and residuefield k, is the inverse image in G(O) of a Borel subgroup of G(k).A reductive group over a local field has a Tits system (B,N), where B is a parahoric group, and the Weyl group of theTits system is an affine Coxeter group.

39.1 References• Bruhat, F.; Tits, Jacques (1972), “Groupes réductifs sur un corps local”, Publications Mathématiques de l'IHÉS41: 5–251, doi:10.1007/bf02715544, ISSN 1618-1913, MR 0327923

• Iwahori, N.; Matsumoto, H. (1965), “On some Bruhat decomposition and the structure of the Hecke rings ofp-adic Chevalley groups”, Publications Mathématiques de l'IHÉS (25): 5–48, ISSN 1618-1913, MR 0185016

• Tits, Jacques (1979), “Reductive groups over local fields”, Automorphic forms, representations and L-functions(Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math.,XXXIII, Providence, R.I.: American Mathematical Society, pp. 29–69, MR 546588

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Chapter 40

Jordan–Chevalley decomposition

In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, ex-presses a linear operator as the sum of its commuting semisimple part and its nilpotent parts. The multiplicativedecomposition expresses an invertible operator as the product of its commuting semisimple and unipotent parts. Thedecomposition is important in the study of algebraic groups. The decomposition is easy to describe when the Jordannormal form of the operator is given, but it exists under weaker hypotheses than the existence of a Jordan normalform.

40.1 Decomposition of endomorphisms

Consider linear operators on a finite-dimensional vector space over a perfect field. An operator T is semisimple ifevery T-invariant subspace has a complementary T-invariant subspace (if the underlying field is algebraically closed,this is the same as the requirement that the operator be diagonalizable). An operator x is nilpotent if some power xmof it is the zero operator. An operator x is unipotent if x − 1 is nilpotent.Now, let x be any operator. A Jordan–Chevalley decomposition of x is an expression of it as a sum:

x = x + xn,

where x is semisimple, x is nilpotent, and x and x commute. If such a decomposition exists it is unique, and xand x are in fact expressible as polynomials in x, (Humphreys 1972, Prop. 4.2, p. 17).If x is an invertible operator, then a multiplicative Jordan–Chevalley decomposition expresses x as a product:

x = x · xᵤ,

where x is semisimple, xᵤ is unipotent, and x and xᵤ commute. Again, if such a decomposition exists it is unique,and x and xᵤ are expressible as polynomials in x.For endomorphisms of a finite dimensional vector space whose characteristic polynomial splits into linear factors overthe ground field (which always happens if that is an algebraically closed field), the Jordan–Chevalley decompositionexists and has a simple description in terms of the Jordan normal form. If x is in the Jordan normal form, then x isthe endomorphism whose matrix on the same basis contains just the diagonal terms of x, and x is the endomorphismwhose matrix on that basis contains just the off-diagonal terms; xᵤ is the endomorphism whose matrix is obtainedfrom the Jordan normal form by dividing all entries of each Jordan block by its diagonal element.

40.2 Decomposition in a real semisimple Lie algebra

In the formulation of Chevalley and Mostow, the additive decomposition states that an element X in a real semisimpleLie algebra g with Iwasawa decomposition g = k ⊕ a ⊕ n can be written as the sum of three commuting elements ofthe Lie algebra X = S + D + N, with S, D and N conjugate to elements in k, a and n respectively. In general the termsin the Iwasawa decomposition do not commute.

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40.3. DECOMPOSITION IN A REAL SEMISIMPLE LIE GROUP 121

40.3 Decomposition in a real semisimple Lie group

The multiplicative decomposition states that if g is an element of the corresponding connected semisimple Lie groupG with corresponding Iwasawa decomposition G = KAN, then g can be written as the product of three commutingelements g = sdu with s, d and u conjugate to elements of K, A and N respectively. In general the terms in the Iwasawadecomposition g = kan do not commute.

40.4 Counterexample

If the ground field is not perfect, then a Jordan–Chevalley decomposition may not exist. Example: Let p be a primenumber, let k be imperfect of characteristic p, and choose a in k that is not a pth power. Let V = k[x]/(xp-a)2, andlet T be the k-linear operator given by multiplication by x on V. This has as its stable k-linear subspaces preciselythe ideals of V viewed as a ring. Suppose T=S+N for commuting k-linear operators S and N that are respectivelysemisimple (just over k, which is weaker than semisimplicity over an algebraic closure of k) and nilpotent. SinceS and N commute, they each commute with T=S+N and hence each acts k[x]-linearly on V. Thus, each preservesthe unique nonzero proper k[x]-submodule J=(xp-a)V in V. But by semisimplicity of S, there would have to be anS-stable k-linear complement to J. However, by k[x]-linearity, S and N are each given by multiplication against therespective polynomials s = S(1) and n =N(1) whose induced effects on the quotient V/(xp-a) must be respectively xand 0 since this quotient is a field. Hence, s = x + (xp-a)h(x) for some polynomial h(x) (which only matters modulo(xp-a)), so it is easily seen that s generates V as a k-algebra and thus the S-stable k-linear subspaces of V are preciselythe k[x]-submodules. It follows that an S-stable complement to J is also a k[x]-submodule of V, contradicting thatJ is the only nonzero proper k[x]-submodule of V. Thus, there is no decomposition of T as a sum of commutingk-linear operators that are respectively semisimple and nilpotent.

40.5 References• Chevalley, Claude (1951), Théorie des groupes de Lie. Tome II. Groupes algébriques, Hermann

• Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN0-8218-2848-7

• Humphreys, James E. (1981), Linear Algebraic Groups, Graduate texts in mathematics 21, Springer, ISBN0-387-90108-6

• Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Springer, ISBN 978-0-387-90053-7

• Lazard, M. (1954), “Théorie des répliques. Critère de Cartan (Exposé No. 6)", Séminaire “Sophus Lie” 1

• Mostow, G. D. (1954), “Factor spaces of solvable groups”, Ann. of Math. 60: 1–27, doi:10.2307/1969700

• Mostow, G. D. (1973), Strong rigidity of locally symmetric spaces, Annals ofMathematics Studies 78, PrincetonUniversity Press

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556

• Serre, Jean-Pierre (1992), Lie algebras and Lie groups: 1964 lectures given at Harvard University, Lecturenotes in mathematics 1500 (2nd ed.), Springer-Verlag, ISBN 978-3-540-55008-2

• Varadarajan, V. S. (1984), Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics102, Springer-Verlag, ISBN 0-387-90969-9

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Chapter 41

Kazhdan–Lusztig polynomial

In the mathematical field of representation theory, aKazhdan–Lusztig polynomial Py,w(q) is a member of a familyof integral polynomials introduced by Kazhdan and Lusztig (1979). They are indexed by pairs of elements y, w of aCoxeter groupW, which can in particular be the Weyl group of a Lie group.

41.1 Motivation and history

In the spring of 1978 Kazhdan and Lusztig were studying Springer representations of the Weyl group of an algebraicgroup on l-adic cohomology groups related to unipotent conjugacy classes. They found a new construction of theserepresentations over the complex numbers (Kazhdan& Lusztig 1980a). The representation had two natural bases, andthe transition matrix between these two bases is essentially given by the Kazhdan–Lusztig polynomials. The actualKazhdan–Lusztig construction of their polynomials is more elementary. Kazhdan and Lusztig used this to constructa canonical basis in the Hecke algebra of the Coxeter group and its representations.In their first paper Kazhdan and Lusztig mentioned that their polynomials were related to the failure of local Poincaréduality for Schubert varieties. In Kazhdan & Lusztig (1980b) they reinterpreted this in terms of the intersectioncohomology of Mark Goresky and Robert MacPherson, and gave another definition of such a basis in terms of thedimensions of certain intersection cohomology groups.The two bases for the Springer representation reminded Kazhdan and Lusztig of the two bases for the Grothendieckgroup of certain infinite dimensional representations of semisimple Lie algebras, given by Verma modules and simplemodules. This analogy, and the work of Jantzen and Joseph relating primitive ideals of enveloping algebras to repre-sentations of Weyl groups, led to the Kazhdan–Lusztig conjectures.

41.2 Definition

Fix a Coxeter group W with generating set S, and write ℓ (w) for the length of an element w (the smallest length ofan expression for w as a product of elements of S). The Hecke algebra of W has a basis of elements T for w ∈ Wover the ring Z[q1/2, q−1/2], with multiplication defined by

TyTw = Tyw, if ℓ(yw) = ℓ(y) + ℓ(w)

(Ts + 1)(Ts − q) = 0, if s ∈ S.

The quadratic second relation implies that each generator T is invertible in the Hecke algebra, with inverse Ts−1= q−1Ts + q−1 − 1. These inverses satisfy the relation (Ts−1 + 1)(Ts−1 − q−1) = 0 (obtained by multiplying thequadratic relation for T by −T −2q−1), and also the braid relations. From this it follows that the Hecke algebra has anautomorphism D that sends q1/2 to q−1/2 and each T to Ts−1. More generally one has D(Tw) = T−1

w−1 ; also D canbe seen to be an involution.The Kazhdan–Lusztig polynomials Pyw(q) are indexed by a pair of elements y, w ofW, and uniquely determined bythe following properties.

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41.3. EXAMPLES 123

• They are 0 unless y ≤ w (in the Bruhat order ofW), 1 if y = w, and for y < w their degree is at most (ℓ (w) −ℓ (y) − 1)/2.

• The elements

C ′w = q−

ℓ(w)2

∑y≤w

Py,wTy

are invariant under the involution D of the Hecke algebra. The elements C ′w form a basis of the Hecke

algebra as a Z[q1/2, q−1/2]-module, called the Kazhdan–Lusztig basis.

To establish existence of the Kazhdan–Lusztig polynomials, Kazhdan and Lusztig gave a simple recursive procedurefor computing the polynomials Pyw(q) in terms of more elementary polynomials denoted Ryw(q). defined by

T−1y−1 =

∑x

D(Rx,y)q−ℓ(x)Tx.

They can be computed using the recursion relations

Rx,y =

0, if x ≤ y

1, if x = y

Rsx,sy, if sx < x and sy < y

Rxs,ys, if xs < x and ys < y

(q − 1)Rsx,y + qRsx,sy, if sx > x and sy < y

The Kazhdan–Lusztig polynomials can then be computed recursively using the relation

q12 (ℓ(w)−ℓ(x))D(Px,w)− q

12 (ℓ(x)−ℓ(w))Px,w =

∑x<y≤w

(−1)ℓ(x)+ℓ(y)q12 (−ℓ(x)+2ℓ(y)−ℓ(w))D(Rx,y)Py,w

using the fact that the two terms on the left are polynomials in q1/2 and q−1/2 without constant terms. These formulasare tiresome to use by hand for rank greater than about 3, but are well adapted for computers, and the only limit oncomputing Kazhdan–Lusztig polynomials with them is that for large rank the number of such polynomials exceedsthe storage capacity of computers.

41.3 Examples

• If y ≤ w then Py,w has constant term 1.

• If y ≤ w and ℓ (w) − ℓ (y) ∈ 0, 1, 2 then Py,w = 1.

• If w = w0 is the longest element of a finite Coxeter group then Py,w = 1 for all y.

• IfW is the Coxeter group A1 or A2 (or more generally any Coxeter group of rank at most 2) then Py,w is 1 ify≤w and 0 otherwise.

• IfW is the Coxeter group A3 with generating set S = a, b, c with a and c commuting then Pb,bacb = 1 + qand Pac,acbca = 1 + q, giving examples of non-constant polynomials.

• The simple values of Kazhdan–Lusztig polynomials for low rank groups are not typical of higher rank groups.For example, for the split form of E8 the most complicated Lusztig–Vogan polynomial (a variation of Kazhdan–Lusztig polynomials: see below) is

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124 CHAPTER 41. KAZHDAN–LUSZTIG POLYNOMIAL

152q22 + 3, 472q21 + 38, 791q20 + 293, 021q19 + 1, 370, 892q18 + 4, 067, 059q17 + 7, 964, 012q16

+ 11, 159, 003q15 + 11, 808, 808q14 + 9, 859, 915q13 + 6, 778, 956q12 + 3, 964, 369q11 + 2, 015, 441q10

+ 906, 567q9 + 363, 611q8 + 129, 820q7 + 41, 239q6 + 11, 426q5 + 2, 677q4 + 492q3 + 61q2 + 3q

• Polo (1999) showed that any polynomial with constant term 1 and non-negative integer coefficients is theKazhdan–Lusztig polynomial for some pair of elements of some symmetric group.

41.4 Kazhdan–Lusztig conjectures

The Kazhdan–Lusztig polynomials arise as transition coefficients between their canonical basis and the natural basisof the Hecke algebra. The Inventiones paper also put forth two equivalent conjectures, known now as Kazhdan–Lusztig conjectures, which related the values of their polynomials at 1 with representations of complex semisimpleLie groups and Lie algebras, addressing a long-standing problem in representation theory.Let W be a finite Weyl group. For each w ∈ W denote by M be the Verma module of highest weight −w(ρ) − ρwhere ρ is the half-sum of positive roots (or Weyl vector), and let L be its irreducible quotient, the simple highestweight module of highest weight −w(ρ) − ρ. Both M and L are locally-finite weight modules over the complexsemisimple Lie algebra g with the Weyl groupW, and therefore admit an algebraic character. Let us write ch(X) forthe character of a g-module X. The Kazhdan-Lusztig conjectures state:

ch(Lw) =∑y≤w

(−1)ℓ(w)−ℓ(y)Py,w(1) ch(My)

ch(Mw) =∑y≤w

Pw0w,w0y(1) ch(Ly)

where w0 is the element of maximal length of the Weyl group.These conjectures were proved independently by Beilinson and Bernstein (1981) and by Brylinski and Kashiwara(1981). The methods introduced in the course of the proof have guided development of representation theorythroughout the 1980s and 1990s, under the name geometric representation theory.

41.4.1 Remarks

1. The two conjectures are known to be equivalent. Moreover, Borho–Jantzen’s translation principle implies thatw(ρ) − ρ can be replaced by w(λ + ρ) − ρ for any dominant integral weight λ. Thus, the Kazhdan-Lusztig conjecturesdescribe the Jordan–Hölder multiplicities of Verma modules in any regular integral block of Bernstein–Gelfand–Gelfand category O.2. A similar interpretation of all coefficients of Kazhdan–Lusztig polynomials follows from the Jantzen conjecture,which roughly says that individual coefficients of P , are multiplicities of L in certain subquotient of the Vermamodule determined by a canonical filtration, the Jantzen filtration. The Jantzen conjecture in regular integral casewas proved in a later paper of Beilinson and Bernstein (1993).3. David Vogan showed as a consequence of the conjectures that

Py,w(q) =∑i

qi dim(Extℓ(w)−ℓ(y)−2i(My, Lw))

and that Extj(My, Lw) vanishes if j + ℓ (w) + ℓ (y) is odd, so the dimensions of all such Ext groups in category O aredetermined in terms of coefficients of Kazhdan–Lusztig polynomials. This result demonstrates that all coefficientsof the Kazhdan–Lusztig polynomials of a finite Weyl group are non-negative integers. However, positivity for thecase of a finite Weyl group W was already known from the interpretation of coefficients of the Kazhdan–Lusztigpolynomials as the dimensions of intersection cohomology groups, irrespective of the conjectures. Conversely, the

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41.5. RELATION TO INTERSECTION COHOMOLOGY OF SCHUBERT VARIETIES 125

relation between Kazhdan–Lusztig polynomials and the Ext groups theoretically can be used to prove the conjectures,although this approach to proving them turned out to be more difficult to carry out.4. Some special cases of the Kazhdan–Lusztig conjectures are easy to verify. For example, M1 is the antidominantVerma module, which is known to be simple. This means that M1 = L1, establishing the second conjecture for w =1, since the sum reduces to a single term. On the other hand, the first conjecture for w = w0 follows from the Weylcharacter formula and the formula for the character of a Verma module, together with the fact that all Kazhdan–Lusztig polynomials Py,w0

are equal to 1.5. Kashiwara (1990) proved a generalization of the Kazhdan–Lusztig conjectures to symmetrizable Kac–Moodyalgebras.

41.5 Relation to intersection cohomology of Schubert varieties

By the Bruhat decomposition the space G/B of the algebraic group G with Weyl groupW is a disjoint union of affinespaces Xw parameterized by elements w of W. The closures of these spaces X are called Schubert varieties, andKazhdan and Lusztig, following a suggestion of Deligne, showed how to express Kazhdan–Lusztig polynomials interms of intersection cohomology groups of Schubert varieties.More precisely, the Kazhdan–Lusztig polynomial Py,w(q) is equal to

Py,w(q) =∑i

qi dim IH2iXy

(Xw)

where each term on the right means: take the complex IC of sheaves whose hyperhomology is the intersection ho-mology of the Schubert variety of w (the closure of the cell X ), take its cohomology of degree 2i, and then takethe dimension of the stalk of this sheaf at any point of the cell X whose closure is the Schubert variety of y. Theodd-dimensional cohomology groups do not appear in the sum because they are all zero.This gave the first proof that all coefficients of Kazhdan–Lusztig polynomials for finite Weyl groups are non-negativeintegers.

41.6 Generalization to real groups

Lusztig–Vogan polynomials (also called Kazhdan–Lusztig polynomials or Kazhdan–Lusztig–Vogan polynomi-als) were introduced in Lusztig & Vogan (1983). They are analogous to Kazhdan–Lusztig polynomials, but aretailored to representations of real semisimple Lie groups, and play major role in the conjectural description of theirunitary duals. Their definition is more complicated, reflecting relative complexity of representations of real groupscompared to complex groups.The distinction, in the cases directly connection to representation theory, is explained on the level of double cosets;or in other terms of actions on analogues of complex flag manifolds G/B where G is a complex Lie group and B aBorel subgroup. The original (K-L) case is then about the details of decomposing

B\G/B,

a classical theme of the Bruhat decomposition, and before that of Schubert cells in a Grassmannian. The L-Vcase takes a real form GR of G, a maximal compact subgroup KR in that semisimple group GR, and makes thecomplexification K of KR. Then the relevant object of study is

K\G/B.

In March 2007, it was announced that the L-V polynomials had been calculated for the split form of E8.

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126 CHAPTER 41. KAZHDAN–LUSZTIG POLYNOMIAL

41.7 Generalization to other objects in representation theory

The second paper of Kazhdan and Lusztig established a geometric setting for definition of Kazhdan–Lusztig poly-nomials, namely, the geometry of singularities of Schubert varieties in the flag variety. Much of the later work ofLusztig explored analogues of Kazhdan–Lusztig polynomials in the context of other natural singular algebraic va-rieties arising in representation theory, in particular, closures of nilpotent orbits and quiver varieties. It turned outthat the representation theory of quantum groups, modular Lie algebras and affine Hecke algebras are all tightly con-trolled by appropriate analogues of Kazhdan–Lusztig polynomials. They admit an elementary description, but thedeeper properties of these polynomials necessary for representation theory follow from sophisticated techniques ofmodern algebraic geometry and homological algebra, such as the use of intersection cohomology, perverse sheavesand Beilinson–Bernstein–Deligne decomposition.The coefficients of the Kazhdan–Lusztig polynomials are conjectured to be the dimensions of some homomorphismspaces in Soergel’s bimodule category. This is the only known positive interpretation of these coefficients for arbitraryCoxeter groups.

41.8 Combinatorial theory

Combinatorial properties of Kazhdan–Lusztig polynomials and their generalizations are a topic of active currentresearch. Given their significance in representation theory and algebraic geometry, attempts have been undertakento develop the theory of Kazhdan–Lusztig polynomials in purely combinatorial fashion, relying to some extent ongeometry, but without reference to intersection cohomology and other advanced techniques. This has led to excitingdevelopments in algebraic combinatorics, such as pattern-avoidance phenomenon. Some references are given in thetextbook of Björner & Brenti (2005). A research monograph on the subject is Billey & Lakshmibai (2000).As of 2005, there is no known combinatorial interpretation of all the coefficients of the Kazhdan–Lusztig polynomials(as the cardinalities of some natural sets) even for the symmetric groups, though explicit formulas exist in many specialcases.

41.9 References

• Beilinson, Alexandre; Bernstein, Joseph (1981), Localisation de g-modules, Sér. I Math. 292 (1), Paris: C. R.Acad. Sci., pp. 15–18.

• Beilinson, Alexandre; Bernstein, Joseph (1993), A proof of the Jantzen conjectures, Advances in Soviet Math-ematics 16 (1), pp. 1–50.

• Billey, Sara; Lakshmibai, V. (2000), Singular loci of Schubert varieties, Progress in Mathematics 182, Boston,MA: Birkhäuser, ISBN 0-8176-4092-4.

• Björner, Anders; Brenti, Francesco (2005), “Ch. 5: Kazhdan–Lusztig and R-polynomials”, Combinatorics ofCoxeter Groups, Graduate Texts in Mathematics 231, Springer, ISBN 978-3-540-44238-7.

• Brenti, Francesco (2003), “Kazhdan-Lusztig Polynomials: History, Problems, and Combinatorial Invariance”,Séminaire Lotharingien de Combinatoire (Ellwangen: Haus Schönenberg) 49: Research article B49b.

• Brylinski, Jean-Luc; Kashiwara, Masaki (October 1981), “Kazhdan-Lusztig conjecture and holonomic sys-tems”, Inventiones Mathematicae (Springer-Verlag) 64 (3): 387–410, doi:10.1007/BF01389272, ISSN 0020-9910.

• Kashiwara, M. (1990), “TheKazhdan-Lusztig conjecture for symmetrizableKacMoody algebras”, TheGrothendieckFestschrift, II, Progress in Mathem. 87, Boston: Birkhauser, pp. 407–433, MR 93a:17026.

• Kazhdan, David; Lusztig, George (June 1979), “Representations of Coxeter groups and Hecke algebras”,Inventiones Mathematicae (Springer-Verlag) 53 (2): 165–184, doi:10.1007/BF01390031, ISSN 0020-9910.

• Kazhdan, David; Lusztig, George (1980a), “A topological approach to Springer’s representations”, Advancesin Mathematics 38 (2): 222–228, doi:10.1016/0001-8708(80)90005-5.

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41.10. EXTERNAL LINKS 127

• Kazhdan, David; Lusztig, George (1980b), “Schubert varieties and Poincaré duality”, Proc. Sympos. PureMath. (American Mathematical Society), XXXVI: 185–203.

• Lusztig, George; Vogan, David (1983), “Singularities of closures of K-orbits on flag manifolds.”, InventionesMathematicae (Springer-Verlag) 71 (2): 365–379, doi:10.1007/BF01389103, ISSN 0020-9910.

• Polo, Patrick (1999), “Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups”, Represen-tation Theory. an Electronic Journal of the American Mathematical Society 3 (4): 90–104, doi:10.1090/S1088-4165-99-00074-6, ISSN 1088-4165, MR 1698201.

• Soergel, Wolfgang (2006), “Kazhdan-Lusztig polynomials and indecomposable bimodules over polynomialrings”, Journal of the Inst. of Math. Jussieu 6 (3): 501–525.

41.10 External links• Readings from Spring 2005 course on Kazhdan-Lusztig Theory at U.C. Davis by Monica Vazirani

• Goresky’s tables of Kazhdan–Lusztig polynomials.

• The GAP programs for computing Kazhdan–Lusztig polynomials.

• Fokko du Cloux’s Coxeter software for computing Kazhdan-Lusztig polynomials for any Coxeter group

• Atlas software for computing Kazhdan–Lusztig-Vogan polynomials.

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Chapter 42

Kempf vanishing theorem

In algebraic geometry, theKempf vanishing theorem, introduced byKempf (1976), states that the higher cohomologygroup H i(G/B,L(λ)) (i > 0) vanishes whenever λ is a dominant weight of B. Here G is a reductive algebraic groupover an algebraically closed field, B a Borel subgroup, and L(λ) a line bundle associated to λ. In characteristic 0 thisis a special case of the Borel–Weil–Bott theorem, but unlike the Borel–Weil–Bott theorem, the Kempf vanishingtheorem still holds in positive characteristic.Andersen (1980) and Haboush (1980) found simpler proofs of the Kempf vanishing theorem using the Frobeniusmorphism.

42.1 References• Andersen, Henning Haahr (1980), “The Frobenius morphism on the cohomology of homogeneous vectorbundles on G/B”, Annals of Mathematics. Second Series 112 (1): 113–121, doi:10.2307/1971322, ISSN 0003-486X, MR 584076

• Hazewinkel, Michiel, ed. (2001), “Kempf_vanishing_theorem”, Encyclopedia ofMathematics, Springer, ISBN978-1-55608-010-4

• Haboush, William J. (1980), “A short proof of the Kempf vanishing theorem”, Inventiones Mathematicae 56(2): 109–112, doi:10.1007/BF01392545, ISSN 0020-9910, MR 558862

• Kempf, George R. (1976), “Linear systems on homogeneous spaces”, Annals of Mathematics. Second Series103 (3): 557–591, doi:10.2307/1970952, ISSN 0003-486X, MR 0409474

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Chapter 43

Kneser–Tits conjecture

In mathematics, the Kneser–Tits problem, introduced by Tits (1964) based on a suggestion by Martin Kneser, askswhether the Whitehead group W(G,K) of a semisimple simply connected isotropic algebraic group G over a fieldK is trivial. The Whitehead group is the quotient of the rational points of G by the normal subgroup generated byK-subgroups isomorphic to the additive group.

43.1 Fields for which the Whitehead group vanishes

A special case of the Kneser–Tits problem asks for which fields the Whitehead group of a semisimple almost simplesimply connected isotropic algebraic group is always trivial. Platonov (1969) showed that this Whitehead group istrivial for local fields K, and gave examples of fields for which it is not always trivial. For global fields the combinedwork of several authors shows that this Whitehead group is always trivial (Gille 2009).

43.2 References• Gille, Philippe (2009), “Le problème de Kneser-Tits”, Astérisque, Séminaire Bourbaki exp. 983 (326): 39–81,ISBN 978-2-85629-269-3, ISSN 0303-1179, MR 2605318

• Platonov, V. P. (1969), “The problem of strong approximation and the Kneser–Tits hypothesis for algebraicgroups”, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 33: 1211–1219, ISSN 0373-2436, MR0258839

• Tits, Jacques (1978), “Groupes de Whitehead de groupes algébriques simples sur un corps (d'après V. P.Platonov et al.)", Séminaire Bourbaki, 29e année (1976/77), Lecture Notes in Math. 677, Berlin, New York:Springer-Verlag, pp. 218–236, MR 521771

• Tits, Jacques (1964), “Algebraic and abstract simple groups”, Annals of Mathematics. Second Series 80: 313–329, ISSN 0003-486X, JSTOR 1970394, MR 0164968

43.3 External links• Hazewinkel, Michiel, ed. (2001), “Kneser-Tits hypothesis”, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

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Chapter 44

Kostant polynomial

In mathematics, the Kostant polynomials, named after Bertram Kostant, provide an explicit basis of the ring ofpolynomials over the ring of polynomials invariant under the finite reflection group of a root system.

44.1 Background

If the reflection groupW corresponds to theWeyl group of a compact semisimple groupK with maximal torus T, thenthe Kostant polynomials describe the structure of the de Rham cohomology of the generalized flag manifold K/T, alsoisomorphic to G/B where G is the complexification of K and B is the corresponding Borel subgroup. Armand Borelshowed that its cohomology ring is isomorphic to the quotient of the ring of polynomials by the ideal generated by theinvariant homogeneous polynomials of positive degree. This ring had already been considered by Claude Chevalleyin establishing the foundations of the cohomology of compact Lie groups and their homogeneous spaces with AndréWeil, Jean-Louis Koszul and Henri Cartan; the existence of such a basis was used by Chevalley to prove that thering of invariants was itself a polynomial ring. A detailed account of Kostant polynomials was given by Bernstein,Gelfand & Gelfand (1973) and independently Demazure (1973) as a tool to understand the Schubert calculus of theflag manifold. The Kostant polynomials are related to the Schubert polynomials defined combinatorially by Lascoux& Schützenberger (1982) for the classical flag manifold, when G = SL(n,C). Their structure is governed by differenceoperators associated to the corresponding root system.Steinberg (1975) defined an analogous basis when the polynomial ring is replaced by the ring of exponentials of theweight lattice. If K is simply connected, this ring can be identified with the representation ring R(T) and the W-invariant subring with R(K). Steinberg’s basis was again motivated by a problem on the topology of homogeneousspaces; the basis arises in describing the T-equivariant K-theory of K/T.

44.2 Definition

Let Φ be a root system in a finite-dimensional real inner product space V with Weyl group W. Let Φ+ be a set ofpositive roots and Δ the corresponding set of simple roots. If α is a root, then sα denotes the corresponding reflectionoperator. Roots are regarded as linear polynomials on V using the inner product α(v) = (α,v). The choice of Δ givesrise to a Bruhat order on the Weyl group determined by the ways of writing elements minimally as products of simpleroot reflection. The minimal length for an elenet s is denoted ℓ(s) . Pick an element v in V such that α(v) > 0 forevery positive root.If αi is a simple root with reflection operator si

six = x− 2(x, αi)

(αi, αi)αi,

then the corresponding divided difference operator is defined by

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44.3. PROPERTIES 131

δif =f − f si

αi.

If ℓ(s) = m and s has reduced expression

s = si1 · · · sim ,

then

δs = δi1 · · · δim

is independent of the reduced expression. Moreover

δsδt = δst

if ℓ(st) = ℓ(s) + ℓ(t) and 0 otherwise.If w0 is the longest element ofW, the element of greatest length or equivalently the element sending Φ+ to −Φ+, then

δw0f =

∑s∈W det s f s∏

α>0 α.

More generally

δsf =det s f s+

∑t<s as,t f t∏

α>0, s−1α<0 α

for some constants as,t.Set

d = |W |−1∏α>0

α.

and

Ps = δs−1w0d.

Then P is a homogeneous polynomial of degree ℓ(s) .These polynomials are the Kostant polynomials.

44.3 Properties

Theorem. The Kostant polynomials form a free basis of the ring of polynomials over the W-invariant polynomials.

In fact the matrix

Nst = δs(Pt)

is unitriangular for any total order such that s ≥ t implies ℓ(s) ≥ ℓ(t) .

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132 CHAPTER 44. KOSTANT POLYNOMIAL

Hence

detN = 1.

Thus if

f =∑s

asPs

with as invariant underW, then

δt(f) =∑s

δt(Ps)as.

Thus

as =∑t

Ms,tδt(f),

where

M = N−1

another unitriangular matrix with polynomial entries. It can be checked directly that as is invariant underW.In fact δi satisfies the derivation property

δi(fg) = δi(f)g + (f si)δi(g).

Hence

δiδs(f) =∑t

δi(δs(Pt))at) =∑t

(δs(Pt) si)δi(at) +∑t

δiδs(Pt)at.

Since

δiδs = δsis

or 0, it follows that

∑t

δs(Pt) δi(at) si = 0

so that by the invertibility of N

δi(at) = 0

for all i, i.e. at is invariant underW.

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44.4. STEINBERG BASIS 133

44.4 Steinberg basis

As above let Φ be a root system in a real inner product space V, and Φ+ a subset of positive roots. From these datawe obtain the subset Δ = α1, α2, ..., αn of the simple roots, the coroots

α∨i = 2(αi, αi)

−1αi,

and the fundamental weights λ1, λ2, ..., λn as the dual basis of the coroots.For each element s inW, let Δs be the subset of Δ consisting of the simple roots satisfying s−1α < 0, and put

λs = s−1∑

αi∈∆s

λi,

where the sum is calculated in the weight lattice P.The set of linear combinations of the exponentials eμ with integer coefficients for μ in P becomes a ring over Zisomorphic to the group algebra of P, or equivalently to the representation ring R(T) of T, where T is a maximal torusin K, the simply connected, connected compact semisimple Lie group with root system Φ. IfW is the Weyl group ofΦ, then the representation ring R(K) of K can be identified with R(T)W .Steinberg’s theorem. The exponentials λs (s in W) form a free basis for the ring of exponentials over the subring ofW-invariant exponentials.Let ρ denote the half sum of the positive roots, and A denote the antisymmetrisation operator

A(ψ) =∑s∈W

(−1)ℓ(s)s · ψ.

The positive roots β with sβ positive can be seen as a set of positive roots for a root system on a subspace of V; theroots are the ones orthogonal to s.λs. The corresponding Weyl group equals the stabilizer of λs inW. It is generatedby the simple reflections sj for which sαj is a positive root.Let M and N be the matrices

Mts = t(λs), Nst = (−1)ℓ(t) · t(ψs),

where ψs is given by the weight s−1ρ - λs. Then the matrix

Bs,s′ = Ω−1(NM)s,s′ =A(ψsλs′)

Ω

is triangular with respect to any total order onW such that s ≥ t implies ℓ(s) ≥ ℓ(t) . Steinberg proved that the entriesof B areW-invariant exponential sums. Moreover its diagonal entries all equal 1, so it has determinant 1. Hence itsinverse C has the same form. Define

φs =∑

Cs,tψt.

If χ is an arbitrary exponential sum, then it follows that

χ =∑s∈W

asλs

with as theW-invariant exponential sum

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134 CHAPTER 44. KOSTANT POLYNOMIAL

as =A(φsχ)

Ω.

Indeed this is the unique solution of the system of equations

tχ =∑s∈W

t(λs) as =∑s

Mt,sas.

44.5 References• Bernstein, I. N.; Gelfand, I. M.; Gelfand, S. I. (1973), “Schubert cells, and the cohomology of the spaces G/P”,Russian Math. Surveys 28: 1–26, doi:10.1070/RM1973v028n03ABEH001557

• Billey, Sara C. (1999), “Kostant polynomials and the cohomology ring for G/B.”, Duke Math. J. 96: 205–224,doi:10.1215/S0012-7094-99-09606-0

• Bourbaki, Nicolas (1981), Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Masson, ISBN 2-225-76076-4

• Cartan, Henri (1950), “Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opèreun groupe de Lie”, Colloque de topologie (espaces fibrés), Bruxelles: 15–27

• Cartan, Henri (1950), “La transgression dans un groupe de Lie et dans un espace fibré principal”, Colloque detopologie (espaces fibrés), Bruxelles: 57–71

• Chevalley, Claude (1955), “Invariants of finite groups generated by reflections”, Amer. J. Math. (The JohnsHopkins University Press) 77 (4): 778–782, doi:10.2307/2372597, JSTOR 2372597

• Demazure, Michel (1973), “Invariants symétriques entiers des groupes de Weyl et torsion”, Invent. Math. 21:287–301, doi:10.1007/BF01418790

• Greub, Werner; Halperin, Stephen; Vanstone, Ray (1976), Connections, curvature, and cohomology. VolumeIII: Cohomology of principal bundles and homogeneous spaces, Pure and Applied Mathematics, 47-III, Aca-demic Press

• Humphreys, James E. (1994), Introduction to Lie Algebras and Representation Theory (2nd ed.), Springer,ISBN 0-387-90053-5

• Kostant, Bertram (1963), “Lie algebra cohomology and generalized Schubert cells”, Ann. Of Math. (Annalsof Mathematics) 77 (1): 72–144, doi:10.2307/1970202, JSTOR 1970202

• Kostant, Bertram (1963), “Lie group representations on polynomial rings”,Amer. J. Math. (The Johns HopkinsUniversity Press) 85 (3): 327–404, doi:10.2307/2373130, JSTOR 2373130

• Kostant, Bertram; Kumar, Shrawan (1986), “The nil Hecke ring and cohomology of G/P for a Kac–Moodygroup G.”, Proc. Nat. Acad. Sci. U.S.A. 83: 1543–1545, doi:10.1073/pnas.83.6.1543

• Alain, Lascoux; Schützenberger, Marcel-Paul (1982), “Polynômes de Schubert [Schubert polynomials]", C. R.Acad. Sci. Paris Sér. I Math. 294: 447–450

• McLeod, John (1979), The Kunneth formula in equivariant K-theory, Lecture Notes in Math. 741, Springer,pp. 316–333

• Steinberg, Robert (1975), “On a theorem of Pittie”, Topology 14: 173–177, doi:10.1016/0040-9383(75)90025-7

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Chapter 45

Lang’s theorem

In algebraic geometry, Lang’s theorem, introduced by Serge Lang, states: if G is a connected smooth algebraicgroup over a finite field Fq , then, writing σ : G→ G, x 7→ xq for the Frobenius, the morphism of varieties

G→ G, x 7→ x−1σ(x)

is surjective. Note that the kernel of this map (i.e., G = G(Fq) → G(Fq) ) is precisely G(Fq) .The theorem implies that H1(Fq, G) = H1

et(SpecFq, G) vanishes,[1] and, consequently, any G-bundle on SpecFq

is isomorphic to the trivial one. Also, the theorem plays a basic role in the theory of finite groups of Lie type.It is not necessary that G is affine. Thus, the theorem also applies to abelian varieties (e.g., elliptic curves.) In fact,this application was Lang’s initial motivation. If G is affine, the Frobenius σ may be replaced by any surjective mapwith finitely many fixed points (see below for the precise statement.)The proof (given below) actually goes through for any σ that induces a nilpotent operator on the Lie algebra of G.[2]

45.1 The Lang–Steinberg theorem

Steinberg (1968) gave a useful improvement to the theorem.Suppose that F is an endomorphism of an algebraic group G. The Lang map is the map from G to G taking g tog−1F(g).The Lang–Steinberg theorem states[3] that if F is surjective and has a finite number of fixed points, and G is aconnected affine algebraic group over an algebraically closed field, then the Lang map is surjective.

45.2 Proof of Lang’s theorem

Define:

fa : G→ G, fa(x) = x−1aσ(x).

Then (identifying the tangent space at a with the tangent space at the identity element) we have:

(dfa)e = d(h (x 7→ (x−1, a, σ(x))))e = dh(e,a,e) (−1, 0, dσe) = −1 + dσe

where h(x, y, z) = xyz . It follows (dfa)e is bijective since the differential of the Frobenius σ vanishes. Sincefa(bx) = ffa(b)(x) , we also see that (dfa)b is bijective for any b.[4] Let X be the closure of the image of f1 . Thesmooth points of X form an open dense subset; thus, there is some b in G such that f1(b) is a smooth point of X.Since the tangent space to X at f1(b) and the tangent space to G at b have the same dimension, it follows that X and

135

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136 CHAPTER 45. LANG’S THEOREM

G have the same dimension, since G is smooth. Since G is connected, the image of f1 then contains an open densesubset U of G. Now, given an arbitrary element a in G, by the same reasoning, the image of fa contains an opendense subset V of G. The intersection U ∩ V is then nonempty but then this implies a is in the image of f1 .

45.3 Notes[1] This is “unwinding definition”. Here, H1(Fq, G) = H1(Gal(Fq/Fq), G(Fq)) is Galois cohomology; cf. Milne, Class

field theory.

[2] Springer 1998, Exercise 4.4.18.

[3] Steinberg 1968, Theorem 10.1

[4] This implies that fa is étale.

45.4 References• T.A. Springer, “Linear algebraic groups”, 2nd ed. 1998.

• Lang, Serge (1956), “Algebraic groups over finite fields”, American Journal of Mathematics 78: 555–563,doi:10.2307/2372673, ISSN 0002-9327, MR 0086367

• Steinberg, Robert (1968), Endomorphisms of linear algebraic groups, Memoirs of the American MathematicalSociety, No. 80, Providence, R.I.: American Mathematical Society, MR 0230728

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Chapter 46

Langlands decomposition

In mathematics, theLanglands decompositionwrites a parabolic subgroup P of a semisimple Lie group as a productP =MAN of a reductive subgroup M, an abelian subgroup A, and a nilpotent subgroup N.

46.1 Applications

See also: Parabolic induction

A key application is in parabolic induction, which leads to the Langlands program: ifG is a reductive algebraic groupand P = MAN is the Langlands decomposition of a parabolic subgroup P, then parabolic induction consists oftaking a representation ofMA , extending it to P by letting N act trivially, and inducing the result from P to G .

46.2 See also• Lie group decompositions

46.3 References• A. W. Knapp, Structure theory of semisimple Lie groups. ISBN 0-8218-0609-2.

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Chapter 47

Lattice (discrete subgroup)

A portion of the discrete Heisenberg group, a discrete subgroup of the continuous Heisenberg Lie group. (The coloring and edges areonly for visual aid.)

In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroupwith the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, thisamounts to the usual geometric notion of a lattice, and both the algebraic structure of lattices and the geometry of thetotality of all lattices are relatively well understood. Deep results of Borel, Harish-Chandra, Mostow, Tamagawa, M.S. Raghunathan, Margulis, Zimmer obtained from the 1950s through the 1970s provided examples and generalizedmuch of the theory to the setting of nilpotent Lie groups and semisimple algebraic groups over a local field. In the1990s, Bass and Lubotzky initiated the study of tree lattices, which remains an active research area.

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47.1. DEFINITION 139

47.1 Definition

Let G be a locally compact topological group with the Haar measure μ. A discrete subgroup Γ is called a lattice inG if the quotient space G/Γ has finite invariant measure, that is, if G is a unimodular group and the volume μ(G/Γ)is finite. The lattice is uniform (or cocompact) if the quotient space is compact, and nonuniform otherwise.

47.2 Arithmetic lattices

An archetypical example of a nonuniform lattice is given by the group SL(2,Z), which is a lattice in the special lineargroup SL(2,R), and by the closely related modular group. This construction admits a far-reaching generalization toa class of lattices in all semisimple algebraic groups over a local field F called arithmetic lattices. For example, letF = R be the field of real numbers. Roughly speaking, the Lie group G(R) is formed by all matrices with entries inR satisfying certain algebraic conditions, and by restricting the entries to the integers Z, one obtains a lattice G(Z).Conversely, Grigory Margulis proved that under certain assumptions on G, any lattice in it essentially arises in thisway. This remarkable statement is known as Arithmeticity of lattices or Margulis Arithmeticity Theorem.

47.3 S-arithmetic lattices

Arithmetic lattices admit an important generalization, known as the S-arithmetic lattices. The first example is givenby the diagonally embedded subgroup

SL

(2,Z

[1

p

])⊂ SL(2,R)× SL(2,Qp), S = p,∞.

This is a lattice in the product of algebraic groups over different local fields, both real and p-adic. It is formed by theunimodular matrices of order 2 with entries in the localization of the ring of integers at the prime p. The set S is afinite set of places of Q which includes all archimedean places and the locally compact group is the direct productof the groups of points of a fixed linear algebraic group G defined over Q (or a more general global field) over thecompletions of Q at the places from S. To form the discrete subgroup, instead of matrices with integer entries, oneconsiders matrices with entries in the localization over the primes (nonarchimedean places) in S. Under fairly generalassumptions, this construction indeed produces a lattice. The class of S-arithmetic lattices is much wider than theclass of arithmetic lattices, but they share many common features.

47.4 Adelic case

A lattice of fundamental importance for the theory of automorphic forms is given by the group G(K) of K-points ofa semisimple (or reductive) linear algebraic group G defined over a global field K. This group diagonally embeds intothe adelic algebraic group G(A), where A is the ring of adeles of K, and is a lattice there. Unlike arithmetic lattices,G(K) is not finitely generated.

47.5 Rigidity

Another group of phenomena concerning lattices in semisimple algebraic groups is collectively known as rigidity. TheMostow rigidity theorem showed that the algebraic structure of a lattice in simple Lie group G of split rank at leasttwo determines G. Thus any isomorphism of lattices in two such groups is essentially induced by an isomorphismbetween the groups themselves. Superrigidity provides a generalization dealing with homomorphisms from a latticein an algebraic group G into another algebraic group H.

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140 CHAPTER 47. LATTICE (DISCRETE SUBGROUP)

47.6 Tree lattices

Let X be a locally finite tree. Then the automorphism group G of X is a locally compact topological group, in whichthe basis of the topology is given by the stabilizers of finite sets of vertices. Vertex stabilizers Gx are thus compactopen subgroups, and a subgroup Γ of G is discrete if Γx is finite for some (and hence, for any) vertex x. The subgroupΓ is an X-lattice if the suitably defined volume of X/Γ is finite, and a uniform X-lattice if this quotient is a finitegraph. In case G\X is finite, this is equivalent to Γ being a lattice (respectively, a uniform lattice) in G.

47.7 See also• Kazhdan’s property (T)

• Graph of groups

47.8 References• Hyman Bass and Alexander Lubotzky, Tree lattices. With appendices by H. Bass, L. Carbone, A. Lubotzky,G. Rosenberg, and J. Tits. Progress in Mathematics, vol 176, Birkhäuser Verlag, Boston, 2001 ISBN 0-8176-4120-3

• Grigory Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Gren-zgebiete (3) [Results in Mathematics and Related Areas (3)], 17. Springer-Verlag, Berlin, 1991. x+388 pp.ISBN 3-540-12179-X MR 1090825

• Dave Witte Morris: Introduction to Arithmetic Groups, draft of a book

• Platonov, Vladimir; Rapinchuk, Andrei (1994), Algebraic groups and number theory. (Translated from the1991 Russian original by Rachel Rowen.), Pure and Applied Mathematics 139, Boston, MA: Academic Press,Inc., ISBN 0-12-558180-7, MR 1278263

• M.S.Raghunathan, Discrete subgroups of Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band68. Springer-Verlag, New York-Heidelberg, 1972 MR 0507234

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Chapter 48

Lazard’s universal ring

In mathematics, Lazard’s universal ring is a ring introduced by Michel Lazard in Lazard (1955) over which theuniversal commutative one-dimensional formal group law is defined.There is a universal commutative one-dimensional formal group law over a universal commutative ring defined asfollows. We let

F(x, y)

be

x + y + Σci,j xiyj

for indeterminates

ci,j,

and we define the universal ring R to be the commutative ring generated by the elements ci,j, with the relations thatare forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring Rhas the following universal property:

For every commutative ring S, one-dimensional formal group laws over S correspond to ring homomor-phisms from R to S.

The commutative ring R constructed above is known as Lazard’s universal ring. At first sight it seems to beincredibly complicated: the relations between its generators are very messy. However Lazard proved that it has avery simple structure: it is just a polynomial ring (over the integers) on generators of degrees 2, 4, 6, … (where ci,jhas degree 2(i + j − 1)). Quillen (1969) proved that the coefficient ring of complex cobordism is naturally isomorphicas a graded ring to Lazard’s universal ring.

48.1 References• Adams, J. Frank (1974), Stable homotopy and generalised homology, University of Chicago Press, ISBN 978-0-226-00524-9

• Lazard, Michel (1955), “Sur les groupes de Lie formels à un paramètre”, Bulletin de la Société Mathématiquede France 83: 251–274, ISSN 0037-9484, MR 0073925

• Lazard, Michel (1975), Commutative formal groups, Lecture Notes in Mathematics 443, Berlin, New York:Springer-Verlag, doi:10.1007/BFb0070554, ISBN 978-3-540-07145-7, MR 0393050

• Quillen, Daniel (1969), “On the formal group laws of unoriented and complex cobordism theory”, Bulletin ofthe American Mathematical Society 75: 1293–1298, doi:10.1090/S0002-9904-1969-12401-8, MR 0253350

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Chapter 49

Lie–Kolchin theorem

In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie’stheorem is the analog for linear Lie algebras.It states that if G is a connected and solvable linear algebraic group defined over an algebraically closed field and

ρ : G→ GL(V )

a representation on a nonzero finite-dimensional vector space V, then there is a one-dimensional linear subspace L ofV such that

ρ(G)(L) = L.

That is, ρ(G) has an invariant line L, on which G therefore acts through a one-dimensional representation. This isequivalent to the statement that V contains a nonzero vector v that is a common (simultaneous) eigenvector for allρ(g), g ∈ G .Because every (nonzero finite-dimensional) representation of G has a one-dimensional invariant subspace accordingto the Lie–Kolchin theorem, every irreducible finite-dimensional representation of a connected and solvable linearalgebraic group G has dimension one, which is another way to state the Lie–Kolchin theorem.Lie’s theorem states that any nonzero representation of a solvable Lie algebra on a finite dimensional vector spaceover an algebraically closed field of characteristic 0 has a one-dimensional invariant subspace.The result for Lie algebras was proved by Sophus Lie (1876) and for algebraic groups was proved by Ellis Kolchin(1948, p.19).The Borel fixed point theorem generalizes the Lie–Kolchin theorem.

49.1 Triangularization

Sometimes the theorem is also referred to as the Lie–Kolchin triangularization theorem because by induction it impliesthat with respect to a suitable basis of V the image ρ(G) has a triangular shape; in other words, the image group ρ(G)is conjugate in GL(n,K) (where n = dim V) to a subgroup of the group T of upper triangular matrices, the standardBorel subgroup of GL(n,K): the image is simultaneously triangularizable.The theorem applies in particular to a Borel subgroup of a semisimple linear algebraic group G.

49.2 Lie’s theorem

Lie’s theorem states that if V is a finite dimensional vector space over an algebraically closed field of characteristic 0,then for any solvable Lie algebra of endomorphisms of V there is a vector that is an eigenvector for every element ofthe Lie algebra.

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49.3. COUNTER-EXAMPLES 143

Applying this result repeatedly shows that there is a basis for V such that all elements of the Lie algebra are rep-resented by upper triangular matrices. This is a generalization of the result of Frobenius that commuting matricesare simultaneously upper triangularizable, as commuting matrices form an abelian Lie algebra, which is a fortiorisolvable.A consequence of Lie’s theorem is that any finite dimensional solvable Lie algebra over a field of characteristic 0 hasa nilpotent derived algebra.

49.3 Counter-examples

If the field K is not algebraically closed, the theorem can fail. The standard unit circle, viewed as the set of complexnumbers x+iy ∈ C |x2+y2 = 1 of absolute value one is a one-dimensional commutative (and therefore solvable)linear algebraic group over the real numbers which has a two-dimensional representation into the special orthogonalgroup SO(2) without an invariant (real) line. Here the image ρ(z) of z = x+ iy is the orthogonal matrix

(x y−y x

).

For algebraically closed fields of characteristic p>0 Lie’s theorem holds provided the dimension of the representationis less than p, but can fail for representations of dimension p. An example is given by the 3-dimensional nilpotentLie algebra spanned by 1, x, and d/dx acting on the p-dimensional vector space k[x]/(xp), which has no eigenvectors.Taking the semidirect product of this 3-dimensional Lie algebra by the p-dimensional representation (considered asan abelian Lie algebra) gives a solvable Lie algebra whose derived algebra is not nilpotent.

49.4 References• Gorbatsevich, V.V. (2001), “l/l058710”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Kolchin, E. R. (1948), “Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinarydifferential equations”, Annals of Mathematics. Second Series 49: 1–42, ISSN 0003-486X, JSTOR 1969111,MR 0024884, Zbl 0037.18701

• Lie, Sophus (1876), “Theorie der Transformationsgruppen. Abhandlung II”, Archiv for Mathematik og Naturv-idenskab 1: 152–193

• WilliamC.Waterhouse, Introduction to AffineGroup Schemes, Graduate Texts inMathematics vol. 66, SpringerVerlag New York, 1979 (chapter 10, in particular section 10.2).

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Chapter 50

Mirabolic group

In mathematics, amirabolic subgroup of the general linear group GLn(k), studied by Gelfand & Kajdan (1975), isa subgroup consisting of automorphisms fixing a given non-zero vector in kn. Its image in the projective general lineargroup is a parabolic subgroup consisting of all elements fixing a given point of projective space. The word “mirabolic”is a portmanteau of “miraculous parabolic”. As an algebraic group, a mirabolic subgroup is the semidirect productof a vector space with its group of automorphisms, and such groups are called mirabolic groups. The mirabolicsubgroup is used to define the Kirillov model of a representation of the general linear group.

• Example: The group of all matrices of the form (**01) is a mirabolic subgroup of the 2-dimensional general linear group.

50.1 References• Bernstein, Joseph N. (1984), “P-invariant distributions on GL(N) and the classification of unitary representa-tions of GL(N) (non-Archimedean case)", Lie group representations, II (College Park, Md., 1982/1983), Lec-ture Notes in Math. 1041, Berlin, New York: Springer-Verlag, pp. 50–102, doi:10.1007/BFb0073145, MR748505

• Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathe-matischenWissenschaften [Fundamental Principles ofMathematical Sciences] 335, Berlin, NewYork: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120

• Finkelberg, Michael; Ginzburg, Victor (2010), “Onmirabolic D-modules”, International Mathematics ResearchNotices (15): 2947–2986, doi:10.1093/imrn/rnp216, ISSN 1073-7928, MR 2673716

• Gelfand, I.M.; Kajdan, D. A. (1975) [1971], “Representations of the groupGL(n,K) where K is a local field”, inGelfand, I. M., Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest,1971), Halsted, New York, pp. 95–118, ISBN 978-0-470-29600-4, MR 0404534

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Chapter 51

Mumford–Tate group

In algebraic geometry, theMumford–Tate group (or Hodge group) MT(F) constructed from a Hodge structure Fis a certain algebraic group G. When F is given by a rational representation of an algebraic torus, the definition of Gis as the Zariski closure of the image in the representation of the circle group, over the rational numbers. Mumford(1966) introduced Mumford–Tate groups over the complex numbers under the name of Hodge groups. Serre (1967)introduced the p-adic analogue of Mumford’s construction for Hodge–Tate modules, using the work of Tate (1967)on p-divisible groups, and named them Mumford–Tate groups.

51.1 Formulation

The algebraic torus T used to describe Hodge structures has a concrete matrix representation, as the 2×2 invertiblematrices of the shape that is given by the action of a+bi on the basis 1,i of the complex numbers C over R:

[a b−b a

].

The circle group inside this group of matrices is the unitary group U(1).Hodge structures arising in geometry, for example on the cohomology groups of Kähler manifolds, have a latticeconsisting of the integral cohomology classes. Not quite so much is needed for the definition of the Mumford–Tategroup, but it does assume that the vector space V underlying the Hodge structure has a given rational structure, i.e. isgiven over the rational numbersQ. For the purposes of the theory the complex vector spaceVC, obtained by extendingthe scalars of V from Q to C, is used.The weight k of the Hodge structure describes the action of the diagonal matrices of T, and V is supposed thereforeto be homogeneous of weight k, under that action. Under the action of the full group VC breaks up into subspacesVpq, complex conjugate in pairs under switching p and q. Thinking of the matrix in terms of the complex numberλ it represents, Vpq has the action of λ by the pth power and of the complex conjugate of λ by the qth power. Herenecessarily

p + q = k.

In more abstract terms, the torus T underlying the matrix group is the Weil restriction of the multiplicative groupGL(1), from the complex field to the real field, an algebraic torus whose character group consists of the two homo-morphisms to GL(1), interchanged by complex conjugation.Once formulated in this fashion, the rational representation ρ of T on V setting up the Hodge structure F determinesthe image ρ(U(1)) in GL(VC); andMT(F) is by definition the Zariski closure, for the Q-Zariski topology on GL(V),of this image.[1]

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146 CHAPTER 51. MUMFORD–TATE GROUP

51.2 Mumford–Tate conjecture

The original context for the formulation of the group in question was the question of the Galois representation onthe Tate module of an abelian variety A. Conjecturally, the image of such a Galois representation, which is an l-adicLie group for a given prime number l, is determined by the corresponding Mumford–Tate group G (coming fromthe Hodge structure on H1(A)), to the extent that knowledge of G determines the Lie algebra of the Galois image.This conjecture is known only in particular cases.[2] Through generalisations of this conjecture, the Mumford–Tategroup has been connected to the motivic Galois group, and, for example, the general issue of extending the Sato–Tateconjecture (now a theorem).

51.3 Period conjecture

A related conjecture on abelian varieties states that the periodmatrix ofA over number field has transcendence degree,in the sense of the field generated by its entries, predicted by the dimension of its Mumford–Tate group, as in theprevious section. Work of Pierre Deligne has shown that the dimension bounds the transcendence degree; so that theMumford–Tate group catches sufficiently many algebraic relations between the periods. This is a special case of thefull Grothendieck period conjecture.[3][4]

51.4 Notes[1] http://www.math.columbia.edu/~thaddeus/seattle/voisin.pdf, pp. 7–9.

[2] http://math.berkeley.edu/~ribet/Articles/mg.pdf, survey by Ken Ribet.

[3] http://math.cts.nthu.edu.tw/Mathematics/preprints/prep2005-6-002.pdf, p. 3.

[4] http://arxiv.org/abs/0805.2569v1, p. 7.

51.5 References• Mumford, David (1966), “Families of abelian varieties”, Algebraic Groups and Discontinuous Subgroups (Proc.Sympos. Pure Math., Boulder, Colo., 1965), Providence, R.I.: American Mathematical Society, pp. 347–351,MR 0206003

• Serre, Jean-Pierre (1967), “Sur les groupes de Galois attachés aux groupes p-divisibles”, in Springer, TonnyA., Proceedings of a Conference on Local Fields (Driebergen, 1966), Berlin, New York: Springer-Verlag, pp.118–131, ISBN 978-3-540-03953-2, MR 0242839

• Tate, John T. (1967), “p-divisible groups.”, in Springer, Tonny A., Proc. Conf. Local Fields( Driebergen,1966), Berlin, New York: Springer-Verlag, MR 0231827

51.6 External links• Lecture slides (PDF) by Phillip Griffiths

• Mumford-Tate groups, families of Calabi-Yau varieties and analogue André-Oort problems I, preprint (PDF)

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Chapter 52

Observable subgroup

In mathematics, in the representation theory of algebraic groups, an observable subgroup is an algebraic subgroup ofa linear algebraic group whose every finite-dimensional rational representation arises as the restriction to the subgroupof a finite-dimensional rational representation of the whole group.An equivalent formulation, in case the base field is closed, is that K is an observable subgroup of G if and only if thequotient variety G/K is a quasi-affine variety.Some basic facts about observable subgroups:

• Every normal algebraic subgroup of an algebraic group is observable.

• Every observable subgroup of an observable subgroup is observable.

52.1 External links• Extensions of Representations of algebraic linear groups

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Chapter 53

Pseudo-reductive group

In mathematics, a pseudo-reductive group or k-reductive group over a field k is a smooth connected affine algebraicgroup defined over k whose unipotent k-radical is trivial. The unipotent k-radical is the largest smooth connectedunipotent normal subgroup defined over k. Over perfect fields these are the same as (connected) reductive groups,but over non-perfect fields Jacques Tits found some examples of pseudo-reductive groups that are not reductive. Ak-reductive group need not be a reductive k-group (a reductive group defined over k). Pseudo-reductive groups arisenaturally in the study of algebraic groups over function fields of positive-dimensional varieties in positive characteristic(even over a perfect field of constants).Springer (1998) gives an exposition of Tits’ results on pseudo-reductive groups, while Conrad, Gabber & Prasad(2010) builds on Tits’ work to develop a general structure theory, including more advanced topics such as constructiontechniques, root systems and root groups and open cells, classification theorems, and applications to rational conjugacytheorems for smooth connected affine groups over arbitrary fields. The general theory is summarized in Rémy (2011).

53.1 Examples of pseudo reductive groups that are not reductive

Suppose that k is a non-perfect field of characteristic 2, and a is an element of k that is not a square. Let G be thegroup of nonzero elements x + y√a in k[√a]. There is a morphism from G to the multiplicative group Gm taking x+ y√a to its norm x2 – ay2, and the kernel is the subgroup of elements of norm 1. The underlying reduced schemeof the geometric kernel is isomorphic to the additive group Ga and is the unipotent radical of the geometric fiber ofG, but this reduced subgroup scheme of the geometric fiber is not defined over k (i.e., it does not arise from a closedsubscheme of G over the ground field) and the unipotent k-radical is trivial. So G is a k-reductive group but is nota reductive k-group. A similar construction works using a primitive nontrivial purely inseparable finite extension ofan imperfect field in any positive characteristic, the only difference being that the formula for the norm map is a bitmore complicated than in the preceding quadratic examples.More generally, ifK is a non-trivial purely inseparable finite extension of k andG is any non-trivial connected reductivegroup defined over K then the Weil restriction H=RK/k(G) is a smooth connected affine algebraic group defined overk for which there is a homomorphism from HK onto G. The kernel of this K-homomorphism descends the unipotentradical of the geometric fiber ofH and is not defined over k (i.e., does not arise from a closed subgroup scheme ofH),so RK/k(G) is pseudo-reductive but not reductive. The previous example is the special case using the multiplicativegroup and the extension K=k[√a].

53.2 Classification and exotic phenomena

Over fields of characteristic greater than 3, all pseudo-reductive groups can be obtained from reductive groups by the“standard construction”, a generalization of the construction above. The standard construction involves an auxiliarychoice of a commutative pseudo-reductive group, which turns out to be a Cartan subgroup of the output of theconstruction, and the main complication for a general pseudo-reductive group is that the structure of Cartan subgroups(which are always commutative and pseudo-reductive) is mysterious. The commutative pseudo-reductive groupsadmit no useful classification (in contrast with the reductive case, for which they are tori and hence are accessible

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53.3. REFERENCES 149

via Galois lattices), but modulo this one has a useful description of the situation away from characteristics 2 and 3 interms of reductive groups over some finite extensions of the ground field.Over imperfect fields of characteristic 2 and 3 there are some extra exotic pseudo-reductive groups coming from theexistence of exceptional isogenies between groups of types B and C in characteristic 2, between groups of type F₄in characteristic 2, and between groups of type G₂ in characteristic 3, using a construction analogous to that of theRee groups. Moreover, in characteristic 2 there are additional possibilities arising not from exceptional isogenies butrather from the fact that for simply connected type C (I.e., symplectic groups) there are roots that are divisible (by2) in the weight lattice; this gives rise to examples whose root system (over a separable closure of the ground field)is non-reduced; such examples exist with a split maximal torus and an irreducible non-reduced root system of anypositive rank. The classification in characteristic 3 is as complete as in larger characteristics, but in characteristic 2the classification provided in the published literature is complete only when [k:k^2]=2 (due to complications causedby the examples with a non-reduced root system, as well as phenomena related to certain regular degenerate quadraticforms that can only exist when [k:k^2]>2). Subsequent work of Conrad and Prasad, building on additional materialincluded in the second edition of Conrad, Gabber & Prasad (2010), completes the classification in characteristic 2(up to a controlled central extension) by providing an exhaustive array of additional constructions that only exist when[k:k^2]>2 , ultimately resting on a notion of special orthogonal group attached to regular but degenerate and not fullydefective quadratic spaces in characteristic 2.

53.3 References• Conrad, Brian; Gabber, Ofer; Prasad, Gopal (2010), Pseudo-reductive groups, NewMathematical Monographs17, Cambridge University Press, doi:10.1017/CBO9780511661143, ISBN 978-0-521-19560-7, MR 2723571

• Rémy, Bertrand (2011), “Groupes algébriques pseudo-réductifs et applications (d'après J. Tits et B. Conrad--O. Gabber--G. Prasad)" (PDF), Astérisque (339): 259–304, ISBN 978-2-85629-326-3, ISSN 0303-1179, MR2906357

• Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics 9 (2nd ed.), Boston, MA:Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713

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Chapter 54

Quasi-split group

In mathematics, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field.Simply connected quasi-split groups over a field correspond to actions of the absolute Galois group on a Dynkindiagram.

54.1 Examples

All split groups (those with a split maximal torus) are quasi-split. These correspond to quasi-split groups where theaction of the Galois group on the Dynkin diagram is trivial.Lang (1956) showed that all simple algebraic groups over finite fields are quasi-split.Over the real numbers, the quasi-split groups include the split groups and the complex groups, together with theorthogonal groups On,n₊₂, the unitary groups SUn,n and SUn,n₊₁, and the form of E6 with signature 2.

54.2 References• Lang, Serge (1956), “Algebraic groups over finite fields”, American Journal of Mathematics 78: 555–563,doi:10.2307/2372673, ISSN 0002-9327, MR 0086367

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Chapter 55

Radical of an algebraic group

The radical of an algebraic group is the identity component of its maximal normal solvable subgroup.

55.1 See also

Semisimple algebraic group

55.2 References• “Radical of a group”, Encyclopaedia of Mathematics

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Chapter 56

Rational representation

In mathematics, in the representation theory of algebraic groups, a linear representation of an algebraic group is saidto be rational if, viewed as a map from the group to the general linear group, it is a rational map of algebraic varieties.Finite direct sums and products of rational representations are rational.A rational G module is a module that can be expressed as a sum (not necessarily direct) of rational representations.Further information: Group representation

56.1 References• Extensions of Representations of Algebraic Linear Groups

• Springer Online Reference Works: Rational Representation

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Chapter 57

Reductive group

In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotentradical of G is trivial (i.e., the group of unipotent elements of the radical of G). Any semisimple algebraic group isreductive, as is any algebraic torus and any general linear group. More generally, over fields that are not necessarilyalgebraically closed, a reductive group is a smooth affine algebraic group such that the unipotent radical of G over thealgebraic closure is trivial. The intervention of an algebraic closure in this definition is necessary to include the caseof imperfect ground fields, such as local and global function fields over finite fields. Algebraic groups over (possiblyimperfect) fields k such that the k-unipotent radical is trivial are called pseudo-reductive groups.The name comes from the complete reducibility of linear representations of such a group, which is a property infact holding only for representations of the algebraic group over fields of characteristic zero. (This only applies torepresentations of the algebraic group: finite-dimensional representations of the underlying discrete group need notbe completely reducible even in characteristic 0.) Haboush’s theorem shows that a certain rather weaker propertycalled geometric reductivity holds for reductive groups in the positive characteristic case.If G ≤ GLn is a smooth closed k -subgroup that acts irreducibly on affine n -space over k , then G is reductive.[1] Itfollows that GLn and SLn are reductive (the latter being even semisimple).

57.1 Lie group case

For more details on this topic, see Reductive Lie algebra.

More generally, in the case of Lie groups, a reductive Lie group G can be defined in terms of its Lie algebra, namelya reductive Lie group is one whose Lie algebra g is a reductive Lie algebra; concretely, a Lie algebra that is the sumof an abelian and a semisimple Lie algebra. Sometimes the condition that the identity component G0 of G is of finiteindex is added.A Lie algebra is reductive if and only if its adjoint representation is completely reducible, but this does not implythat all of its finite-dimensional representations are completely reducible. The concept of reductive is not quite thesame for Lie groups as it is for algebraic groups because a reductive Lie group can be the group of real points of aunipotent algebraic group.For example, the one-dimensional, abelian Lie algebra R is obviously reductive, and is the Lie algebra of both areductive algebraic groupGm (the multiplicative group of nonzero real numbers) and also a unipotent (non-reductive)algebraic group Ga (the additive group of real numbers). These are not isomorphic as algebraic groups; at the Liealgebra level we see the same structure, but this is not enough to make any stronger assertion (essentially because theexponential map is not an algebraic function).

57.2 See also• Luna’s slice theorem

• Root datum

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• Pseudo-reductive group

57.3 Notes[1] See Springer 1998, exercise 2.4.15

57.4 References• Borel, Armand (1991), Linear Algebraic Groups, Graduate Texts in Mathematics 126 (2nd ed.), New York:Springer-Verlag, ISBN 978-0-387-97370-8.

• A. Borel, J. Tits,Groupes réductifs Publ. Math. IHES, 27 (1965) pp. 55–150; Compléments à l'article «Groupesréductifs». Publications Mathématiques de l'IHÉS, 41 (1972), p. 253–276

• Bruhat, François; Tits, Jacques Groupes réductifs sur un corps local : I. Données radicielles valuées. Pub-lications Mathématiques de l'IHÉS, 41 (1972), p. 5–251 II. Schémas en groupes. Existence d'une donnéeradicielle valuée. Publications Mathématiques de l'IHÉS, 60 (1984), p. 5–184

• V.L. Popov (2001), “Reductive group”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• A.L.Onishchik (2001), “Lie algebra, reductive”, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

• Springer, Tonny A. (1979), “Reductive groups” (PDF), Automorphic forms, representations, and L-functions1, pp. 3–27, ISBN 0-8218-3347-2

• Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics 9 (2nd ed.), Boston, MA:Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713

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Chapter 58

Restricted Lie algebra

In mathematics, a restricted Lie algebra is a Lie algebra together with an additional "p operation.”

58.1 Definition

Let L be a Lie algebra over a field k of characteristic p>0. A p operation on L is a map X 7→ X [p] satisfying

• ad(X [p]) = ad(X)p for all X ∈ L ,

• (tX)[p] = tpX [p] for all t ∈ k,X ∈ L ,

• (X + Y )[p] = X [p] + Y [p] +∑p−1

i=1si(X,Y )

i , for allX,Y ∈ L , where si(X,Y ) is the coefficient of ti−1 inthe formal expression ad(tX + Y )p−1(X) .

If the characteristic of k is 0, then L is a restricted Lie algebra where the p operation is the identity map.

58.2 Examples

For any associative algebra A defined over a field of characteristic p, the bracket operation [X,Y ] := XY − Y Xand p operation X [p] := Xp make A into a restricted Lie algebra Lie(A) .Let G be an algebraic group over a field k of characteristic p, and Lie(G) be the Zariski tangent space at the identityelement of G. Each element of Lie(G) uniquely defines a left-invariant vector field on G, and the commutator ofvector fields defines a Lie algebra structure on Lie(G) just as in the Lie group case. If p>0, the Frobenius mapx 7→ xp defines a p operation on Lie(G) .

58.3 Restricted universal enveloping algebra

The functor A 7→ Lie(A) has a left adjoint L 7→ U [p](L) called the restricted universal enveloping algebra. Toconstruct this, let U(L) be the universal enveloping algebra of L forgetting the p operation. Letting I be the two-sidedideal generated by elements of the form xp − x[p] , we set U [p](L) = U(L)/I . It satsfies a form of the PBWtheorem.

58.4 See also

Restricted Lie algebras are used in Jacobson's Galois correspondence for purely inseparable extensions of fields ofexponent 1.

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58.5 References• Borel, Armand (1991), Linear Algebraic Groups, Graduate Texts in Mathematics 126 (2nd ed.), Springer-Verlag, Zbl 0726.20030.

• Block, Richard E.; Wilson, Robert Lee (1988), “Classification of the restricted simple Lie algebras”, Journalof Algebra 114 (1): 115–259, doi:10.1016/0021-8693(88)90216-5, ISSN 0021-8693, MR 931904.

• Montgomery, Susan (1993), Hopf algebras and their actions on rings. Expanded version of ten lectures givenat the CBMS Conference on Hopf algebras and their actions on rings, which took place at DePaul University inChicago, USA, August 10-14, 1992, Regional Conference Series in Mathematics 82, Providence, RI: AmericanMathematical Society, p. 23, ISBN 978-0-8218-0738-5, Zbl 0793.16029.

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Chapter 59

Root datum

In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a gener-alization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazurein SGA III, published in 1970.

59.1 Definition

A root datum consists of a quadruple

(X∗,Φ, X∗,Φ∨)

where

• X∗ and X∗ are free abelian groups of finite rank together with a perfect pairing between them with values inZ which we denote by ( , ) (in other words, each is identified with the dual lattice of the other).

• Φ is a finite subset ofX∗ and Φ∨ is a finite subset ofX∗ and there is a bijection from Φ onto Φ∨ , denoted byα 7→ α∨ .

• For each α , (α, α∨) = 2 .

• For each α , the map x 7→ x− (x, α∨)α induces an automorphism of the root datum (in other words it mapsΦ to Φ and the induced action onX∗ maps Φ∨ to Φ∨ )

The elements of Φ are called the roots of the root datum, and the elements of Φ∨ are called the coroots. Theelements of X∗ are sometimes called weights and those of X∗ accordingly coweights.If Φ does not contain 2α for any α ∈ Φ , then the root datum is called reduced.

59.2 The root datum of an algebraic group

If G is a reductive algebraic group over an algebraically closed field K with a split maximal torus T then its rootdatum is a quadruple

(X*, Φ, X*, Φv),

where

• X* is the lattice of characters of the maximal torus,

• X* is the dual lattice (given by the 1-parameter subgroups),

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158 CHAPTER 59. ROOT DATUM

• Φ is a set of roots,

• Φv is the corresponding set of coroots.

A connected split reductive algebraic group over K is uniquely determined (up to isomorphism) by its root datum,which is always reduced. Conversely for any root datum there is a reductive algebraic group. A root datum containsslightly more information than the Dynkin diagram, because it also determines the center of the group.For any root datum (X*, Φ,X*, Φv), we can define a dual root datum (X*, Φv,X*, Φ) by switching the characterswith the 1-parameter subgroups, and switching the roots with the coroots.If G is a connected reductive algebraic group over the algebraically closed field K, then its Langlands dual group LGis the complex connected reductive group whose root datum is dual to that of G.

59.3 References• Michel Demazure, Exp. XXI in SGA 3 vol 3

• T. A. Springer, Reductive groups, in Automorphic forms, representations, and L-functions vol 1 ISBN 0-8218-3347-2

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Chapter 60

Rost invariant

In mathematics, the Rost invariant is a cohomological invariant of an absolutely simple simply connected algebraicgroup G over a field k, which associates an element of the Galois cohomology group H3(k, Q/Z(2)) to a principalhomogeneous space for G. Here the coefficient group Q/Z(2) is the tensor product of the group of roots of unity ofan algebraic closure of k with itself. Markus Rost (1991) first introduced the invariant for groups of type F4 and laterextended it to more general groups in unpublished work that was summarized by Serre (1995).The Rost invariant is a generalization of the Arason invariant.

60.1 Definition

Suppose that G is an absolutely almost simple simply connected algebraic group over a field k. The Rost invariantassociates an element a(P) of the Galois cohomology group H3(k,Q/Z(2)) to a G-torsor P.The element a(P) is constructed as follows. For any extension K of k there is an exact sequence

0 → H3(K,Q/Z(2)) → H3et(PK ,Q/Z(2)) → Q/Z

where the middle group is the étale cohomology group and Q/Z is the geometric part of the cohomology. Choosea finite extension K of k such that G splits over K and P has a rational point over K. Then the exact sequence splitscanonically as a direct sum so the étale cohomology group containsQ/Z canonically. The invariant a(P) is the imageof the element 1/[K:k] of Q/Z under the trace map from H3et(PK,Q/Z(2)) to H3et(P,Q/Z(2)), which lies in the subgroup H3(k,Q/Z(2)).These invariants a(P) are functorial in field extensions K of k; in other words the fit together to form an element ofthe cyclic group Inv3(G,Q/Z(2)) of cohomological invariants of the group G, which consists of morphisms of thefunctor K→H1(K,G) to the functor K→H3(K,Q/Z(2)). This element of Inv3(G,Q/Z(2)) is a generator of the groupand is called the Rost invariant of G.

60.2 References

• Garibaldi, Ryan Skip (2001), “The Rost invariant has trivial kernel for quasi-split groups of low rank”, Com-ment. Math. Helv. 76 (4): 684–711, doi:10.1007/s00014-001-8325-8, MR 1881703

• Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003), “Rost invariants of simply connected alge-braic groups”, Cohomological invariants in Galois cohomology, University Lecture Series 28, Providence, RI:American Mathematical Society, ISBN 0-8218-3287-5, MR 1999383, Zbl 1159.12311

• Rost, Markus (1991), “A (mod 3) invariant for exceptional Jordan algebras”, C. R. Acad. Sci. Paris Sér. IMath. 313 (12): 823–827, MR 1138557

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• Serre, Jean-Pierre (1995), “Cohomologie galoisienne: progrès et problèmes”, Astérisque, Séminaire BourbakiExp. No. 783 227 (4): 229–257, MR 1321649

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Chapter 61

Semisimple algebraic group

In mathematics, especially in the areas of abstract algebra and algebraic geometry studying linear algebraic groups,a semisimple algebraic group is a type of matrix group which behaves much like a semisimple Lie algebra orsemisimple ring.

61.1 Definition

A linear algebraic group is called semisimple if and only if the (solvable) radical of the identity component is trivial.Equivalently, a semisimple linear algebraic group has no non-trivial connected, normal, abelian subgroups.

61.2 Examples• Over an algebraically closed field k , the special linear group SLn(k) is semisimple.

• Every direct sum of simple algebraic groups is semisimple.

61.3 Properties

61.4 References• Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics 126 (2nd ed.), Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012

• Humphreys, James E. (1972), Linear Algebraic Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90108-4, MR 0396773

• Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics 9 (2nd ed.), Boston, MA:Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713

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Chapter 62

Serre group

In mathematics, the Serre group S is the pro-algebraic group whose representations correspond to CM-motives overthe algebraic closure of the rationals, or to polarizable rational Hodge structures with abelian Mumford–Tate groups.It is a projective limit of finite dimensional tori, so in particular is abelian. It was introduced by Serre (1968). It is asubgroup of the Taniyama group.There are two different but related groups called the Serre group, one the connected component of the identity inthe other. This article is mainly about the connected group, usually called the Serre group but sometimes called theconnected Serre group. In addition one can define Serre groups of algebraic number fields, and the Serre group is theinverse limit of the Serre groups of number fields.

62.1 Definition

The Serre group is the projective limit of the Serre groups of SL of finite Galois extensions of the rationals, and eachof these groups SL is a torus, so is determined by its module of characters, a finite free Z-module with an action ofthe finite Galois group Gal(L/Q). If L* is the algebraic group with L*(A) the units of A⊗L, then L* is a torus withthe same dimension as L, and its characters can be identified with integral functions on Gal(L/Q). The Serre groupSL is a quotient of this torus L*, so can be described explicitly in terms of the module X*(SL) of rational characters.This module of rational characters can be identified with the integral functions λ on Gal(L/Q) such that

(σ−1)(ι+1)λ = (ι+1)(σ−1)λ = 0

for all σ in Gal(L/Q), where ι is complex conjugation. It is acted on by the Galois group.The full Serre group S can be described similarly in terms of its module X*(S) of rational characters. This moduleof rational characters can be identified with the locally constant integral functions λ on Gal(Q/Q) such that

(σ−1)(ι+1)λ = (ι+1)(σ−1)λ = 0

for all σ in Gal(Q/Q), where ι is complex conjugation.

62.2 References• Deligne, Pierre; Milne, James S.; Ogus, Arthur; Shih, Kuang-yen (1982), Hodge cycles, motives, and Shimuravarieties., Lecture Notes in Mathematics 900, Berlin-New York: Springer-Verlag, ISBN 3-540-11174-3, MR0654325

• Serre, Jean-Pierre (1968), Abelian l-adic representations and elliptic curves., McGill University lecture notes,New York-Amsterdam: W. A. Benjamin, Inc., MR 0263823

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Chapter 63

Severi–Brauer variety

In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to aprojective space over an algebraic closure of K. The varieties are associated to central simple algebras in such a waythat the algebra splits over K if and only if the variety has a point rational over K.[1] Francesco Severi (1932) studiedthese varieties, and they are also named after Richard Brauer because of their close relation to the Brauer group.In dimension one, the Severi–Brauer varieties are conics. The corresponding central simple algebras are the quaternionalgebras. The algebra (a,b)K corresponds to the conic C(a,b) with equation

z2 = ax2 + by2

and the algebra (a,b)K splits, that is, (a,b)K is isomorphic to a matrix algebra over K, if and only if C(a,b) has a pointdefined over K: this is in turn equivalent to C(a,b) being isomorphic to the projective line over K.[1][2]

Such varieties are of interest not only in diophantine geometry, but also in Galois cohomology. They represent (atleast if K is a perfect field) Galois cohomology classes in

H1(PGLn)

in the projective linear group, where n is the dimension of V. There is a short exact sequence

1 → GL1 → GLn→ PGLn→ 1

of algebraic groups. This implies a connecting homomorphism

H1(PGLn) → H2(GL1)

at the level of cohomology. Here H2(GL1) is identified with the Brauer group of K, while the kernel is trivial because

H1(GLn) = 1

by an extension of Hilbert’s Theorem 90.[3][4] Therefore the Severi–Brauer varieties can be faithfully represented byBrauer group elements, i.e. classes of central simple algebras.Lichtenbaum showed that if X is a Severi–Brauer variety over K then there is an exact sequence

0 → Pic(X) → Z δ→Br(K) → Br(K)(X) → 0 .

Here the map δ sends 1 to the Brauer class corresponding to X.[2]

As a consequence, we see that if the class of X has order d in the Brauer group then there is a divisor class of degreed on X. The associated linear system defines the d-dimensional embedding of X over a splitting field L.[5]

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63.1 References[1] Jacobson (1996) p.113

[2] Gille & Szamuely (2006) p.129

[3] Gille & Szamuely (2006) p.26

[4] Berhuy, Grégory (2010), An Introduction to Galois Cohomology and its Applications, LondonMathematical Society LectureNote Series 377, Cambridge University Press, p. 113, ISBN 0-521-73866-0, Zbl 1207.12003

[5] Gille & Szamuely (2006) p.131

• Artin, Michael (1982), “Brauer-Severi varieties”, Brauer groups in ring theory and algebraic geometry (Wilrijk,1981), Lecture Notes in Math. 917, Notes by A. Verschoren, Berlin, New York: Springer-Verlag, pp. 194–210, doi:10.1007/BFb0092235, ISBN 978-3-540-11216-7, MR 657430, Zbl 0536.14006

• Hazewinkel, Michiel, ed. (2001), “Brauer–Severi variety”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Gille, Philippe; Szamuely, Tamás (2006), “Severi–Brauer varieties”, Central Simple Algebras and Galois Coho-mology, Cambridge Studies in Advanced Mathematics 101, Cambridge University Press, pp. 114–134, ISBN0-521-86103-9, MR 2266528, Zbl 1137.12001

• Jacobson, Nathan (1996), Finite-dimensional division algebras over fields, Berlin: Springer-Verlag, ISBN 3-540-57029-2, Zbl 0874.16002

• Saltman, David J. (1999), Lectures on division algebras, Regional Conference Series in Mathematics 94, Prov-idence, RI: American Mathematical Society, ISBN 0-8218-0979-2, Zbl 0934.16013

• Severi, Francesco (1932), “Un nuovo campo di ricerche nella geometria sopra una superficie e sopra una varietàalgebrica”, Memorie della Reale Accademia d'Italia (in Italian) 3 (5), Reprinted in volume 3 of his collectedworks

63.2 Further reading• Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions,Colloquium Publications 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, ISBN0-8218-0904-0, MR MR1632779, Zbl 0955.16001

63.3 External links• Expository paper on Galois descent (PDF)

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Chapter 64

Siegel parabolic subgroup

In mathematics, the Siegel parabolic subgroup, named after Carl Ludwig Siegel, is the parabolic subgroup of thesymplectic group with abelian radical, given by the matrices of the symplectic group whose lower left quadrant is 0(for the standard symplectic form).

64.1 References

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Chapter 65

Spaltenstein variety

In algebraic geometry, a Spaltenstein variety is a variety given by the fixed point set of a nilpotent transformationon a flag variety. They were introduced by Nicolas Spaltenstein (1976, 1982). In the special case of full flag varietiesthe Spaltenstein varieties are Springer varieties.

65.1 References• Spaltenstein, N. (1976), “The fixed point set of a unipotent transformation on the flag manifold”, IndagationesMathematicae 38 (5): 452–456, MR 0485901

• Spaltenstein, Nicolas (1982), Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics 946,Berlin, New York: Springer-Verlag, ISBN 978-3-540-11585-4, MR 672610

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Chapter 66

Special group (algebraic group theory)

In the theory of algebraic groups, a special group is a linear algebraic group G with the property that every principalG-bundle is locally trivial in the Zariski topology. Special groups include the general linear group, the special lineargroup, and the symplectic group. Special groups are necessarily connected. Products of special groups are special.The projective linear group is not special. There exist Azumaya algebras, which are trivial over a finite separableextension, but not over the base field.

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Chapter 67

Springer resolution

In mathematics, the Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Liealgebra, or the unipotent elements of a reductive algebraic group, introduced by Springer (1969). The fibers of thisresolution are called Springer fibers.If U is the variety of unipotent elements in a reductive group G, and X the variety of Borel subgroups B, then theSpringer resolution of U is the variety of pairs (u,B) of U×X such that u is in the Borel subgroup B. The map to U isthe projection to the first factor. The Springer resolution for Lie algebras is similar, except that U is replaced by thenilpotent elements of the Lie algebra of G and X replaced by the variety of Borel subalgebrasTheGrothendieck–Springer resolution is defined similarly, except that U is replaced by the whole group G (or thewhole Lie algebra of G). When restricted to the unipotent elements of G it becomes the Springer resolution.

67.1 References• Chriss, Neil; Ginzburg, Victor (1997), Representation theory and complex geometry, Boston, MA: BirkhäuserBoston, Inc., ISBN 0-8176-3792-3, MR 1433132

• Dolgachev, I.; Goldstein, N. (1984), “On the Springer resolution of the minimal unipotent conjugacy class”, J.Pure Appl. Algebra 32 (1): 33–47, doi:10.1016/0022-4049(84)90012-4, MR 0739636

• Ginzburg, Victor (1998), “Geometric methods in the representation theory of Hecke algebras and quantumgroups”, Representation theories and algebraic geometry (Montreal, PQ, 1997), NATO Advanced Science In-stitutes Series C: Mathematical and Physical Sciences 514, Kluwer Acad. Publ., Dordrecht, pp. 127–183,arXiv:math/9802004, ISBN 0-7923-5193-2, MR 1649626

• Springer, T. A. (1969), “The unipotent variety of a semi-simple group”, Algebraic Geometry (Internat. Colloq.,Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, pp. 373–391, ISBN 978-0-19-635281-7,MR 0263830

• Springer, T. A. (1976), “Trigonometric sums, Green functions of finite groups and representations of Weylgroups”, Invent. Math. 36: 173–207, doi:10.1007/BF01390009, MR 0442103

• Steinberg, Robert (1974), Conjugacy classes in algebraic groups., Lecture Notes in Mathematics 366, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0067854, ISBN 978-3-540-06657-6, MR 0352279

• Steinberg, Robert (1976), “On the desingularization of the unipotent variety”, Invent. Math. 36: 209–224,doi:10.1007/BF01390010, MR 0430094

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Chapter 68

Steinberg representation

In mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by St, is aparticular linear representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional sign representation ε of a Coxeter or Weyl group that takes all reflectionsto –1.For groups over finite fields, these representations were introduced by Robert Steinberg (1951, 1956, 1957), firstfor the general linear groups, then for classical groups, and then for all Chevalley groups, with a construction thatimmediately generalized to the other groups of Lie type that were discovered soon after by Steinberg, Suzuki andRee. Over a finite field of characteristic p, the Steinberg representation has degree equal to the largest power of pdividing the order of the group.The Steinberg representation is the Alvis–Curtis dual of the trivial 1-dimensional representation.Matsumoto (1969), Shalika (1970), and Harish-Chandra (1973) defined analogous Steinberg representations (some-times called special representations) for algebraic groups over local fields.

68.1 The Steinberg representation of a finite group• The character value of St on an element g equals, up to sign, the order of a Sylow subgroup of the centralizerof g if g has order prime to p, and is zero if the order of g is divisible by p.

• The Steinberg representation is equal to an alternating sum over all parabolic subgroups containing a Borelsubgroup, of the representation induced from the identity representation of the parabolic subgroup.

• The Steinberg representation is both regular and unipotent, and is the only irreducible regular unipotent rep-resentation (for the given prime p).

• The Steinberg representation is used in the proof of Haboush’s theorem (the Mumford conjecture).

Most finite simple groups have exactly one Steinberg representation. A few have more than one because they aregroups of Lie type in more than one way. For symmetric groups (and other Coxeter groups) the sign representationis analogous to the Steinberg representation. Some of the sporadic simple groups act as doubly transitive permutationgroups so have a BN-pair for which one can define a Steinberg representation, but for most of the sporadic groupsthere is no known analogue of it.

68.2 The Steinberg representation of a p-adic group

Matsumoto (1969), Shalika (1970), and Harish-Chandra (1973) introduced Steinberg representations for algebraicgroups over local fields. Casselman (1973) showed that the different ways of defining Steinberg representationsare equivalent. Borel & Serre (1976) and Borel (1976) showed how to realize the Steinberg representation in thecohomology group Hlc(X) of the Bruhat–Tits building of the group.

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170 CHAPTER 68. STEINBERG REPRESENTATION

68.3 References• Borel, Armand (1976), “Admissible representations of a semi-simple group over a local field with vectorsfixed under an Iwahori subgroup”, Inventiones Mathematicae 35: 233–259, doi:10.1007/BF01390139, ISSN0020-9910, MR 0444849

• Borel, Armand; Serre, Jean-Pierre (1976), “Cohomologie d'immeubles et de groupes S-arithmétiques”, Topology.an International Journal of Mathematics 15 (3): 211–232, doi:10.1016/0040-9383(76)90037-9, ISSN 0040-9383, MR 0447474

• Finite Groups of Lie Type: Conjugacy Classes and Complex Characters (Wiley Classics Library) by Roger W.Carter, John Wiley & Sons Inc; New Ed edition (August 1993) ISBN 0-471-94109-3

• Casselman, W. (1973), “The Steinberg character as a true character”, in Moore, Calvin C., Harmonic analysison homogeneous spaces (Williams Coll., Williamstown, Mass., 1972), Proc. Sympos. Pure Math. XXVI,Providence, R.I.: American Mathematical Society, pp. 413–417, ISBN 978-0-8218-1426-0, MR 0338273

• Harish-Chandra (1973), “Harmonic analysis on reductive p-adic groups”, in Moore, Calvin C., Harmonicanalysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass.,1972), Proc. Sympos. Pure Math. XXVI, Providence, R.I.: American Mathematical Society, pp. 167–192,ISBN 978-0-8218-1426-0, MR 0340486

• Matsumoto, Hideya (1969), “Fonctions sphériques sur un groupe semi-simple p-adique”, Comptes Rendus Heb-domadaires des Séances de l'Académie des Sciences. Séries a et B 269: A829––A832, ISSN 0151-0509, MR0263977

• Shalika, J. A. (1970), “On the space of cusp forms of a P-adic Chevalley group”, Annals of Mathematics.Second Series 92 (2): 262–278, doi:10.2307/1970837, ISSN 0003-486X, JSTOR 1970837, MR 0265514

• Steinberg, Robert (2001), “Steinbergmodule”, in Hazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

• Steinberg, Robert (1951), “A geometric approach to the representations of the full linear group over a Galoisfield”, Transactions of the American Mathematical Society 71 (2): 274–282, doi:10.1090/S0002-9947-1951-0043784-0, ISSN 0002-9947, JSTOR 1990691, MR 0043784

• Steinberg, Robert (1956), “Prime power representations of finite linear groups”, Canadian Journal of Mathe-matics 8: 580–591, doi:10.4153/CJM-1956-063-3, ISSN 0008-414X, MR 0080669

• Steinberg, R. (1957), “Prime power representations of finite linear groups II”, Canad. J. Math. 9: 347–351,doi:10.4153/CJM-1957-041-1

• R. Steinberg, Collected Papers, Amer. Math. Soc. (1997) ISBN 0-8218-0576-2 pp. 580–586

• Humphreys, J.E. (1987), “The Steinberg representation”, Bull. Amer. Math. Soc. (N.S.) 16 (2): 237–263,doi:10.1090/S0273-0979-1987-15512-1, MR 876960

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Chapter 69

Superstrong approximation

Superstrong approximation is a generalisation of strong approximation in algebraic groups G, to provide “spectralgap” results. The spectrum in question is that of the Laplacian matrix associated to a family of quotients of a discretegroup Γ; and the gap is that between the first and second eigenvalues (normalisation so that the first eigenvaluecorresponds to constant functions as eigenvectors). Here Γ is a subgroup of the rational points of G, but need not be alattice: it may be a so-called thin group. The “gap” in question is a lower bound (absolute constant) for the differenceof those eigenvalues.A consequence and equivalent of this property, potentially holding for Zariski dense subgroups Γ of the special lineargroup over the integers, and in more general classes of algebraic groups G, is that the sequence of Cayley graphs forreductions Γp modulo prime numbers p, with respect to any fixed set S in Γ that is a symmetric set and generatingset, is an expander family.[1]

In this context “strong approximation” is the statement that S when reduced generates the full group of points of Gover the prime fields with p elements, when p is large enough. It is equivalent to the Cayley graphs being connected(when p is large enough), or that the locally constant functions on these graphs are constant, so that the eigenspace forthe first eigenvalue is one-dimensional. Superstrong approximation therefore is a concrete quantitative improvementon these statements.

69.1 Background

Property (τ) is an analogue in discrete group theory of Kazhdan’s property (T), and was introduced by AlexanderLubotzky.[2] For a given family of normal subgroups N of finite index in Γ, one equivalent formulation is that theCayley graphs of the groups Γ/N, all with respect to a fixed symmetric set of generators S, form an expander family.[3]Therefore superstrong approximation is a formulation of property (τ), where the subgroups N are the kernels ofreduction modulo large enough primes p.The Lubotzky–Weiss conjecture states (for special linear groups and reduction modulo primes) that an expansionresult of this kind holds independent of the choice of S. For applications, it is also relevant to have results where themodulus is not restricted to being a prime.[4]

69.2 Proofs of superstrong approximation

Results on superstrong approximation have been found using techniques on approximate subgroups, and growth ratein finite simple groups.[5]

69.3 Notes[1] (Breuillard & Oh 2014, pages x, 343)

[2] http://www.ams.org/notices/200506/what-is.pdf

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[3] Alexander Lubotzky (1 January 1994). Discrete Groups, Expanding Graphs and Invariant Measures. Springer. p. 49.ISBN 978-3-7643-5075-8.

[4] (Breuillard & Oh 2014, pages 3-4)

[5] (Breuillard & Oh 2014, page xi)

69.4 References• Breuillard, Emmanuel; Oh, Hee, eds. (2014), Thin Groups and Superstrong Approximation, Cambridge Uni-versity Press, ISBN 978-1-107-03685-7

• Matthews, C. R.; Vaserstein, L. N.; Weisfeiler, B. (1984), “Congruence properties of Zariski-dense subgroups.I.”, Proc. London Math. Soc. (3) 48 (3): 514–532, doi:10.1112/plms/s3-48.3.514, MR 0735226

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Chapter 70

Taniyama group

In mathematics, the Taniyama group is a group that is an extension of the absolute Galois group of the rationals bythe Serre group. It was introduced by Langlands (1977) using an observation by Deligne, and named after YutakaTaniyama. It was intended to be the group schemewhose representations correspond to the (hypothetical) CMmotivesover the field Q of rational numbers.

70.1 References• Deligne, Pierre; Milne, James S.; Ogus, Arthur; Shih, Kuang-yen (1982), “Langlands’s Construction of theTaniyama Group”, Hodge cycles, motives, and Shimura varieties., Lecture Notes in Mathematics 900, Berlin-New York: Springer-Verlag, doi:10.1007/978-3-540-38955-2_14, ISBN 3-540-11174-3, MR 0654325

• Langlands, R. P., “Automorphic representations, Shimura varieties, and motives. Ein Märchen”, Automorphicforms, representations and L-functions, Proc. Sympos. Pure Math. 33, pp. 205–246, ISBN 0-8218-1437-0,MR 0546619

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Chapter 71

Tannakian category

Inmathematics, a tannakian category is a particular kind ofmonoidal categoryC, equippedwith some extra structurerelative to a given field K. The role of such categories C is to approximate, in some sense, the category of linearrepresentations of an algebraic group G defined over K. A number of major applications of the theory have beenmade, or might be made in pursuit of some of the central conjectures of contemporary algebraic geometry andnumber theory.The name is taken from Tannaka–Krein duality, a theory about compact groups G and their representation theory.The theory was developed first in the school of Alexander Grothendieck. It was later reconsidered by Pierre Deligne,and some simplifications made. The pattern of the theory is that of Grothendieck’s Galois theory, which is a theoryabout finite permutation representations of groups G which are profinite groups.The gist of the theory, which is rather elaborate in detail in the exposition of Saavedra Rivano, is that the fiber functorΦ of the Galois theory is replaced by a tensor functor T from C to K-Vect. The group of natural transformationsof Φ to itself, which turns out to be a profinite group in the Galois theory, is replaced by the group (a priori only amonoid) of natural transformations of T into itself, that respect the tensor structure. This is by nature not an algebraicgroup, but an inverse limit of algebraic groups (pro-algebraic group).

71.1 Applications

The construction is used in cases where a Hodge structure or l-adic representation is to be considered in the light ofgroup representation theory. For example the Mumford–Tate group and motivic Galois group are potentially to berecovered from one cohomology group or Galois module, by means of a mediating tannakian category it generates.Those areas of application are closely connected to the theory ofmotives. Another place in which tannakian categorieshave been used is in connection with the Grothendieck–Katz p-curvature conjecture; in other words, in boundingmonodromy groups.

71.2 Formal definition

A neutral Tannakian category is a rigid abelian tensor category, such that there exists a K-tensor functor to thecategory of finite dimensional K-vector spaces that is exact and faithful.

71.3 References

• N. Saavedra Rivano, Catégories Tannakiennes, Springer LNM 265, 1972

• Pierre Deligne and J. S. Milne, Tannakian categories, inHodge Cycles, Motives, and Shimura Varieties by PierreDeligne, James S. Milne, Arthur Ogus, Kuang-yen Shih, Lecture Notes in Math. 900, Springer-Verlag, 1982,414pp. An annotated version of this article can be found on J. Milne’s homepage.

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71.3. REFERENCES 175

• Pierre Deligne, Catégories tannakiennes. In The Grothendieck Festschrift, Volume 2, 111–195. Birkhauser,1990.

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Chapter 72

Thin group (algebraic group theory)

In algebraic group theory, a thin group is a discrete Zariski-dense subgroup of G(R) that has infinite covolume,where G is a semisimple algebraic group over the reals. This is in contrast to a lattice, which is a discrete subgroupof finite covolume.The theory of “group expansion” (expander graph properties of related Cayley graphs) for particular thin groups hasbeen applied to arithmetic properties of Apollonian circles and in Zaremba’s conjecture.[1]

72.1 References[1] http://gauss.math.yale.edu/~ho2/MSRI_Bourgain.pdf

• Breuillard, Emmanuel; Oh, Hee, eds. (2014), Thin Groups and Superstrong Approximation, Cambridge Uni-versity Press, ISBN 978-1-107-03685-7

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Chapter 73

Unipotent

This article is about the algebraic term. For a biological cell having the capacity to develop into only one cell type,see Cell potency#Unipotency.

In mathematics, a unipotent element, r, of a ring, R, is one such that r − 1 is a nilpotent element; in other words, (r− 1)n is zero.In particular, a square matrix,M, is a unipotent matrix, if and only if its characteristic polynomial, P(t), is a powerof t − 1. Equivalently, M is unipotent if all its eigenvalues are 1.The term quasi-unipotentmeans that some power is unipotent, for example for a diagonalizablematrix with eigenvaluesthat are all roots of unity.In an unipotent affine algebraic group all elements are unipotent (see below for the definition of an element beingunipotent in such a group).

73.1 Unipotent algebraic groups

An element, x, of an affine algebraic group is unipotent when its associated right translation operator, rx, on the affinecoordinate ring A[G] of G is locally unipotent as an element of the ring of linear endomorphism of A[G]. (Locallyunipotent means that its restriction to any finite-dimensional stable subspace of A[G] is unipotent in the usual ringsense.)An affine algebraic group is called unipotent if all its elements are unipotent. Any unipotent algebraic group isisomorphic to a closed subgroup of the group of upper triangular matrices with diagonal entries 1, and conversely anysuch subgroup is unipotent. In particular any unipotent group is a nilpotent group, though the converse is not true(counterexample: the diagonal matrices of GLn(k)).If a unipotent group acts on an affine variety, all its orbits are closed, and if it acts linearly on a finite-dimensionalvector space then it has a non-zero fixed vector. In fact, the latter property characterizes unipotent groups.Unipotent groups over an algebraically closed field of any given dimension can in principle be classified, but in practicethe complexity of the classification increases very rapidly with the dimension, so people tend to give up somewherearound dimension 6.Over the real numbers (or more generally any field of characteristic 0) the exponential map takes any nilpotent squarematrix to a unipotent matrix. Moreover, if U is a commutative unipotent group, the exponential map induces anisomorphism from the Lie algebra of U to U itself.

73.2 Unipotent radical

The unipotent radical of an algebraic group G is the set of unipotent elements in the radical of G. It is a connectedunipotent normal subgroup of G, and contains all other such subgroups. A group is called reductive if its unipotentradical is trivial. If G is reductive then its radical is a torus.

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178 CHAPTER 73. UNIPOTENT

73.3 Jordan decomposition

Main article: Jordan–Chevalley decomposition

Any element g of a linear algebraic group over a perfect field can be written uniquely as the product g = gugs ofcommuting unipotent and semisimple elements gu and gs. In the case of the group GLn(C), this essentially says thatany invertible complex matrix is conjugate to the product of a diagonal matrix and an upper triangular one, which is(more or less) the multiplicative version of the Jordan–Chevalley decomposition.There is also a version of the Jordan decomposition for groups: any commutative linear algebraic group over a perfectfield is the product of a unipotent group and a semisimple group.

73.4 See also• unipotent representation

• Deligne–Lusztig theory

73.5 References• A. Borel, Linear algebraic groups, ISBN 0-387-97370-2

• Borel, Armand (1956), “Groupes linéaires algébriques”, Annals of Mathematics. Second Series (Annals ofMathematics) 64 (1): 20–82, doi:10.2307/1969949, JSTOR 1969949

• Popov, V.L. (2001), “unipotent element”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Popov, V.L. (2001), “unipotent group”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Suprunenko, D.A. (2001), “unipotent matrix”, in Hazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

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Chapter 74

Weil conjecture on Tamagawa numbers

In mathematics, theWeil conjecture on Tamagawa numbers is the statement that the Tamagawa number τ(G) ofa simply connected simple algebraic group defined over a number field is 1. Weil (1959) did not explicitly conjecturethis, but calculated the Tamagawa number in many cases and observed that in the cases he calculated it was aninteger, and equal to 1 when the group is simply connected. The first observation does not hold for all groups: Ono(1963) found some examples whose Tamagawa numbers are not integers. The second observation, that the Tamagawanumbers of simply connected semisimple groups seem to be 1, became known as theWeil conjecture. Several authorschecked this in many cases, and finally Kottwitz proved it for all groups in 1988.Ono (1965) used the Weil conjecture to calculate the Tamagawa numbers of all semisimple algebraic groups.Tamagawa numbers were introduced by Tamagawa (1966), and named after him by Weil (1959).Here simply connected is in the algebraic group theory sense of not having a proper algebraic covering, which is notalways the topologists’ meaning.

74.1 Tamagawa measure and Tamagawa numbers

Let k be a global field, A its ring of adeles, and G an algebraic group defined over k.The Tamagawa measure on the adelic algebraic group G(A) is defined as follows. Take a left-invariant n-form ω onG(k) defined over k, where n is the dimension of G. This induces Haar measures on G(ks) for all places of s, andhence a Haar measure on G(A), if the product over all places converges. This Haar measure on G(A) does not dependon the choice of ω, because multiplying ω by an element of k* multiplies the Haar measure on G(A) by 1, using theproduct formula for valuations.The Tamagawa number τ(G) is the Tamagawa measure of G(A)/G(k).

74.2 History

Weil checked this in enough classical group cases to propose the conjecture. In particular for spin groups it impliesthe known Smith–Minkowski–Siegel mass formula.Robert Langlands (1966) introduced harmonic analysis methods to show it for Chevalley groups. J. G. M. Mars gavefurther results during the 1960s.K. F. Lai (1980) extended the class of known cases to quasisplit reductive groups. Kottwitz (1988) proved it for allgroups satisfying the Hasse principle, which at the time was known for all groups without E8 factors. V. I. Chernousov(1989) removed this restriction, by proving the Hasse principle for the resistant E8 case (see strong approximation inalgebraic groups), thus completing the proof of Weil’s conjecture.In 2011, Jacob Lurie and Dennis Gaitsgory announced a proof of the conjecture for algebraic groups over functionfields over finite fields.Lurie (2011) Lurie (2014)

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74.3 See also• Ran space

74.4 References• Hazewinkel, Michiel, ed. (2001), “Tamagawa number”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Chernousov, V. I. (1989), “The Hasse principle for groups of type E8”, Soviet Math. Dokl. 39: 592–596, MR1014762

• Kottwitz, Robert E. (1988), “Tamagawa numbers”, Ann. Of Math. (2) (Annals of Mathematics) 127 (3):629–646, doi:10.2307/2007007, JSTOR 2007007, MR 0942522.

• Lai, K. F. (1980), “Tamagawa number of reductive algebraic groups”, Compositio Mathematica 41 (2): 153–188, MR 581580

• Langlands, R. P. (1966), “The volume of the fundamental domain for some arithmetical subgroups of Chevalleygroups”, Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., Providence, R.I.: Amer.Math. Soc., pp. 143–148, MR 0213362

• Ono, Takashi (1963), “On the Tamagawa number of algebraic tori”, Annals of Mathematics. Second Series 78:47–73, doi:10.2307/1970502, ISSN 0003-486X, MR 0156851

• Ono, Takashi (1965), “On the relative theory of Tamagawa numbers”, Annals of Mathematics. Second Series82: 88–111, doi:10.2307/1970563, ISSN 0003-486X, MR 0177991

• Tamagawa, Tsuneo (1966), “Adèles”, Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. PureMath. IX, Providence, R.I.: American Mathematical Society, pp. 113–121, MR 0212025

• Voskresenskii, V. E. (1991), Algebraic Groups and their Birational Invariants, AMS translation

• Weil, André (1959), Exp. No. 186, Adèles et groupes algébriques, Séminaire Bourbaki 5, pp. 249–257

• Weil, André (1982) [1961], Adeles and algebraic groups, Progress inMathematics 23, Boston, MA: BirkhäuserBoston, ISBN 978-3-7643-3092-7, MR 670072

• Lurie, Jacob (2011), Tamagawa Numbers via Nonabelian Poincaré Duality

• Lurie, Jacob (2014), Tamagawa Numbers via Nonabelian Poincaré Duality

74.5 Further reading• Aravind Asok, Brent Doran and Frances Kirwan, “Yang-Mills theory and Tamagawa Numbers: the fascinationof unexpected links in mathematics”, February 22, 2013

• J. Lurie, The Siegel Mass Formula, Tamagawa Numbers, and Nonabelian Poincaré Duality posted June 8,2012.

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Chapter 75

Weyl module

In algebra, aWeyl module is a representation of a reductive algebraic group, introduced by Carter and Lusztig (1974,1974b) and named after Hermann Weyl. In characteristic 0 these representations are irreducible, but in positivecharacteristic they can be reducible, and their decomposition into irreducible components can be hard to determine.

75.1 Further reading• Carter, RogerW.; Lusztig, George (1974), “On themodular representations of the general linear and symmetricgroups”,Mathematische Zeitschrift 136: 193–242, doi:10.1007/BF01214125, ISSN 0025-5874, MR 0354887

• Carter, Roger W.; Lusztig, G. (1974b), “On the modular representations of the general linear and symmetricgroups”, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ.,Canberra, 1973), Lecture Notes in Mathematics 372, Berlin, New York: Springer-Verlag, pp. 218–220,doi:10.1007/BFb0065172, MR 0369503

• Dipper, R. (2001), “Weyl_module”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

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Chapter 76

Witt vector

In mathematics, aWitt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how toput a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of orderp is the ring of p-adic integers.

76.1 Motivation

Any p -adic integer (an element of Zp ) can be written as a power series a0 + a1p1 + a2p

2 + · · · , where the a 's areusually taken from the set 0, 1, 2, ..., p − 1 . However, it is hard to figure out an algebraic expression for additionand multiplication, as one faces the problem of carry. Luckily, this set of representatives is not the only possiblechoice, and Teichmüller suggested an alternative set consisting of 0 together with the p − 1 st roots of 1 : in otherwords, the p roots of

xp − x = 0 in Zp .

These Teichmüller representatives can be identified with the elements of the finite field Fp of order p (by takingresidues modulo p ), and elements of F×

p are taken to their representatives by the Teichmüller character ω : F×p → Z×

p

. This identifies the set of p -adic integers with infinite sequences of elements of ω(F×p ) ∪ 0 .

We now have the following problem: given two infinite sequences of elements of ω(F×p ) ∪ 0 , describe their sum

and product as p -adic integers explicitly. This problem was solved by Witt using Witt vectors.

76.1.1 Details

We basically want to derive the ring p -adic integers Zp from the finite field with p elements, Fp , by some generalconstruction.A p -adic integer is a sequence (n0, n1, ...) with ni ∈ Z/p(i+1)Z ,such that ni ≡ nj mod pi if i < j . Theycan be expanded as a power series in p : a0 + a1p

1 + a2p2 + · · · , where the a 's are usually taken from the set

0, 1, 2, ..., p− 1 (The equation is happening in Zp , with ai and pj all images from Z to Zp ). Set-theoretically itis Fp . But Zp and

∏N Fp are not isomorphic as rings. If we denote a+ b = c , then the addition should instead be:

c0 ≡ a0 + b0 mod p

c0 + c1p ≡ a0 + a1p+ b0 + b1p mod p2

c0 + c1p+ c2p2 ≡ a0 + a1p+ a2p

2 + b0 + b1p+ b2p2 mod p3

But we lack some properties of the coefficients to produce a general formula.Luckily, there is an alternative subset of Zp which can work as the coefficient set. This is the set of Teichmüllerrepresentatives of elements of Fp . Without 0 they form a subgroup ofZ∗

p , identified with F∗p through the Teichmüller

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76.2. CONSTRUCTION OF WITT RINGS 183

character ω : F∗p → Z∗

p . Note that ω is not additive, as the sum need not be a representative. Despite this, ifω(k) = ω(i) + ω(j) mod p in Zp , then i + j = k in Fp . This is conceptually justified by m ω = idFp if wedenotem : Zp → Zp/pZp

∼= Fp .Teichmüller representatives are explicitly calculated as roots of xp−1 − 1 = 0 through Hensel lifting. For example,in Z3 , to calculate the representative of 2 , you first find the unique solution of x2 − 1 = 0 in Z/9Z with x ≡ 2mod 3 ; you get 8 , then repeat it in Z/27Z , with conditions x2 − 1 = 0 and x ≡ 2 mod 9 ; this time it is 26 , andso on. The existence of lift in each step is guaranteed by (xp−1 − 1, (p− 1)xp−2) = 1 in every Z/pnZ .We can also write the representatives as a0 + a1p

1 + a2p2 + ... . Note for every j ∈ 0, 1, 2, ..., p − 1 , there is

exactly one representative, namely ω(j) , with a0 = j , so we can also expand every p -adic integer as a power seriesin p , with coefficients from the Teichmüller representatives.Explicitly, if b = a0 + a1p

1 + a2p2 + ... , then b − ω(a0) = a′1p

1 + a′2p2 + ... . Then you subtract ω(a′1)p and

proceed similarly. Note the coefficients you get most probably differ from the ai 's modulo p , except the first one.This time we have additional properties of the coefficients like api = ai , so we can make some changes to get a neatformula. Since the Teichmüller character is not additive, we don't have c0 = a0 + b0 in Zp . But it happens in Fp ,as the first congruence implies. So actually cp0 ≡ (a0 + b0)

p mod p2 , thus c0 − a0 − b0 ≡ (a0 + b0)p − a0 − b0 ≡(

p1

)ap−10 b0 + ... +

(p1

)a0b

p−10 mod p2 . Since

(pi

)is divisible by p , this resolves the p -coefficient problem of c1

and gives c1 ≡ a1 + b1 − ap−10 b0 − p−1

2 ap−20 b20 − ...− a0b

p−10 mod p . Note this completely determines c1 by the

lift. Moreover, the mod p indicates that the calculation can actually be done in Fp , satisfying our basic aim.Now for c2 . It is already very cumbersome at this step. c1 = cp1 ≡ (a1+b1−ap−1

0 b0− p−12 ap−2

0 b20− ...−a0bp−10 )p

mod p . As for c0 , a single p th power is not enough: actually we take c0 = cp2

0 ≡ (a0 + b0)p2 .

(p2

i

)is not always

divisible by p2 , but that only happens when i = pd , in which case aibp2−i = adbp−d combined with similarmonomials in cp1 would make a multiple of p2 .At this step, we see that we are actually working with something like

c0 ≡ a0 + b0 mod p

cp0 + c1p ≡ ap0 + a1p+ bp0 + b1p mod p2

cp2

0 + cp1p+ c2p2 ≡ ap

2

0 + ap1p+ a2p2 + bp

2

0 + bp1p+ b2p2 mod p3

This motivates the definition of Witt vectors.

76.2 Construction of Witt rings

Fix a prime number p. AWitt vector over a commutative ring R is a sequence : (X0, X1, X2, ...) of elements of R.Define theWitt polynomialsWi by

1. W0 = X0

2. W1 = Xp0 + pX1

3. W2 = Xp2

0 + pXp1 + p2X2

and in general

Wn =∑i

piXpn−i

i .

(W0,W1,W2, ...) is called the ghost components of the Witt vector (X0, X1, X2, ...) , and is usually denoted by(X(0), X(1), X(2), ...) .Then Witt showed that there is a unique way to make the set of Witt vectors over any commutative ring R into a ring,called the ring of Witt vectors, such that

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184 CHAPTER 76. WITT VECTOR

• the sum and product are given by polynomials with integral coefficients that do not depend on R, and

• Every Witt polynomial is a homomorphism from the ring of Witt vectors over R to R.

In other words, if

• (X + Y )i and (XY )i are given by polynomials with integral coefficients that do not depend on R, and

• X(i) + Y (i) = (X + Y )(i) , X(i)Y (i) = (XY )(i) .

The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example,

• (X0, X1, ...) + (Y0, Y1, ...) = (X0 + Y0, X1 + Y1 + (Xp0 + Y p

0 − (X0 + Y0)p)/p, ...)

• (X0, X1, ...)× (Y0, Y1, ...) = (X0Y0, Xp0Y1 +X1Y

p0 + pX1Y1, ...) .

76.3 Examples

• The Witt ring of any commutative ring R in which p is invertible is just isomorphic to RN (the product of acountable number of copies of R). In fact the Witt polynomials always give a homomorphism from the ring ofWitt vectors to RN, and if p is invertible this homomorphism is an isomorphism.

• The Witt ring of the finite field of order p is the ring of p-adic integers, as is demonstrated above.

• The Witt ring of a finite field of order pn is the unramified extension of degree n of the ring of p-adic integers.

76.4 Universal Witt vectors

The Witt polynomials for different primes p are special cases of universal Witt polynomials, which can be used toform a universal Witt ring (not depending on a choice of prime p). Define the universal Witt polynomials Wn forn≥1 by

1. W1 = X1

2. W2 = X21 + 2X2

3. W3 = X31 + 3X3

4. W4 = X41 + 2X2

2 + 4X4

and in general

Wn =∑d|n

dXn/dd .

Again, (W1,W2,W3, ...) is called the ghost components of theWitt vector (X1, X2, X3, ...) , and is usually denotedby (X(1), X(2), X(3), ...) .We can use these polynomials to define the ring of universal Witt vectors over any commutative ring R in much thesame way as above (so the universal Witt polynomials are all homomorphisms to the ring R).

76.5 Generating Functions

Later Witt orally stated another approach using generating functions.[1]

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76.5. GENERATING FUNCTIONS 185

76.5.1 Definition

Let X be a Witt vector and define

fX(t) =∏n≥1

(1−Xntn) =

∑n≥0

Antn

For n ≥ 1 let Sn denote the collection of subsets of 1, 2, ..., n whose elements add up to n . Then An =∑S∈S(−1)|S|∑

i∈S Xi .We can get the ghost components by taking the logarithmic derivative:

d

dtlog fX(t) =

∑n≥1

d

dt(1−Xnt

n) = −∑n≥1

∑d≥1

Xdnt

nd

d= −

∑m≥1

∑d|m

md X

dmd

mtm = −

∑m≥1

X(m)tm

m

76.5.2 Sum

Now we can see fZ(t) = fX(t)fY (t) ifZ = X+Y . So thatCn =∑

0≤i≤nAnBn−i ifAn, Bn, Cn are respectivecoefficients in the power series for fX(t), fY (t), fZ(t) . ThenZn =

∑0≤i≤nAnBn−i−

∑S∈S,S =n(−1)|S|∑

i∈S Zi

. Since An is a polynomial in X1, ..., Xn and likely for Bn , we can show by induction that Zn is a polynomial inX1, ..., Xn, Y1, ..., Yn .

76.5.3 Product

If we setW = XY then

d

dtlog fW (t) = −

∑m≥1

X(m)Y (m)tm

m

But

∑m≥1

X(m)Y (m)

mtm =

∑m≥1

∑d|m dX

m/dd

∑e|m eY

m/ee

mtm

Now 3-tuples m, d, e with m ∈ Z+, d|m, e|m are in bijection with 3-tuples d, e, n with d, e, n ∈ Z+ , via n =m/[d, e] ( [d, e] is the Least common multiple), our series becomes

∑d,e≥1

de[d,e]

∑n≥1(X

[d,e]/dd Y

[d,e]/ee t[d,e])n

n

So that

fW (t) =∏

d,e≥1

(1−X[d,e]/dd Y [d,e]/e

e t[d,e])de/[d,e] =∑n≥0

Dntn

whereDn s are polynomials ofX1, ..., Xn, Y1, ..., Yn . So by similar induction, suppose fW (t) =∏

n≥1(1−Wntn)

, thenWn can be solved as polynomials of X1, ..., Xn, Y1, ..., Yn .

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186 CHAPTER 76. WITT VECTOR

76.6 Ring schemes

The map taking a commutative ring R to the ring of Witt vectors over R (for a fixed prime p) is a functor fromcommutative rings to commutative rings, and is also representable, so it can be thought of as a ring scheme, calledthe Witt scheme, over Spec(Z). The Witt scheme can be canonically identified with the spectrum of the ring ofsymmetric functions.Similarly the rings of truncated Witt vectors, and the rings of universal Witt vectors, correspond to ring schemes,called the truncated Witt schemes and the universal Witt scheme.Moreover, the functor taking the commutative ring R to the set Rn is represented by the affine space An

Z , and thering structure on Rn makes An

Z into a ring scheme denoted On . From the construction of truncated Witt vectors itfollows that their associated ring schemeWn is the schemeAn

Z with the unique ring structure such that the morphismWn → On given by the Witt polynomials is a morphism of ring schemes.

76.7 Commutative unipotent algebraic groups

Over an algebraically closed field of characteristic 0, any unipotent abelian connected algebraic group is isomorphic toa product of copies of the additive groupGa . The analogue of this for fields of characteristic p is false: the truncatedWitt schemes are counterexamples. (We make them into algebraic groups by forgetting the multiplication and justusing the additive structure.) However these are essentially the only counterexamples: over an algebraically closedfield of characteristic p, any unipotent abelian connected algebraic group is isogenous to a product of truncated Wittgroup schemes.

76.8 See also• Formal group

• Artin–Hasse exponential

76.9 References[1] Lang, Serge (September 19, 2005). “Chapter VI: Galois Theory”. Algebra (3rd ed.). Springer. p. 330. ISBN 978-0-387-

95385-4.

• Dolgachev, I.V. (2001), “Witt vector”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Hazewinkel, Michiel (2009), “Witt vectors. I.”, Handbook of algebra. Vol. 6, Amsterdam: Elsevier/North-Holland, pp. 319–472, arXiv:0804.3888, ISBN 978-0-444-53257-2, MR 2553661

• Mumford, David, Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies 59, Princeton,NJ: Princeton University Press, ISBN 978-0-691-07993-6

• Serre, Jean-Pierre (1979), Local fields, Graduate Texts inMathematics 67, Berlin, NewYork: Springer-Verlag,ISBN 978-0-387-90424-5, MR 554237, section II.6

• Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics 117, Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-96648-9, MR 918564

• Witt, Ernst (1936), “Zyklische Körper und Algebren der Characteristik p vom Grad pn. Struktur diskretbewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik pn", Journal für Reineund Angewandte Mathematik (in German) 176: 126–140, doi:10.1515/crll.1937.176.126

• Greenberg, M. J. (1969), Lectures on Forms in Many Variables, New York and Amsterdam, Benjamin, MR241358, ASIN: B0006BX17M

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Chapter 77

Wonderful compactification

In algebraic group theory, a wonderful compactification of a variety acted on by an algebraic group G is a G-equivariant compactification such that the closure of each orbit is smooth. C. De Concini and C. Procesi (1983)constructed a wonderful compactification of any symmetric variety given by a quotient G/Gσ of an algebraic group Gby the subgroup Gσ fixed by some involution σ of G over the complex numbers, sometimes called the De Concini–Procesi compactification, and Strickland (1987) generalized this to arbitrary characteristic. In particular, by writinga group G itself as a symmetric homogeneous space G=(G×G)/G (modulo the diagonal subgroup) this gives a won-derful compactification of the group G itself.

77.1 References• De Concini, C.; Procesi, C. (1983), “Complete symmetric varieties”, in Gherardelli, Francesco, Invarianttheory (Montecatini, 1982), Lecture Notes in Mathematics 996, Berlin, New York: Springer-Verlag, pp. 1–44,doi:10.1007/BFb0063234, ISBN 978-3-540-12319-4, MR 718125

• Evens, Sam; Jones, Benjamin F. (2008), On the wonderful compactification, Lecture notes

• Springer, Tonny A. (2006), “Some results on compactifications of semisimple groups”, International Congressof Mathematicians. Vol. II, Eur. Math. Soc., Zürich, pp. 1337–1348, MR 2275648

• Strickland, Elisabetta (1987), “A vanishing theorem for group compactifications”,Mathematische Annalen 277(1): 165–171, doi:10.1007/BF01457285, ISSN 0025-5831, MR 884653

187

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Chapter 78

Étale group scheme

In mathematics, more precisely in algebra, an étale group scheme is a certain kind of group scheme.

78.1 Definition

A finite group scheme G over a field K is called an étale group scheme if it is represented by an étale K-algebra R ,i.e. if R⊗K K is isomorphic to K × ...× K .

188

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78.2. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 189

78.2 Text and image sources, contributors, and licenses

78.2.1 Text• (B, N) pair Source: https://en.wikipedia.org/wiki/(B%2C_N)_pair?oldid=622690924 Contributors: Michael Hardy, Charles Matthews,

Giftlite, Linas, R.e.b., SmackBot, Jon Awbrey, Cronholm144, Mathsci, Redrocket, JackSchmidt, Addbot, Roentgenium111, EmausBot,L'altro giocoliere, Spectral sequence, Mobrie06, Teddyktchan and Anonymous: 4

• Adelic algebraic group Source: https://en.wikipedia.org/wiki/Adelic_algebraic_group?oldid=643060864Contributors: Edward,MichaelHardy, TakuyaMurata, Charles Matthews, Drbreznjev, Oleg Alexandrov, R.e.b., SmackBot, Bluebot, CRGreathouse, Headbomb, Rob-Har, Deflective, Ahuskay, JackSchmidt, Alecobbe, Addbot, Roentgenium111, Wohingenau, Omnipaedista, Concord113, Deltahedronand Anonymous: 10

• Algebraic group Source: https://en.wikipedia.org/wiki/Algebraic_group?oldid=670511317 Contributors: AxelBoldt, Michael Hardy,TakuyaMurata, Charles Matthews, Dysprosia, Giftlite, Fropuff, Waltpohl, Vivacissamamente, Paul August, Zaslav, Linas, R.e.b., Wave-length, Crasshopper, Ppntori, Nbarth, Lesnail, JoergWinkelmann~enwiki, Cronholm144, Jim.belk,Michael Kinyon, Krasnoludek, BprsoltQaoddz, Thijs!bot, Headbomb, Turgidson, Jakob.scholbach, David Eppstein, LokiClock, Hesam7, JackSchmidt, Mr. Stradivarius, Dea-conJohnFairfax, Alexbot, Addbot, Luckas-bot, FrescoBot, LucienBOT, Artem M. Pelenitsyn, ChuispastonBot, Mgvongoeden, Yasinza-ehringer, Mark viking, CsDix, Jodosma, Danneks, K9re11 and Anonymous: 12

• Algebraic torus Source: https://en.wikipedia.org/wiki/Algebraic_torus?oldid=656029552Contributors: SimonP,Michael Hardy, CharlesMatthews, Rvollmert, Giftlite, BD2412, Chenxlee, MZMcBride, Masnevets, SmackBot, TimBentley, S.H.C., Gwern, Reedy Bot, Loki-Clock, Kreizhn, Stca74, Henry Delforn (old), AnomieBOT, D.Lazard and Anonymous: 5

• Approximation in algebraic groups Source: https://en.wikipedia.org/wiki/Approximation_in_algebraic_groups?oldid=637681597Con-tributors: Michael Hardy, CharlesMatthews, GregorB,MZMcBride, R.e.b., Headbomb, Turgidson, ReedyBot, Roentgenium111, AnomieBOT,D.Lazard, Brirush and Anonymous: 1

• Arason invariant Source: https://en.wikipedia.org/wiki/Arason_invariant?oldid=635370630Contributors: R.e.b., David Eppstein, Rjwilm-siBot, Wcherowi, Deltahedron, Spectral sequence, K9re11 and CaptainLama

• Arithmetic group Source: https://en.wikipedia.org/wiki/Arithmetic_group?oldid=636761153 Contributors: Michael Hardy, CharlesMatthews, Gauge, R.e.b., RussBot, SmackBot, Bluebot, E946, Jim.belk, Dedd-morozz, Turgidson, Hesam7, Arcfrk, Crazy1880, Erik9bot,Suhagja, Brirush and Anonymous: 2

• Borel subgroup Source: https://en.wikipedia.org/wiki/Borel_subgroup?oldid=659850603 Contributors: AxelBoldt, Charles Matthews,Giftlite, Fropuff, Curps, Rich Farmbrough, Cedders, Salix alba, Ligulem, R.e.b., Mathbot, Abu Amaal, Wavelength, SmackBot, Nbarth,Ligulembot, Ulner, Mathsci, David Eppstein, TubularWorld, Nilradical, Roentgenium111, Citation bot, Tbunke, RjwilmsiBot, Café Bene,Monkbot and Anonymous: 11

• Borel–de Siebenthal theory Source: https://en.wikipedia.org/wiki/Borel%E2%80%93de_Siebenthal_theory?oldid=670634335 Con-tributors: Salix alba, Bgwhite, Wavelength, Mathsci, Bob1960evens, Yobot, Dijkschneier, John of Reading, Ceradon and Anonymous:3

• Bott–Samelson variety Source: https://en.wikipedia.org/wiki/Bott%E2%80%93Samelson_variety?oldid=670998635Contributors: Takuya-Murata, Rjwilmsi, R.e.b., John Baez, Dea13, Yobot, AnomieBOT, Trappist the monk and Anonymous: 1

• Bruhat decomposition Source: https://en.wikipedia.org/wiki/Bruhat_decomposition?oldid=646851784Contributors: CharlesMatthews,Giftlite, R.e.b., Masnevets, Nbarth, Ulner, Leyo, JackSchmidt, Addbot, Luckas-bot, Yobot, Omnipaedista, Negi(afk) and Anonymous: 4

• Cartan subgroup Source: https://en.wikipedia.org/wiki/Cartan_subgroup?oldid=607354783Contributors: Tango, TakuyaMurata, CharlesMatthews, Momotaro, Zetawoof, R.e.b., SmackBot, CBM, Dimple83, David Eppstein, Werieth, Francis digeon and Monkbot

• Chevalley’s structure theorem Source: https://en.wikipedia.org/wiki/Chevalley’s_structure_theorem?oldid=647529574Contributors:Giftlite, Rjwilmsi, R.e.b., Magioladitis, WQUlrich, Trappist the monk, Deltahedron, K9re11 and Anonymous: 1

• Cohomological invariant Source: https://en.wikipedia.org/wiki/Cohomological_invariant?oldid=666916939Contributors: Giftlite, R.e.b.,Guy Macon, Deltahedron and K9re11

• Complexification (Lie group) Source: https://en.wikipedia.org/wiki/Complexification_(Lie_group)?oldid=641249517Contributors: MichaelHardy, TakuyaMurata, Rjwilmsi, Wavelength, Mathsci, JaGa, Malinaccier, YohanN7, Yobot, John of Reading, Stal potaten and Anony-mous: 1

• Cuspidal representation Source: https://en.wikipedia.org/wiki/Cuspidal_representation?oldid=597581810Contributors: CharlesMatthews,Trace (usurped), RJFJR, Koavf, Salix alba, Shell Kinney, SmackBot, BlackFingolfin, KPWM Spotter, RobHar, STBot, MystBot, Addbot,GoingBatty and Anonymous: 1

• Diagonalizable group Source: https://en.wikipedia.org/wiki/Diagonalizable_group?oldid=536704945Contributors: Michael Hardy, Takuya-Murata, Bearcat, Malcolma, Dan131m, ShelfSkewed, RobHar, Mild Bill Hiccup, Ozob, Slawekb, Werieth, Qetuth and FoCuSandLeArN

• Dieudonnémodule Source: https://en.wikipedia.org/wiki/Dieudonn%C3%A9_module?oldid=575146780 Contributors: Michael Hardy,Charles Matthews, Giftlite, Sam Derbyshire, R.e.b., Headbomb, Yobot, Moswento and Anonymous: 1

• Differential algebraic group Source: https://en.wikipedia.org/wiki/Differential_algebraic_group?oldid=647529911Contributors: MichaelHardy, R.e.b., Headbomb, Addbot and K9re11

• Differential Galois theory Source: https://en.wikipedia.org/wiki/Differential_Galois_theory?oldid=638925369 Contributors: MichaelHardy, GTBacchus, Charles Matthews, Wikiborg, Fredrik, Giftlite, Lethe, Jason Quinn, Shanes, EmilJ, Army1987, Woodstone, OlegAlexandrov, Joriki, Linas, Shreevatsa, XaosBits, Rjwilmsi, R.e.b., [email protected], Pmdboi, SmackBot, Nbarth, Emurphy42,Jim.belk, Goens, Paddles, Xantharius, Headbomb, Warut, Squids and Chips, VolkovBot, Strcpy~enwiki, Mike4ty4, Gamesguru2, BO-Tarate, Addbot, Ettrig, Unara, UncleSpacker, Galoa2804~enwiki, Levochik, CaroleHenson, Spectral sequence, Brirush and Anonymous:19

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190 CHAPTER 78. ÉTALE GROUP SCHEME

• E6 (mathematics) Source: https://en.wikipedia.org/wiki/E6_(mathematics)?oldid=670720552 Contributors: Zundark, Michael Hardy,Charles Matthews, Phys, Giftlite, Fropuff, Gro-Tsen, Tomruen, Almit39, Lumidek, Ukexpat, Giraffedata, Arthena, Oleg Alexandrov,BD2412, Rjwilmsi, Koavf, Salix alba, R.e.b., John Baez, Algebraist, YurikBot, Gaius Cornelius, Nbarth, Vanished User 0001, Cron-holm144, Jim.belk, Phuzion, Dan Gluck, Headbomb, .anacondabot, Magioladitis, Exceptg, Ludvikus, Rocchini, Remember the dot,Drschawrz, Ioverka, Mr. Stradivarius, Nilradical, Addbot, DOI bot, Eall Ân Ûle, Apaul00, Luckas-bot, Yobot, Jgmoxness, Citation bot,OgreBot, Trappist the monk, The tree stump, WildBot, GoingBatty, Joel B. Lewis, Helpful Pixie Bot, Bilingsley, Jimw338, Mark LMacDonald, Cjean42, CsDix, Paul2520, Monkbot, Ilia Smilga and Anonymous: 12

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• F4 (mathematics) Source: https://en.wikipedia.org/wiki/F4_(mathematics)?oldid=664720332 Contributors: Zundark, Michael Hardy,Charles Matthews, Phys, Giftlite, Fropuff, Gro-Tsen, Tomruen, Lumidek, Oleg Alexandrov, Rjwilmsi, Koavf, R.e.b., John Baez, Wave-length, Tamfang, Vanished User 0001, Daqu, Headbomb, RobHar, Exceptg, Connor Behan, Vitor 1234, Remember the dot, Drschawrz,Mr. Stradivarius, Nilradical, Muhandes, SockPuppetForTomruen, Addbot, Yobot, Amirobot, Jgmoxness, The tree stump, Akarvilhe,Paolo328, Mark L MacDonald, CsDix, Monkbot and Anonymous: 13

• Fixed-point subgroup Source: https://en.wikipedia.org/wiki/Fixed-point_subgroup?oldid=633861852 Contributors: Michael Hardy,TakuyaMurata, Bearcat, Postcard Cathy, Yobot and MaryGaulke

• Formal group Source: https://en.wikipedia.org/wiki/Formal_group?oldid=672793856 Contributors: Michael Hardy, Charles Matthews,Giftlite, Waltpohl, Gauge, Mdd, Rjwilmsi, R.e.b., Wavelength, JosephSilverman, SmackBot, Lhf, FelisLeo, SashatoBot, S.H.C., MichaelKinyon, Krasnoludek, Headbomb, Darklilac, David Eppstein, Gwern, Cardano~enwiki, Saibod, Brenont, JackSchmidt, Addbot, AnomieBOT,Udoh, Freebirth Toad, Tathagata84, HawaiianEarring, BG19bot, ChrisGualtieri, Deltahedron, Spectral sequence and Anonymous: 17

• Fundamental lemma (Langlands program) Source: https://en.wikipedia.org/wiki/Fundamental_lemma_(Langlands_program)?oldid=622465549 Contributors: Michael Hardy, Dominus, TakuyaMurata, Charles Matthews, Tobias Bergemann, Giftlite, Stevey7788, R.e.b.,Sodin, Bgwhite, Newone, Headbomb, RobHar, David Eppstein, Arcfrk, Rumping, Sławomir Biały, GoingBatty, Slawekb, Phanjuy,EdoBot, LJosil, BattyBot and Anonymous: 12

• G2 (mathematics) Source: https://en.wikipedia.org/wiki/G2_(mathematics)?oldid=664306968Contributors: Moly,Michael Hardy, CharlesMatthews, Phys, Gandalf61, Giftlite, Fropuff, Gro-Tsen, Tomruen, Lumidek, Susvolans, Oleg Alexandrov, Rjwilmsi, R.e.b., John Baez,Wavelength, Jemebius, Pred, Silly rabbit, Nbarth, Vanished User 0001, Headbomb, RobHar, Exceptg, David Eppstein, Connor Behan,Remember the dot, JohnBlackburne, Drschawrz, Cheesefondue, Mr. Stradivarius, Jjauregui, Addbot, LaaknorBot, Yobot, Jgmoxness,Citation bot 1, Cnwilliams, ZéroBot, Quondum, Vladimirdx, Paolo328, ChrisGualtieri, CsDix, Hamiltonjacobi22, Teddyktchan andAnonymous: 19

• Geometric invariant theory Source: https://en.wikipedia.org/wiki/Geometric_invariant_theory?oldid=641869024Contributors: Takuya-Murata, CharlesMatthews, Altenmann, Giftlite, Paul August, Gauge, BD2412, R.e.b., Hillman, Crasshopper, Arthur Rubin, Eigenlambda,JCSantos, RyanEberhart, RobHar, David Eppstein, LokiClock, Arcfrk, Stca74, JackSchmidt, Citation bot, Locobot, Citation bot 1,RobinK, Trappist the monk, Basemaze, D.Lazard, Brad7777, Jeremy112233, ChrisGualtieri, Enyokoyama, Mark viking and Anony-mous: 6

• Glossary of algebraic groups Source: https://en.wikipedia.org/wiki/Glossary_of_algebraic_groups?oldid=655539884Contributors: Smack-Bot, Jakob.scholbach, Plevyman, Addbot, Trappist the monk, BG19bot, Jeremy112233 and Anonymous: 1

• Good filtration Source: https://en.wikipedia.org/wiki/Good_filtration?oldid=626886550 Contributors: R.e.b., David Eppstein and Trap-pist the monk

• Grosshans subgroup Source: https://en.wikipedia.org/wiki/Grosshans_subgroup?oldid=606945261Contributors: Michael Hardy, CharlesMatthews, Vipul, NatusRoma, Srice13 and David Eppstein

• Group of Lie type Source: https://en.wikipedia.org/wiki/Group_of_Lie_type?oldid=661649624 Contributors: Michael Hardy, Kidburla,Charles Matthews, Greenrd, MathMartin, Giftlite, DemonThing, Rjwilmsi, R.e.b., Mathbot, RFBailey, CaliforniaAliBaba, SmackBot,Moocowpong1, Nbarth, Jim.belk, CBM, Thijs!bot, Headbomb, Turgidson, Scott Tillinghast, Houston TX, SH 111, Zowzuri2, Yecril,Arcfrk, Lemadezorn, JackSchmidt, Addbot, Yobot, AnomieBOT, Neyagawa, ELipz and Anonymous: 12

• Group scheme Source: https://en.wikipedia.org/wiki/Group_scheme?oldid=659294050 Contributors: Zundark, Michael Hardy, Takuya-Murata, Charles Matthews, UtherSRG, Tobias Bergemann, Giftlite, Fropuff, Vivacissamamente, Oleg Alexandrov, SDC, Salix alba,R.e.b., John Z, SmackBot, Commander Keane bot, Gutworth, Vina-iwbot~enwiki, S.H.C., Harryboyles, Michael Kinyon, Noleander,Jakob.scholbach, David Eppstein, Cardano~enwiki, LokiClock, Stca74, Addbot, Lightbot, Citation bot, Ebony Jackson, Trappist themonk, CsDix and Anonymous: 14

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• Haboush’s theorem Source: https://en.wikipedia.org/wiki/Haboush’s_theorem?oldid=665536134 Contributors: Charles Matthews,Giftlite, Jeff3000, Rjwilmsi, R.e.b., Sodin, SmackBot, BeteNoir, RobHar, David Eppstein, R'n'B, DavidCBryant, Addbot, Yobot, Citationbot, Ringspectrum, Citation bot 1, ZéroBot, Brad7777, ChrisGualtieri, Kernel-spaceman and Anonymous: 3

• Hochschild–Mostow group Source: https://en.wikipedia.org/wiki/Hochschild%E2%80%93Mostow_group?oldid=569591400 Contrib-utors: R.e.b., Headbomb, Pjoef and Yobot

• Hyperspecial subgroup Source: https://en.wikipedia.org/wiki/Hyperspecial_subgroup?oldid=549879284 Contributors: RobHar, Yobotand Anonymous: 1

• Inner form Source: https://en.wikipedia.org/wiki/Inner_form?oldid=626902364 Contributors: Michael Hardy, R.e.b., TexasAndroid,David Eppstein, Trappist the monk and Anonymous: 1

• Iwahori subgroup Source: https://en.wikipedia.org/wiki/Iwahori_subgroup?oldid=607873711 Contributors: Rjwilmsi, R.e.b. and Head-bomb

• Jordan–Chevalley decomposition Source: https://en.wikipedia.org/wiki/Jordan%E2%80%93Chevalley_decomposition?oldid=609980581Contributors: Michael Hardy, Charles Matthews, Jitse Niesen, Giftlite, Jeremy Henty, Edudobay, Pavel Vozenilek, Rjwilmsi, Algebraist,Nbarth, Michael Kinyon, Makyen, Mathsci, CBM, RobHar, Vanish2, Jmath666, Kmhkmh, JackSchmidt, DFRussia, Qwfp, Marc vanLeeuwen, Addbot, Yobot, GeometryGirl, Milolance, Dr.QD and Anonymous: 8

• Kazhdan–Lusztig polynomial Source: https://en.wikipedia.org/wiki/Kazhdan%E2%80%93Lusztig_polynomial?oldid=662813470Con-tributors: Nonenmac, Michael Hardy, TakuyaMurata, Charles Matthews, Giftlite, Bender235, Wikiklrsc, Rjwilmsi, R.e.b., Wavelength,KSmrq, Nbarth, Mhym, Akriasas, Lambiam, Cronholm144, Jeffreyadams, Myasuda, Changbao, R'n'B, Lantonov, Arcfrk, Marc vanLeeuwen, MystBot, Addbot, Yobot, Kilom691, Libedins, Citation bot 1, Trappist the monk, Tition1, Khazar2, Monkbot and Anony-mous: 13

• Kempf vanishing theorem Source: https://en.wikipedia.org/wiki/Kempf_vanishing_theorem?oldid=637486029 Contributors: MichaelHardy, Rjwilmsi, R.e.b., Trappist the monk and K9re11

• Kneser–Tits conjecture Source: https://en.wikipedia.org/wiki/Kneser%E2%80%93Tits_conjecture?oldid=638004691Contributors: MichaelHardy, R.e.b., Myasuda, Headbomb, Policron, Yobot, KonradVoelkel, Mark viking and K9re11

• Kostant polynomial Source: https://en.wikipedia.org/wiki/Kostant_polynomial?oldid=626940105Contributors: Michael Hardy, Giftlite,BD2412, Mathsci, David Eppstein, Veddharta, Makotoy, Yobot, Citation bot 1, EdoDodo, Chronulator, Helpful Pixie Bot, Citation-CleanerBot, Violapaul, Mark viking, The Disambiguator and Anonymous: 1

• Lang’s theorem Source: https://en.wikipedia.org/wiki/Lang’s_theorem?oldid=633363464Contributors: TakuyaMurata, Bearcat, Giftlite,Imaginatorium, Rjwilmsi, Salix alba, Biscuittin, AnomieBOT, Trappist the monk, Malerooster and K9re11

• Langlands decomposition Source: https://en.wikipedia.org/wiki/Langlands_decomposition?oldid=633100246 Contributors: Giftlite,Oleg Alexandrov, R.e.b., Nbarth, Ulner, JackSchmidt, Roentgenium111, Yobot, K9re11 and Anonymous: 1

• Lattice (discrete subgroup) Source: https://en.wikipedia.org/wiki/Lattice_(discrete_subgroup)?oldid=648705147 Contributors: Zun-dark, Michael Hardy, Charles Matthews, Giftlite, BD2412, Headbomb, Arcfrk, Mr. Stradivarius, Addbot, Yobot, EmausBot, Eransoko,Handsofftibet, Brirush, CsDix and Anonymous: 4

• Lazard’s universal ring Source: https://en.wikipedia.org/wiki/Lazard’s_universal_ring?oldid=614968696Contributors: Michael Hardy,R.e.b., Headbomb, Yobot, D.Lazard, Mark viking and Anonymous: 1

• Lie–Kolchin theorem Source: https://en.wikipedia.org/wiki/Lie%E2%80%93Kolchin_theorem?oldid=645238013 Contributors: Zun-dark, Michael Hardy, Gabbe, TakuyaMurata, Charles Matthews, Psychonaut, Giftlite, Gauge, Gene Nygaard, Ceyockey, Natalya, Linas,R.e.b., Nowhither, Hillman, RDBury, BeteNoir, Nbarth, Headbomb, West Brom 4ever, David Eppstein, JackSchmidt, CrackerJack7891,Addbot, Kilom691, FactSpewer, Xqbot, Brad7777, Deltahedron and Anonymous: 2

• Mirabolic group Source: https://en.wikipedia.org/wiki/Mirabolic_group?oldid=606132841 Contributors: R.e.b., Headbomb and Jone-sey95

• Mumford–Tate group Source: https://en.wikipedia.org/wiki/Mumford%E2%80%93Tate_group?oldid=627019913Contributors: MichaelHardy, Charles Matthews, R.e.b., Ilmari Karonen, Yobot and Trappist the monk

• Observable subgroup Source: https://en.wikipedia.org/wiki/Observable_subgroup?oldid=649818618Contributors: Michael Hardy, CharlesMatthews, Vipul, K9re11 and Anonymous: 1

• Pseudo-reductive group Source: https://en.wikipedia.org/wiki/Pseudo-reductive_group?oldid=665812171 Contributors: R.e.b., Trap-pist the monk, Josve05a and Anonymous: 7

• Quasi-split group Source: https://en.wikipedia.org/wiki/Quasi-split_group?oldid=634413923 Contributors: Michael Hardy, TakuyaMu-rata, Giftlite, Rjwilmsi, R.e.b., Trappist the monk and Anonymous: 1

• Radical of an algebraic group Source: https://en.wikipedia.org/wiki/Radical_of_an_algebraic_group?oldid=606962764 Contributors:Charles Matthews, Trace (usurped), Ceyockey, John Broughton, J. Finkelstein, Lahiru k, Catgut, David Eppstein, Creative1984, Roent-genium111, Sonia, Anne Bauval, Shadowjams, ClueBot NG and Anonymous: 9

• Rational representation Source: https://en.wikipedia.org/wiki/Rational_representation?oldid=467000046Contributors: CharlesMatthews,Vipul, Devourer09, Qetuth and Anonymous: 1

• Reductive group Source: https://en.wikipedia.org/wiki/Reductive_group?oldid=645392490 Contributors: The Anome, Michael Hardy,TakuyaMurata, Charles Matthews, Giftlite, Trace (usurped), R.e.b., Wavelength, Kinser, Nbarth, Michael Kinyon, Wikid77, RobHar,OrenBochman, David Eppstein, R'n'B, Plclark, Alexbot, He7d3r, Addbot, Ringspectrum, Sławomir Biały, Trappist the monk, ZéroBot,Nosuchforever, ChrisGualtieri, Cjustinc and Anonymous: 9

• Restricted Lie algebra Source: https://en.wikipedia.org/wiki/Restricted_Lie_algebra?oldid=615861126 Contributors: TakuyaMurata,Giftlite, Rjwilmsi, R.e.b., Masnevets, Headbomb, Darij and Deltahedron

• Root datum Source: https://en.wikipedia.org/wiki/Root_datum?oldid=660597647 Contributors: Giftlite, Woohookitty, R.e.b., Nbarth,Vina-iwbot~enwiki, Michael Kinyon, David Cherney, Sagaciousuk, David Eppstein, Hesam7, Geometry guy, Addbot, Yobot, JochenBurghardt and Anonymous: 7

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• Rost invariant Source: https://en.wikipedia.org/wiki/Rost_invariant?oldid=610223313 Contributors: TakuyaMurata, R.e.b., Myasuda,David Eppstein and Spectral sequence

• Semisimple algebraic group Source: https://en.wikipedia.org/wiki/Semisimple_algebraic_group?oldid=628692576Contributors: MichaelHardy, Crasshopper, Arcfrk, JackSchmidt, Yobot, Trappist the monk, Qetuth and Anonymous: 3

• Serre group Source: https://en.wikipedia.org/wiki/Serre_group?oldid=635261950 Contributors: R.e.b., David Eppstein and K9re11• Severi–Brauer variety Source: https://en.wikipedia.org/wiki/Severi%E2%80%93Brauer_variety?oldid=658116093Contributors: Michael

Hardy, TakuyaMurata, Charles Matthews, Shanes, R.e.b., User24, SmackBot, Gutworth, Frap, Yobot, Unara, Eugene-elgato, Erik9bot,John of Reading, Deltahedron and Anonymous: 3

• Siegel parabolic subgroup Source: https://en.wikipedia.org/wiki/Siegel_parabolic_subgroup?oldid=635261523 Contributors: R.e.b.,Dawynn and K9re11

• Spaltenstein variety Source: https://en.wikipedia.org/wiki/Spaltenstein_variety?oldid=654615991 Contributors: TakuyaMurata, R.e.b.,MenoBot II, Trappist the monk and K9re11

• Special group (algebraic group theory) Source: https://en.wikipedia.org/wiki/Special_group_(algebraic_group_theory)?oldid=606964766Contributors: Discospinster, R.e.b., SmackBot, 345Kai, David Eppstein, Eeekster, AnomieBOT, Micmac95 and Anonymous: 2

• Springer resolution Source: https://en.wikipedia.org/wiki/Springer_resolution?oldid=662955402 Contributors: Michael Hardy, R.e.b.,Headbomb, R'n'B, Helpful Pixie Bot and Anonymous: 2

• Steinberg representation Source: https://en.wikipedia.org/wiki/Steinberg_representation?oldid=581189687 Contributors: TakuyaMu-rata, Charles Matthews, R.e.b., SmackBot, Headbomb, Addbot, Citation bot, Citation bot 1, Luizpuodzius and Anonymous: 1

• Superstrong approximation Source: https://en.wikipedia.org/wiki/Superstrong_approximation?oldid=639322824Contributors: CharlesMatthews, R.e.b., David Eppstein, AnomieBOT and SporkBot

• Taniyama group Source: https://en.wikipedia.org/wiki/Taniyama_group?oldid=635279259 Contributors: Giftlite, R.e.b., David Epp-stein and K9re11

• Tannakian category Source: https://en.wikipedia.org/wiki/Tannakian_category?oldid=667169105Contributors: CharlesMatthews, Kinser,Colonies Chris, RobHar, David Eppstein, Tomo suzuki, Addbot, AnomieBOT, Bci2, Mark viking and Anonymous: 7

• Thin group (algebraic group theory) Source: https://en.wikipedia.org/wiki/Thin_group_(algebraic_group_theory)?oldid=646545642Contributors: Michael Hardy, Charles Matthews, R.e.b., Yobot, SporkBot and K9re11

• Unipotent Source: https://en.wikipedia.org/wiki/Unipotent?oldid=670240925 Contributors: Michael Hardy, TakuyaMurata, CharlesMatthews, Jason Quinn, 4pq1injbok, Ceyockey, Gatewaycat, Ryan Reich, Rjwilmsi, R.e.b., Wavelength, SmackBot, Nbarth, Krasnoludek,David Cherney, Jakob.scholbach, David Eppstein, JackSchmidt, Addbot, Luckas-bot, Citation bot, Citation bot 1, HRoestBot, MiraclePen, Lendormi, Nosuchforever, Snow Blizzard, Stephan Alexander Spahn and Anonymous: 9

• Weil conjecture onTamagawanumbers Source: https://en.wikipedia.org/wiki/Weil_conjecture_on_Tamagawa_numbers?oldid=670873896Contributors: TakuyaMurata, Charles Matthews, Giftlite, Bender235, Rjwilmsi, R.e.b., Bgwhite, BeteNoir, Calc rulz, Urs Schreiber, Lo-kiClock, Geometry guy, AnomieBOT, Citation bot, Citation bot 1, Trappist the monk and Anonymous: 1

• Weyl module Source: https://en.wikipedia.org/wiki/Weyl_module?oldid=670498423 Contributors: Michael Hardy, Charles Matthews,R.e.b., Headbomb, George Ponderevo and K9re11

• Witt vector Source: https://en.wikipedia.org/wiki/Witt_vector?oldid=660594079Contributors: Michael Hardy, CharlesMatthews, Giftlite,Waltpohl, Rich Farmbrough, Rgdboer, Oleg Alexandrov, Ryan Reich, Rjwilmsi, R.e.b., SmackBot, PDD, Chef aka Pangloss, Eastfrisian,Ntsimp, Headbomb, RobHar, Darklilac, Jakob.scholbach, Mjg0, Typometer, LokiClock, Addbot, Citation bot, Eric Rowland, Charvest,HRoestBot, MastiBot, Helpful Pixie Bot, BG19bot, Zhongmou Zhang, Teddyktchan, Some1Redirects4You and Anonymous: 14

• Wonderful compactification Source: https://en.wikipedia.org/wiki/Wonderful_compactification?oldid=638675925Contributors: MichaelHardy, TakuyaMurata, R.e.b., Hebrides, Trappist the monk and K9re11

• Étale group scheme Source: https://en.wikipedia.org/wiki/%C3%89tale_group_scheme?oldid=649826231Contributors: Xezbeth, Calaka,DrilBot, ElNuevoEinstein, Wcherowi, K9re11 and Andrei Marzan

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