Algebraic Groups Ak

Algebraic groups akFrom Wikipedia, the free encyclopedia

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 American Mathematical Society 105.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Meetings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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5.3 Fellows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.5 Prizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.6 Typesetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.7 Presidents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.7.1 18881900 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.7.2 19011950 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.7.3 19512000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.7.4 2001present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6 Approximation in algebraic groups 156.1 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.3 Formal definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7 Arason invariant 177.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8 Arithmetic group 188.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9 Borel subgroup 199.1 Parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.2 Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

10 Borelde Siebenthal theory 2110.1 Connected subgroups of maximal rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2110.2 Maximal connected subgroups of maximal rank . . . . . . . . . . . . . . . . . . . . . . . . . . . 2110.3 Closed subsystems of roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.4 Applications to symmetric spaces of compact type . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.5 Applications to hermitian symmetric spaces of compact type . . . . . . . . . . . . . . . . . . . . . 25

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10.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11 BottSamelson variety 2811.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

12 Bruhat decomposition 2912.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.4 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

13 Cartan subgroup 3113.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3113.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3113.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3113.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

14 Chevalleys structure theorem 3214.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

15 Cohomological invariant 3315.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3315.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3315.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

16 Diagonalizable group 3516.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

17 Dieudonn module 3617.1 Dieudonn rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3617.2 Dieudonn modules and groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3717.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3717.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

18 Differential algebraic group 3818.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

19 Differential Galois theory 3919.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3919.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3919.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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19.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

20 E6 (mathematics) 4120.1 Real and complex forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4120.2 E6 as an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4220.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

20.3.1 Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4220.3.2 Roots of E6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4220.3.3 Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4520.3.4 Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

20.4 Important subalgebras and representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4620.5 E6 polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4620.6 Chevalley and Steinberg groups of type E6 and 2E6 . . . . . . . . . . . . . . . . . . . . . . . . . . 4620.7 Importance in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4720.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4820.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

21 E7 (mathematics) 5021.1 Real and complex forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5021.2 E7 as an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5121.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

21.3.1 Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5121.3.2 Root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5121.3.3 Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5321.3.4 Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

21.4 Important subalgebras and representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.4.1 E7 Polynomial Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

21.5 Chevalley groups of type E7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.6 Importance in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5621.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

22 E8 (mathematics) 5722.1 Basic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5722.2 Real and complex forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5722.3 E8 as an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5822.4 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5822.5 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5922.6 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5922.7 E8 root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

22.7.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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22.7.2 Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6022.7.3 Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6122.7.4 Simple roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6222.7.5 Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322.7.6 E8 root lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322.7.7 Simple subalgebras of E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

22.8 Chevalley groups of type E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322.9 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322.10Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6522.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6522.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

23 F4 (mathematics) 6823.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

23.1.1 Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.1.2 Weyl/Coxeter group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.1.3 Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.1.4 F4 lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.1.5 Roots of F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923.1.6 F4 polynomial invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

23.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

24 Fixed-point subgroup 73

25 Formal group 7425.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7425.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.3 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.4 The logarithm of a commutative formal group law . . . . . . . . . . . . . . . . . . . . . . . . . . 7625.5 The formal group ring of a formal group law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7625.6 Formal group laws as functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7725.7 The height of a formal group law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7725.8 Lazard ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7725.9 Formal groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7825.10LubinTate formal group laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

26 Fundamental lemma (Langlands program) 8126.1 Motivation and history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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26.2 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8126.3 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

27 G2 (mathematics) 8427.1 Real forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8427.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

27.2.1 Dynkin diagram and Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8427.2.2 Roots of G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8527.2.3 Weyl/Coxeter group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8527.2.4 Special holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

27.3 Polynomial Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8527.4 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8527.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8527.6 Finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8627.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8627.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

28 Geometric invariant theory 8828.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8828.2 Mumfords book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8928.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9028.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9128.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

29 Glossary of algebraic groups 9229.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

30 Good filtration 9330.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

31 Group of Lie type 9431.1 Classical groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9431.2 Chevalley groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9431.3 Steinberg groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9431.4 SuzukiRee groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9531.5 Relations with finite simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9631.6 Small groups of Lie type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9631.7 Notation issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9731.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9831.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

CONTENTS vii

31.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

32 Group scheme 10032.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10032.2 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10032.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10132.4 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10232.5 Finite flat group schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10232.6 Cartier duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10332.7 Dieudonn modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10332.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

33 HochschildMostow group 10533.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

34 Hyperspecial subgroup 10634.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

35 Inner form 10735.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

36 Iwahori subgroup 10836.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

37 JordanChevalley decomposition 10937.1 Decomposition of endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10937.2 Decomposition in a real semisimple Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10937.3 Decomposition in a real semisimple Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11037.4 Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11037.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

38 KazhdanLusztig polynomial 11138.1 Motivation and history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11138.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11138.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11238.4 KazhdanLusztig conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

38.4.1 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11338.5 Relation to intersection cohomology of Schubert varieties . . . . . . . . . . . . . . . . . . . . . . 11438.6 Generalization to real groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11438.7 Generalization to other objects in representation theory . . . . . . . . . . . . . . . . . . . . . . . 11538.8 Combinatorial theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11538.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11538.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

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39 Kempf vanishing theorem 11739.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

40 KneserTits conjecture 11840.1 Fields for which the Whitehead group vanishes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11840.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11840.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

41 Kostant polynomial 11941.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11941.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11941.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12041.4 Steinberg basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12241.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

42 tale group scheme 12442.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12442.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 125

42.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12542.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12742.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

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.

1

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 46. 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.

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 idles /dlz/)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, t1)},

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).

3

4 CHAPTER 2. ADELIC ALGEBRAIC GROUP

Tsuneo Tamagawa's observation was that, starting from an invariant differential form on G, defined over K, themeasure involved was well-defined: while could be replaced by c 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 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 idles were introduced by Chevalley (1936) under the name "lment idal, which is ideal elementin French, which Chevalley (1940) then abbreviated to idle 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 Chevalleys 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 adle(short for additive idles, and also a French womans name) was in use shortly afterwards (Jaffard 1953) and mayhave been introduced by Andr 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), Gnralisation de la thorie du corps de classes pour les extensions infinies.,Journal de Mathmatiques Pures et Appliques (in French) 15: 359371, JFM 62.1153.02

Chevalley, Claude (1940), La thorie du corps de classes, Annals of Mathematics. Second Series 41: 394418, 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'adles (d'aprs Iwasawa), Sminaire Bourbaki, Secrtariat mathmatique,Paris, MR 0157859

Ono, Takashi (1957), Sur une proprit arithmtique des groupes algbriques commutatifs, Bulletin de laSocit Mathmatique de France 85: 307323, ISSN 0037-9484, MR 0094362

Tate, John T. (1950), Fourier analysis in number fields, and Heckes zeta-functions, Algebraic Number The-ory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305347, ISBN 978-0-9502734-2-6, MR 0217026

Weil, Andr (1938), Zur algebraischen Theorie der algebraischen Funktionen., Journal fr Reine und Ange-wandte Mathematik (in German) 179: 129133, 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

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 Chevalleys 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 22 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.

5

6 CHAPTER 3. ALGEBRAIC GROUP

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

CherlinZilber conjecture

Adelic algebraic group

Glossary of algebraic groups

3.5 Notes

3.6 References Chevalley, Claude, ed. (1958), Sminaire C. Chevalley, 1956-1958. Classification des groupes de Lie al-gbriques, 2 vols, Paris: Secrtariat Mathmatique, MR 0106966, Reprinted as volume 3 of Chevalleys 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:Birkhuser 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, Andr (1971), Courbes algbriques et varits abliennes, Paris: Hermann, OCLC 322901

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 GL,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

7

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 ) = IndGKGL X(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 extensionsare 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.

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 TateShafarevich 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. VIIIX

T. Ono, On Tamagawa Numbers

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

Chapter 5

American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to theinterests of mathematical research and scholarship, and serves the national and international community through itspublications, meetings, advocacy and other programs.The society is one of the four parts of the Joint Policy Board forMathematics (JPBM) and amember of the ConferenceBoard of the Mathematical Sciences (CBMS).

5.1 History

It was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressedby the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president andFiske became secretary.[1] The society soon decided to publish a journal, but ran into some resistance, due to concernsabout competing with theAmerican Journal ofMathematics. The result was theBulletin of the NewYorkMathematicalSociety, with Fiske as editor-in-chief. The de facto journal, as intended, was influential in increasing membership.The popularity of the Bulletin soon led to Transactions of the American Mathematical Society and Proceedings of theAmerican Mathematical Society, which were also de facto journals. In 1891 Charlotte Scott became the first womanto join the society. The society reorganized under its present name and became a national society in 1894, and thatyear Scott served as the first woman on the first Council of the AmericanMathematical Society. In 1951, the societysheadquarters moved from New York City to Providence, Rhode Island. In 1954 the society called for the creationof a new teaching degree, a Doctor of Arts in Mathematics, similar to a PhD but without a research thesis.[2] In the1970s, as reported in A Brief History of the Association for Women in Mathematics: The Presidents Perspectives,by Lenore Blum, In those years the AMS [American Mathematical Society] was governed by what could only becalled an old boys network, closed to all but those in the inner circle. Mary challenged that by sitting in on theCouncil meeting in Atlantic City. When she was told she had to leave, she refused saying she would wait until thepolice came. (Mary relates the story somewhat differently: When she was told she had to leave, she responded shecould find no rules in the by-laws restricting attendance at Council meetings. She was then told it was by gentlemensagreement. Naturally Mary replied Well, obviously I'm no gentleman.) After that time, Council meetings wereopen to observers and the process of democratization of the Society had begun. [3] Julia Robinson was the firstfemale president of the American Mathematical Society (19831984), but was unable to complete her term as shewas suffering from leukemia. [4] The society also added an office in Ann Arbor, Michigan in 1984 and an office inWashington, D.C. in 1992. In 1988 the Journal of the American Mathematical Society was created, with the intentof being the flagship journal of the AMS.

5.2 Meetings

The AMS, along with the Mathematical Association of America and other organizations, holds the largest annualresearch mathematics meeting in the world, the Joint Mathematics Meeting held in early January. The 2013 JointMathematics Meeting in San Diego drew over 6,600 attendees. Each of the four regional sections of the AMS(Central, Eastern, Southeastern and Western) hold meetings in the spring and fall of each year. The society alsoco-sponsors meetings with other international mathematical societies.

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5.3. FELLOWS 11

5.3 Fellows

See also: Category:Fellows of the American Mathematical Society

TheAMS selects an annual class of Fellowswho havemade outstanding contributions to the advancement ofmathematics.[5]

5.4 Publications

The AMS publishes Mathematical Reviews, a database of reviews of mathematical publications, various journals,and books. In 1997 the AMS acquired the Chelsea Publishing Company, which it continues to use as an imprint.Journals:

General Bulletin of the American Mathematical Society - published quarterly, Electronic Research Announcements of the American Mathematical Society - online only, Journal of the American Mathematical Society - published quarterly, Memoirs of the American Mathematical Society - published six times per year, Notices of the American Mathematical Society - published monthly, one of the most widely read math-ematical periodicals,

Proceedings of the American Mathematical Society - published monthly, Transactions of the American Mathematical Society - published monthly,

Subject-specific Mathematics of Computation - published quarterly, Mathematical Surveys and Monographs Conformal Geometry and Dynamics - online only, Representation Theory - online only.

Blogs:

Blog on Blogs e-Mentoring Network in the Mathematical Sciences AMS Graduate Student Blog PhD + Epsilon On the Market

5.5 Prizes

Some prizes are awarded jointly with other mathematical organizations. See specific articles for details.

Bcher Memorial Prize Cole Prize Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student Fulkerson Prize Leroy P. Steele Prizes Norbert Wiener Prize in Applied Mathematics Oswald Veblen Prize in Geometry

12 CHAPTER 5. AMERICAN MATHEMATICAL SOCIETY

5.6 Typesetting

The AMS was an early advocate of the typesetting program TeX, requiring that contributions be written in it andproducing its own packages AMS-TeX and AMS-LaTeX. TeX and LaTeX are now ubiquitous in mathematical pub-lishing.

5.7 Presidents

The AMS is led by the President, who is elected for a two-year term, and cannot serve for two consecutive terms.[6]

5.7.1 18881900

John Howard Van Amringe (New York Mathematical Society) (18881890)

Emory McClintock (New York Mathematical Society) (189194)

George Hill (189596)

Simon Newcomb (189798)

Robert Woodward (18991900)

5.7.2 19011950

Eliakim Moore (190102)

Thomas Fiske (190304)

William Osgood (190506)

Henry White (190708)

Maxime Bcher (190910)

Henry Fine (191112)

Edward Van Vleck (191314)

Ernest Brown (191516)

Leonard Dickson (191718)

Frank Morley (191920)

Gilbert Bliss (192122)

Oswald Veblen (192324)

George Birkhoff (192526)

Virgil Snyder (192728)

Earle Raymond Hedrick (192930)

Luther Eisenhart (193132)

Arthur Byron Coble (193334)

Solomon Lefschetz (193536)

Robert Moore (193738)

Griffith C. Evans (193940)

5.7. PRESIDENTS 13

Marston Morse (194142)

Marshall Stone (194344)

Theophil Hildebrandt (194546)

Einar Hille (194748)

Joseph L. Walsh (194950)

5.7.3 19512000

John von Neumann (195152)

Gordon Whyburn (195354)

Raymond Wilder (195556)

Richard Brauer (195758)

Edward McShane (195960)

Deane Montgomery (196162)

Joseph Doob (196364)

Abraham Albert (196566)

Charles B. Morrey, Jr. (196768)

Oscar Zariski (196970)

Nathan Jacobson (197172)

Saunders Mac Lane (197374)

Lipman Bers (197576)

R. H. Bing (197778)

Peter Lax (197980)

Andrew Gleason (198182)

Julia Robinson (198384)

Irving Kaplansky (198586)

George Mostow (198788)

William Browder (198990)

Michael Artin (199192)

Ronald Graham (199394)

Cathleen Morawetz (199596)

Arthur Jaffe (199798)

Felix Browder (19992000)

14 CHAPTER 5. AMERICAN MATHEMATICAL SOCIETY

5.7.4 2001present

Hyman Bass (200102)

David Eisenbud (200304)

James Arthur (200506)

James Glimm (200708)

George E. Andrews (200910)

Eric M. Friedlander (201112)

David Vogan (201314)

Robert L. Bryant (201516)

5.8 See also Mathematical Association of America

European Mathematical Society

London Mathematical Society

List of Mathematical Societies

5.9 References[1] Archibald, Raymond Clare (1939). History of the American Mathematical Society, 1888-1938. Bull. Amer. Math. Soc.

45 (1): 3146. doi:10.1090/s0002-9904-1939-06908-5.

[2] Journal of Proceedings and Addresses of the Annual Conference 1960. Association of Graduate Schools

[3] A Brief History of the Association for Women in Mathematics (from Notices): How it was. Awm-math.org. Retrieved2015-05-28.

[4] http://www.encyclopedia.com/topic/Julia_Bowman_Robinson.aspx

[5] Fellows of the American Mathematical Society. Retrieved 21 May 2013.

[6] Bylaws (as amended December 2003)". American Mathematical Society.

5.10 External links The AMS website

A Semicentennial History of the American Mathematical Society, 18881938 by Raymond Clare Archibald

MacTutor: The New York Mathematical Society

MacTutor: The American Mathematical Society

This article incorporates material from American Mathematical Society on PlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

Chapter 6

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.

6.1 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.

6.2 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 KneserTitsconjecture.

6.3 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).

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16 CHAPTER 6. APPROXIMATION IN ALGEBRAIC GROUPS

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.

6.4 See also Superstrong approximation

6.5 References Eichler, Martin (1938), Allgemeine Kongruenzklasseneinteilungen der Ideale einfacher Algebren ber alge-braischen Zahlkrpern und ihre L-Reihen., Journal fr Reine und Angewandte Mathematik (in German) 179:227251, 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. 187196, MR0213361

Margulis, G. A. (1977), Cobounded subgroups in algebraic groups over local fields, Akademija Nauk SSSR.Funkcional'nyi Analiz i ego Priloenija 11 (2): 4557, 95, ISSN 0374-1990, MR 0442107

Platonov, V. P. (1969), The problem of strong approximation and the KneserTits hypothesis for algebraicgroups, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 33: 12111219, 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): 553572, ISSN 0003-486X, JSTOR 1970924, MR 0444571

Chapter 7

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.

7.1 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 Pfisterforma,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).

7.2 References Arason, Jn Kr. (1975), Cohomologische Invarianten quadratischer Formen, J. Algebra (in German) 36 (3):448491, doi:10.1016/0021-8693(75)90145-3, ISSN 0021-8693, MR 0389761, Zbl 0314.12104

Esnault, Hlne; Kahn, Bruno; Levine, Marc; Viehweg, Eckart (1998), The Arason invariant and mod 2algebraic cycles, J. Amer. Math. Soc. 11 (1): 73118, 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

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