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Bibliography
Adams, J. F. [1] Lectures on Lie Groups. New York, Amsterdam: Benjamin, 1969. Atiyah, M. F. [l] Characters and cohomology of finite groups. lnst. Hautes Etudes Sci. Pub/.
Math., 9, 23-64 (1961). Atiyah, M. F., and R. Bott [1) A Lefschetz fixed point formula for elliptic complexes: II. Applications. Ann. of
Math., 88, 451-491 (1968). Atiyah, M. F., R. Bott, and A. Shapiro [I] Clifford modules. Topology, 3, 3-38 (1964). Atiyah, M. F., and F. Hirzebruch [I) Vector bundles and homogeneous spaces. Amer. Math. Soc. Symposium in Pure
Math., III, 7-38, 1961. Atiyah, M. F., and G. B. Segal [I] Equivariant K-theory and completion. J. Differential Geometry, 3, 1-18 (1969). Atiyah, M. F., and D. 0. Tall [I] Group representations, A.-rings, and the J-homomorphism. Topology, 8, 253-297
(1969). Beyl, F. R., and J. Tappe [l] Group Extensions, Representations and the Schur Multiplicator. Lecture
Notes in Mathematics, 958. New York, Berlin, Heidelberg: Springer-Verlag, 1982.
Boerner, H. [I] Darstellungen von Gruppen. Berlin, Heidelberg, New York: Springer-Verlag,
1967, 2. Auftage. Bott, R. [1] Homogeneous vector bundles. Ann. of Math., 66, 203-248 (1957). [2] The index theorem for homogeneous differential operators. In: Differential and
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Symbol Index
'A [X, Y] Gx X/G Qk(M) A®B r(Q) C[G] !t * !2 V*
"i si VG Xv A.,(V)
IJik S,(V)
H
transpose of A 4 Lie product 18 isotropy group 31 orbit space 31 alternating k-forms 42 graded tensor product 56 Clifford group 57 group nng 66 convolution 67 dual of V 75 exterior power 75 exterior power of standard
representation 265 ff symmetric power 7 5 fixed point set 77 character of V 80 = L N(V)t' 103
• Adams operation 104 = L s·(v)t• 101
r
= L2 (G) 129 convolution 1 ~0 143 complexification 151 root system 189 lattice of integral forms
195 positive roots 204 fundamental Weyl
chamber 204 half sum of positive roots
207 group generated by R*
208 = n (e(rx/2)- e( -rx/2))
aeR+
241 orders in L T* 250 Cartan composite 253 half-spin representations
279
Subject Index
Words in italics indicate the relevant main key word with further references.
A
A(n) (alternating group) 10 A(A.) (alternating sum) 240 A. (Dynkin diagram) 212, 216, 218 abelian Lie group 25, 30, 39 -, representations 69, 107 If abelian subalgebra, maximal 188 abstract subgroup 28 action of a group on a space 30 If Ad, ad: adjoint representation 18 AdG/T: T-+ Aut L(G/T) 49, 160 Adams operation 104 If affine variety 152 affine Weyl group: extended Weyl group
226 adjoint representation 18, 71, 183 If, 222 alcove 226 algebra -,associative 54 -, Clifford 54 -,Lie 14 If, 19, 20 f algebraic group 156 AJtk V (alternating k-forms) 41 alternating 41, 240, 244 -,group 10 -,sum 240 -,tensor 77 analytic structure on Lie groups 138 angle of roots 199, 205
B. (Dynkin diagram) 212, 216, 220 base space 32 basis of root system 204 Borel-Weil-Bott theorem 256 boundary of manifold 50 bounded operator 130 bundle H, 188 -,atlas 33 -,chart 32 - -,tangent 14 -,map 32
c C(Q), c.: Clifford algebra 55 c. (Dynkin diagram) 212, 216, 221 CO(G, K) as a representation 123 If c.>(X) 40
308
C[G]: group ring 66 c(y + (!) E lrr(G, T) 241 canonical decomposition of
representation 70, 101 canonical metric of root system 213,
215 f Car tan - composite 253, 262 -matrix 210 --,determinant 216 -number 198 - subgroup 176 ff, 297 f Casimir operator 122 center 165, 189, 201, 229 f, 235 - ofD-11 5, 11 central element 230 centralizer 165, 169, 189 chamber 192, 227 change of coordinates 2 character of a representation 80 If, 107 - formula 239 ff - group 82, 83, 107 f - ring: representation ring 102 ff chart - of bundle 32 -domain 2 -, manifold 2 class function 81, 134, 166 classical groups 1 ff, 169 ff -, complexification 155 -,definitions 4 ff -,maximal torus 169 ff -, representations 265 ff -,root systems 216 ff -, Weyl groups 169 ff Clebsch-Gordan formula 87, 92,260 f Clifford algebra 7, 54 ff, 283 ff -,classification 286, 291 f -,modules 287 f Clifford group 57 coassociative 147 codimension 27 coinverse 146 compact group, structure 232 ff compact-open topology 142 compact operator 130 ff compact semisimple Lie algebra 209 complex root: root 185 complex type 97 complexification 151 ff comultiplication 146 Con( G) (conjugation classes) 166, 180 conjugate -,homomorphisms 67
Subject Index
-,representations 75 -,elements in G and T 166, 180 -,tori 159 ff conjugation 95 -,complex 75 - in Clifford algebra 57 - of quaternions 6 -,theorem for tori 159 ff, 177 ii connected component of unit 10 contained (a representation in another)
general element of a Lie group 168, 181 general position 231 generator of a torus 38 germ 12,169 GL(n, C), GL(n, IHI), GL(n, IR) 3, 7 global - root 185, 189, 195 - weight: weight 108 G-manifold 31 graded 56 Gram-Schmidt orthogonalization 11 Grassmann manifold 38 group ring 66, 83, 129 G-space 30 ff
H
IHI 5 Jr, = ker a, a e R 192, 198 $.;1: harmonic polynomials 88 H( G) 264, 277 f half-space 206 half-spin representation 279 ff -.type 290 half sum of positive roots 207, 241 f harmonic polynomial 88 ff, 117 ff Hermitian 4, 21, 153 higher (order on LT*) 250 Hom(U, V) 15 Homa(U, V) 67 homogeneous - polynomial 84, 117 - space 30 ff, 35 homomorphism of Lie groups 2, 29 homotopic 51, 53, 179 homotopy group 36 f, 187 f, 223 ff Hopf -algebra 147 - fibration 37, 40
I
/:integra/lattice 195 ig V: induced representation 143 immersion 27, 40 indecomposable 204, 254 index of self-conjugate irreducible
s s+,s_ 94 S1 (circle) 3 -,representations 125 s· (sphere) 36 S(n) (symmetric group) 10 Si(V) (symmetric power) 75 s« (reflection ass. to ex e R) 192, 197 scalar product - canonical of root system 213, 215 f - Euclidean 4 - on function space 79, 245 - Hermitian 4 - symplectic 8, 11 Schur's Lemma 69, 72 seff-adjoint 21, 131 seff-conjugate 98, 261 semisimple 71 ff, 201, 214, 229, 237, 249,
256 separation of points 132 short root 213 sign(a) 41 similar 67 simple - module 72, 287 - reflection 205 -root 204 singular element in Lie group 168,
SO(n): special orthogonal group 4 S0(3)-representations 84 ff, 115 ff Sp(n): symplectic group 8 Sp(n, C) 9, 155 special 168, 181 special linear group 4 special orthogonal group 4, 36 -, complexification 155 -,maximal torus 171 -, representation ring of S0(2n) 282 -,representations 272 ff -,root system 219 f, 222 -, VVeylgroup 171 special A.-ring 104 special unitary group 5, 36 -, complexification 155 -, maximal torus 170 -,representations 265 -,root system 218 -, VVeyl group 170 spectral theorem 132 sphere 36 spherical - coordinates 120 - harmonic 88 ff Spin(n): spinor group 54 ff Spin(4) ~ Spin(3) x Spin(3) 292 Spin(6) ~ SU(4) 292 spinor group 54 ff, 91 -,maximal torus 173 ff -,representation ring 279 f -,representations 278 ff -, types of representations 290 -, root system 222 -, VVeyl group 173 ff splitting principle 106 standard basis of quaternions -over C 5 -over IR 6 standard isomorphism - C2 -.IHI 6 -IR4 -.IHI 6 standard representation 69, 71 star operator 274 Steinberg's formula 259 Stiefel - diagram 223 - manifold 37 Stokes' Theorem 50 Stone-VVeierstrass 132 string of roots 201 structure group 32
Subject Index
SU(2)-representations 84 ff SU(n): special unitary group 5 SU(4) ~ Spin(6) 292 subgroup 27 -,closed 28 submanifold 27, 28 submodule 68, 72 subrepresentation 68 sum of - characters 80 - representations 69 - representative functions 125 - root systems 211 Supp(f): support 43 support 43, 257 symmetric - character 240 -group 10 - operator 131 ff -power 75 -sum 252 -tensor 77 symmetry 197 symplectic 7 ff -bilinear form J 9 - group 8, 11, 37, 155 - -,complex 9, 155 - -,maximal torus 173 - -,representations 269 ff - -,root system 221 - -, Weyl group 173 - representation 93 system of generators 254
T
,q/{G, K) (representative functions) 125 ff
TPM, TPJ: tangent space 11 ff tangent -bundle 14 -space 4ff -map 12 Tannaka-Kreln duality 146 ff tensor product 74, 80, 84, 87, 137 -,graded 56 -, multiplicities of weights 259 topological group 29 topologically cyclic 38, 190 torus 3, 25, 38 -, maximal 157 ff, 223 total space of tangent bundle 14 Tr{A), trace 76 transformation of integrals 41, 51
transitive group action 31, 35 triangular matrix 11 trigonometric polynomial 125, 136 trivial representation 69, 137 twisted 293 type 93 If, 261 If type I, type II 293
u U(n): unitary group 4 U(2)-representations 84 If unitary group 4, 11, 36, 153 -, maximal torus 170 -,representation ring 175 -, representations 267 -,root system 217 f -, Weyl group 170 unitary representation 68
v V,.(IC"), V,.(IR") 37 f V(k, l) = V0 k ® V01 137 vector bundle 14 vector field 11 ff, 15 vector product 11, 22, 118 velocity vector 11 virtual character: representation ring
103 volume form 43, 49
w wall 192, 202, 226 wedge product 42 weight 108 ff, 116, 184, 239 - space 108, 112 -vector 114 Weyl - chamber 192, 202 - character formula 239 ff
313
-group 158, 165 ff, 177, 193 ff, 198 If --,extended 226 -,integral formula 163
z Z(H): centralizer 165 ll..fn: cyclic group 10 ll..(I*> 257
Graduate Texts in Mathematics
continued from page It
61 WHITEHEAD. Elements of Homotopy 92 DIBSTEL. Sequences and Series in Banach Theory. Spaces.
62 KARGAPOLOV/MERLZJAKOV. Fundamentals 93 DuBRoVINIFoMENKoiNOVIKOV. Modem of the Theory of Groups. Geometry-Methods and Applications.
63 Bou.oBAS. Graph Theory. Part I. 2nd ed. 64 EDWARDS. Fourier Series. Vol. I. 2nd ed. 94 W ARNBR. Foundations of Differentiable 65 WELLS. Differential Analysis on Complex Manifolds and lie Groups.
Manifolds. 2nd ed. 95 SHIRYABV. Probability. 2nd ed. 66 W ATERHOUSB. Introduction to Affine 96 CONWAY. A Course in Functional
Group Schemes. Analysis. 2nd ed. 67 SERRE. Local Fields. 97 KoBLITZ. Introduction to Elliptic Curves 68 WEIDMANN. Linear Operators in Hilbert and Modular Forms. 2nd ed.
Combinatorial Group Theory. 2nd ed. Analysis on Semigroups: Theory of 73 HUNGERFORD. Algebra. Positive Definite and Related Functions. 74 DAVENPORT. Multiplicative Number 101 EDWARDS. Galois Theory.
Theory. 2nd ed. 102 V ARADARAJAN. Lie Groups, Lie Algebras 75 HOCHSCHILD. Basic Theory of Algebraic and Their Representations.
Groups and Lie Algebras. 103 LANG. Complex Analysis. 3rd ed. 76 lrrAKA. Algebraic Geometry. 104 DUBROVINIFOMENKoiNOVIKOV. Modem 77 HEeKE. Lectures on the Theory of Geometry-Methods and Applications.
Algebraic Numbers. Part II. 78 BURRIS!SANKAPPANAVAR. A Course in lOS LANG. Sf.,(R).
Universal Algebra. 106 SILVERMAN. The Arithmetic of Elliptic 79 WALTERS. An Introduction to Ergodic Curves.
Theory. 107 OLVER. Applications of Lie Groups to 80 RoBINSON. A Course in the Theory of Differential Equations. 2nd ed.
Groups. 2nd ed. 108 RANGE. Holomorphic Functions and 81 FORSTER. Lectures on Riemann Surfaces. Integral Representations in Several 82 Borr/Tu. Differential Forms in Algebraic Complex Variables.
Topology. 109 LEHTO. Univalent Functions and 83 WASHINGTON. Introduction to Cyclotomic Teichmiiller Spaces.
Fields. 110 LANG. Algebraic Number Theory. 84 IRI!LANDIROS!!N. A Classical Introduction Ill HUSI!MOLLI!R. Elliptic Curves.
to Modem Number Theory. 2nd ed. 112 LANG. Elliptic Functions. 85 EDWARDS. Fourier Series. Vol. II. 2nd ed. 113 KARATZASISHREVE. Brownian Motion and 86 vAN LINT. Introduction to Coding Theory. Stochastic Calculus. 2nd ed.
2nd ed. 114 KoBLITZ. A Course in Number Theory 87 BROWN. Cohomology of Groups. and Cryptography. 2nd ed. 88 PIERCE. Associative Algebras. 115 BI!RG!!RIGOSTIAUX. Differential Geometry: 89 LANo. Introduction to Algebraic and Manifolds, Curves, and Surfaces.
Abelian Functions. 2nd ed. 116 KELLEY/SRINIVASAN. Measure and 90 BRI!lNDSTBD. An Introduction to Convex Integral. Vol. I.
Polytopes. 117 SERRE. Algebraic Groups and Class 91 B!!ARDON. On the Geometry of Discrete Fields.
Groups. 118 PEDERSEN. Analysis Now.
119 ROTMAN. An Introduction to Algebraic 142 LANO. Real and Functional Analysis. Topology. 3rd ed.
120 ZIEMER. Weakly Differentiable Functions: 143 DooB. Measure Theory. Sobolev Spaces and Functions of 144 DENNls/FARB. Noncommutative Bounded Variation. Algebra.
121 LANO. Cyclotomic Fields I and II. 145 VICK. Homology Theory. An Combined 2nd ed. Introduction to Algebraic Topology.
122 REMMERT. Theory of Complex Functions. 2nd ed. Readings in Mathematics 146 BRIDGES. Computability: A
123 EBBINOHAUsiHBRMBS et al. Numbers. Mathematical Sketchbook. Readings in Mathematics 147 ROSENBERG. Algebraic K-Theory
124 DusRoVINIFoMENKo/NOVIKov. Modem and Its Applications. Geometry-Methods and Applications. 148 ROTMAN. An Introduction to the Part III. Theory of Groups. 4th ed.
125 BBRBNSTBJN!GAY. Complex Variables: An 149 RATCLIFFE. foundations of Introduction. Hyperbolic Manifolds.
126 BORBL. Linear Algebraic Groups. 150 BISENBUD. Commutative Algebra 127 MASSEY. A Basic Course in Algebraic with a View Toward Algebraic
Topology. Geometry. 128 RAUCH. Partial Differential Equations. 151 SILVERMAN. Advanced Topics in 129 fuLTON/HARRIS. Representation Theory: the Arithmetic of Elliptic Curves.
A first Course. 152 ZIBGLER. Lectures on Polytopes. Readings in Mathematics 153 FULTON. Algebraic Topology: A
130 DoDsoN/PoSTON. Tensor Geometry. first Course. 131 LAM. A First Course in Noncommutative 154 BROWNIPEARCY. An Introduction to
Rings. Analysis. 132 BEAROON. Iteration of Rational Functions. 155 KAsSEL. Quantum Groups. 133 HARRis. Algebraic Geometry: A f'mt 156 KECHRJS. Classical Descriptive Set
Course. Theory. 134 RoMAN. Coding and Information Theory. 157 MALLIAVIN. Integration and 135 ROMAN. Advanced Linear Algebra. Probability. 136 ADKINs/WEJNIRAUB. Algebra: An 158 ROMAN. field Theory.
Approach via Module Theory. 159 CONWAY. Functions of One 137 AxLmVBOURDONIRAMEY.Harmomc Complex Variable II.
Function Theory. 160 LANG. Differential and Riemannian 138 CoHEN. A Course in Computational Manifolds.
Algebraic Number Theory. 161 BoRWEJNIERDaYI. Polynomials and 139 BREOON. Topology and Geometry. Polynomial Inequalities. 140 AUBIN. Optima and Equilibria. An 162 ALPERIN/BELL. Groups and