Clifford algebras and Lie theory

Cover......Page 1
Clifford Algebras and Lie Theory......Page 3
Preface......Page 6
Acknowledgments......Page 9
Conventions......Page 10
Contents......Page 11
List of Symbols......Page 16
1.1 Quadratic vector spaces......Page 18
1.2 Isotropic subspaces......Page 20
1.3 Split bilinear forms......Page 22
1.4 E. Cartan-Dieudonné Theorem......Page 24
1.5 Witt's Theorem......Page 28
1.6 Orthogonal groups for K=R,C......Page 29
1.7 Lagrangian Grassmannians......Page 35
2.1.1 Definition......Page 39
2.1.2 Universal property, functoriality......Page 40
2.1.3 Derivations......Page 41
2.1.5 Duality pairings......Page 42
2.2.1 Definition and first properties......Page 43
2.2.2 Universal property, functoriality......Page 45
2.2.3 The Clifford algebras Cl(n,m)......Page 46
2.2.4 The Clifford algebras Cl(n)......Page 47
2.2.5 Symbol map and quantization map......Page 48
2.2.6 Transposition......Page 50
2.2.7 Chirality element......Page 51
2.2.8 The trace and the super-trace......Page 52
2.2.9 Lie derivatives and contractions......Page 53
2.2.10 The Lie algebra q(2(V))......Page 55
2.2.11 A formula for the Clifford product......Page 57
2.3.1 Differential operators......Page 58
2.3.2 Graded Poisson algebras......Page 60
2.3.3 Graded super Poisson algebras......Page 61
2.3.4 Poisson structures on (V)......Page 62
3.1.1 The Clifford group......Page 65
3.1.2 The groups Pin(V) and Spin(V)......Page 67
3.2.1 Basic constructions......Page 70
3.2.2 The spinor module SF......Page 72
3.2.3 The dual spinor module SF......Page 74
3.2.4 Irreducibility of the spinor module......Page 75
3.2.5 Abstract spinor modules......Page 76
3.3 Pure spinors......Page 78
3.4 The canonical bilinear pairing on spinors......Page 81
3.5 The character chi: Gamma(V)F->Kx......Page 85
3.6 Cartan's triality principle......Page 86
3.7.1 The Clifford algebra Cl(V)......Page 90
3.7.2 The groups Spinc(V) and Pinc(V)......Page 91
3.7.3 Spinor modules over Cl(V)......Page 93
3.7.4 Classification of irreducible Cl(V)-modules......Page 95
3.7.5 Spin representation......Page 96
Spin(5)......Page 99
Spin(7)......Page 100
Spin(8)......Page 101
4.1 Pull-backs and push-forwards of spinors......Page 102
4.2.1 The Lie algebra o(V*V)......Page 105
4.2.2 The group SO(V*V)......Page 106
4.2.3 The group Spin(V*V)......Page 107
4.3.1 The symbol map in terms of the spinor module......Page 109
4.3.2 The symbol of elements in the spin group......Page 110
4.3.3 Another factorization......Page 112
4.3.5 Clifford exponentials versus exterior algebra exponentials......Page 114
4.3.6 The symbol of elements exp(gamma(A)-i ei taui)......Page 116
4.4 Volume forms on conjugacy classes......Page 118
5.1.1 Construction......Page 123
5.1.3 Augmentation map, anti-automorphism......Page 124
5.1.6 Unitary representations......Page 125
5.1.8 Further remarks......Page 126
5.2 The Poincaré-Birkhoff-Witt Theorem......Page 127
5.3 U(g) as left-invariant differential operators......Page 130
5.4.1 Hopf algebras......Page 132
5.4.2 Hopf algebra structure on S(E)......Page 134
5.4.3 Hopf algebra structure on U(g)......Page 135
5.4.4 Primitive elements......Page 137
5.4.5 Coderivations......Page 138
5.4.6 Coderivations of S(E)......Page 139
5.5 Petracci's proof of the Poincaré-Birkhoff-Witt Theorem......Page 140
5.5.1 A g-representation by coderivations......Page 141
5.5.2 The formal vector fields Xzeta(phi)......Page 142
5.5.3 Proof of Petracci's Theorem......Page 144
5.6 The center of the enveloping algebra......Page 145
6.1 Differential spaces......Page 148
6.3 Homotopies......Page 150
6.4 Koszul algebras......Page 153
6.5 Symmetrization......Page 154
6.6 g-differential spaces......Page 156
6.7 The g-differential algebra g*......Page 158
6.9 The Weil algebra......Page 161
6.10 Chern-Weil homomorphisms......Page 164
6.11 The non-commutative Weil algebra Wg......Page 166
6.12 Equivariant cohomology of g-differential spaces......Page 169
6.13 Transgression in the Weil algebra......Page 171
7.1 The g-differential algebra Cl(g)......Page 176
7.2.1 Poisson structure on the Weil algebra......Page 180
7.2.2 Definition of the quantum Weil algebra......Page 182
7.2.3 The cubic Dirac operator......Page 184
7.2.4 W(g) as a level 1 enveloping algebra......Page 185
7.2.5 Conjugation......Page 186
7.3 Application: Duflo's Theorem......Page 187
7.4 Relative Dirac operators......Page 189
7.5.1 Enveloping algebras......Page 195
7.5.2 Clifford algebras......Page 197
7.5.3 Quantum Weil algebras......Page 201
8.1 Notation......Page 204
8.2.1 Harish-Chandra projection for U(g)......Page 205
8.2.2 Harish-Chandra projection of the quadratic Casimir......Page 207
8.2.3 Harish-Chandra projection for Cl(g)......Page 208
8.3 Equal rank subalgebras......Page 210
8.4 The kernel of DV......Page 216
8.5 q-dimensions......Page 219
8.6 The shifted Dirac operator......Page 221
8.7.1 Central extensions of compact Lie groups......Page 222
8.7.2 Twisted representations......Page 224
8.7.3 The rho-representation of g as a twisted representation of G......Page 225
8.7.4 Definition of the induction map......Page 226
8.7.5 The kernel of DM......Page 228
9.1 Differential operators on homogeneous spaces......Page 231
9.2.1 Linear connections......Page 234
9.2.2 Principal connections......Page 235
9.2.3 Dirac operators......Page 237
9.3 Dirac operators on homogeneous spaces......Page 239
10.1 Lie algebra cohomology......Page 242
10.2.1 Definition and basic properties......Page 244
10.2.2 Schouten bracket......Page 246
10.3 Lie algebra homology for reductive Lie algebras......Page 249
10.3.1 Hopf algebra structure on (g)g......Page 251
10.4 Primitive elements......Page 252
10.5 Hopf-Koszul-Samelson Theorem......Page 253
10.6 Consequences of the Hopf-Koszul-Samelson Theorem......Page 255
10.7 Transgression Theorem......Page 256
11.1 Cl(g) and the rho-representation......Page 260
11.2 Relation with extremal projectors......Page 266
11.3 The isomorphism (Clg)g=Cl(P(g))......Page 271
11.4.1 The space Homg(g,lambda(Sg))......Page 273
11.4.2 The space Homg(g,gamma(Ug))......Page 276
11.5 The Harish-Chandra projection of q(P(g))Clg......Page 280
11.6 Relation with the principal TDS......Page 282
A.1 Super vector spaces......Page 286
A.2 Graded super vector spaces......Page 288
A.3 Filtered super vector spaces......Page 290
B.1 Definitions and basic properties......Page 292
B.2 Cartan subalgebras......Page 293
B.3 Representation theory of sl(2,C)......Page 294
B.4 Roots......Page 295
B.5 Simple roots......Page 298
B.6 The Weyl group......Page 299
B.7 Weyl chambers......Page 302
B.8 Weights of representations......Page 304
B.9 Highest weight representations......Page 306
B.10 Extremal weights......Page 309
B.11 Multiplicity computations......Page 310
C.1 Preliminaries......Page 312
C.2 Group actions on manifolds......Page 313
C.3 The exponential map......Page 314
C.4 The vector field 12(xiL+xiR)......Page 317
C.5 Maurer-Cartan forms......Page 318
C.6 Quadratic Lie groups......Page 320
References......Page 322
Index......Page 327