Book by E. Meinrenken: Clifford Algebras and Lie Theory


Eckhard Meinrenken: Clifford Algebras and Lie Theory [hardcover+e-book], Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, Vol. 58, Springer, Berlin, 2013, 321 pages. ISBN 978-3-642-36215-6. Publisher homepage: http://www.springer.com/mathematics/algebra/book/978-3-642-36215-6

  • Convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics
  • Included are many developments from the last 15 years, drawn in part from the author’s research
  • Largely self-contained exposition

This monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré–Birkhoff–Witt theorem.

This is followed by discussions of Weil algebras, Chern–Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra.

Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics.

Content Level » Graduate

Keywords » Clifford algebras – Dirac operators – Lie algebras – Lie groups – Spinors

Table of contents: http://www.springer.com/cda/content/document/cda_downloaddocument/9783642362156-t1.pdf?SGWID=0-0-45-1379823-p174886453

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
1 Symmetric bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Quadratic vector spaces . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Isotropic subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Split bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 E. Cartan–Dieudonné Theorem . . . . . . . . . . . . . . . . . . . 7
1.5 Witt’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Orthogonal groups for K = R,C . . . . . . . . . . . . . . . . . . 12
1.7 Lagrangian Grassmannians . . . . . . . . . . . . . . . . . . . . . 18
2 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Exterior algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.2 Universal property, functoriality . . . . . . . . . . . . . . . 24
2.1.3 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.4 Transposition . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.5 Duality pairings . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Definition and first properties . . . . . . . . . . . . . . . . 27
2.2.2 Universal property, functoriality . . . . . . . . . . . . . . . 29
2.2.3 The Clifford algebras Cl(n,m) . . . . . . . . . . . . . . . 30
2.2.4 The Clifford algebras Cl(n) . . . . . . . . . . . . . . . . . 31
2.2.5 Symbol map and quantization map . . . . . . . . . . . . . 32
2.2.6 Transposition . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.7 Chirality element . . . . . . . . . . . . . . . . . . . . . . 35
2.2.8 The trace and the super-trace . . . . . . . . . . . . . . . . 36
2.2.9 Lie derivatives and contractions . . . . . . . . . . . . . . . 37
2.2.10 The Lie algebra q(∧2(V )) . . . . . . . . . . . . . . . . . . 39
2.2.11 A formula for the Clifford product . . . . . . . . . . . . . 41
2.3 The Clifford algebra as a quantization . . . . . . . . . . . . . . . . 42
2.3.1 Differential operators . . . . . . . . . . . . . . . . . . . . 42
2.3.2 Graded Poisson algebras . . . . . . . . . . . . . . . . . . . 44
2.3.3 Graded super Poisson algebras . . . . . . . . . . . . . . . 45
2.3.4 Poisson structures on ∧(V ) . . . . . . . . . . . . . . . . . 46
3 The spin representation . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1 TheClifford groupandthe spingroup . . . . . . . . . . . . . . . 49
3.1.1 TheClifford group . . . . . . . . . . . . . . . . . . . . . . 49
3.1.2 The groups Pin(V ) and Spin(V ) . . . . . . . . . . . . . . 51
3.2 Clifford modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.1 Basic constructions . . . . . . . . . . . . . . . . . . . . . 54
3.2.2 The spinor module SF . . . . . . . . . . . . . . . . . . . . 56
3.2.3 The dual spinor module SF . . . . . . . . . . . . . . . . . 58
3.2.4 Irreducibility of the spinor module . . . . . . . . . . . . . 59
3.2.5 Abstract spinor modules . . . . . . . . . . . . . . . . . . . 60
3.3 Pure spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4 The canonical bilinear pairing on spinors . . . . . . . . . . . . . . 65
3.5 The character χ : Γ (V)F →K × . . . . . . . . . . . . . . . . . . 69
3.6 Cartan’s triality principle . . . . . . . . . . . . . . . . . . . . . . 70
3.7 The Clifford algebra Cl(V ) . . . . . . . . . . . . . . . . . . . . . 74
3.7.1 The Clifford algebra Cl(V ) . . . . . . . . . . . . . . . . . 74
3.7.2 The groups Spinc(V ) and Pinc(V ) . . . . . . . . . . . . . 75
3.7.3 Spinor modules over Cl(V ) . . . . . . . . . . . . . . . . . 77
3.7.4 Classification of irreducible Cl(V )-modules . . . . . . . . 79
3.7.5 Spin representation . . . . . . . . . . . . . . . . . . . . . 80
3.7.6 Applications to compact Lie groups . . . . . . . . . . . . . 83
4 Covariant and contravariant spinors . . . . . . . . . . . . . . . . . . 87
4.1 Pull-backs and push-forwards of spinors . . . . . . . . . . . . . . 87
4.2 Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.1 The Lie algebra o(V∗ ⊕V ) . . . . . . . . . . . . . . . . . 90
4.2.2 The group SO(V∗ ⊕V ) . . . . . . . . . . . . . . . . . . . 91
4.2.3 The group Spin(V∗ ⊕V ) . . . . . . . . . . . . . . . . . . 92
4.3 The quantization map revisited . . . . . . . . . . . . . . . . . . . 94
4.3.1 The symbol map in terms of the spinor module . . . . . . . 94
4.3.2 The symbol of elements in the spingroup . . . . . . . . . . 95
4.3.3 Another factorization . . . . . . . . . . . . . . . . . . . . 97
4.3.4 The symbol of elements exp(γ (A)) . . . . . . . . . . . . . 99
4.3.5 Clifford exponentials versus exterior algebra exponentials . 99
4.3.6 The symbol of elements exp(γ (A) −i eiτ i ) . . . . . . . 101
4.3.7 The function A →S(A) . . . . . . . . . . . . . . . . . . 103
4.4 Volume forms on conjugacy classes . . . . . . . . . . . . . . . . . 103
5 Enveloping algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.1 The universal enveloping algebra . . . . . . . . . . . . . . . . . . 109
5.1.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.1.2 Universal property . . . . . . . . . . . . . . . . . . . . . . 110
5.1.3 Augmentationmap, anti-automorphism . . . . . . . . . . . 110
5.1.4 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1.5 Modules over U(g) . . . . . . . . . . . . . . . . . . . . . 111
5.1.6 Unitaryrepresentations . . . . . . . . . . . . . . . . . . . 111
5.1.7 Graded or filtered Lie algebras and super Lie algebras . . . 112
5.1.8 Further remarks . . . . . . . . . . . . . . . . . . . . . . . 112
5.2 The Poincaré–Birkhoff–Witt Theorem . . . . . . . . . . . . . . . 113
5.3 U(g) as left-invariant differential operators . . . . . . . . . . . . . 116
5.4 The enveloping algebra as a Hopf algebra . . . . . . . . . . . . . . 118
5.4.1 Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4.2 Hopf algebra structure on S(E) . . . . . . . . . . . . . . . 120
5.4.3 Hopf algebra structure on U(g) . . . . . . . . . . . . . . . 121
5.4.4 Primitive elements . . . . . . . . . . . . . . . . . . . . . . 123
5.4.5 Coderivations . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4.6 Coderivations of S(E) . . . . . . . . . . . . . . . . . . . . 125
5.5 Petracci’s proof of the Poincaré–Birkhoff–Witt Theorem . . . . . . 126
5.5.1 A g-representation by coderivations . . . . . . . . . . . . . 127
5.5.2 The formal vector fields Xζ (φ) . . . . . . . . . . . . . . . 128
5.5.3 Proof of Petracci’s Theorem . . . . . . . . . . . . . . . . . 130
5.6 The center of the enveloping algebra . . . . . . . . . . . . . . . . 131
6 Weil algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.1 Differential spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2 Symmetric and tensor algebra over differential spaces . . . . . . . 137
6.3 Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.4 Koszul algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.5 Symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.6 g-differential spaces . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.7 The g-differential algebra ∧g∗ . . . . . . . . . . . . . . . . . . . . 145
6.8 g-homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.9 The Weil algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.10 Chern–Weilhomomorphisms . . . . . . . . . . . . . . . . . . . . 151
6.11 The non-commutative Weil algebra Wg . . . . . . . . . . . . . . . 153
6.12 Equivariant cohomology of g-differential spaces . . . . . . . . . . 156
6.13 Transgression in the Weil algebra . . . . . . . . . . . . . . . . . . 158
7 QuantumWeil algebras . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.1 The g-differential algebra Cl(g) . . . . . . . . . . . . . . . . . . . 163
7.2 The quantum Weil algebra . . . . . . . . . . . . . . . . . . . . . . 167
7.2.1 Poisson structure on the Weil algebra . . . . . . . . . . . . 167
7.2.2 Definition of the quantum Weil algebra . . . . . . . . . . . 169
7.2.3 The cubic Dirac operator . . . . . . . . . . . . . . . . . . 171
7.2.4 W (g) as a level 1 enveloping algebra . . . . . . . . . . . . 172
7.2.5 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.3 Application: Duflo’s Theorem . . . . . . . . . . . . . . . . . . . . 174
7.4 Relative Dirac operators . . . . . . . . . . . . . . . . . . . . . . . 176
7.5 Harish-Chandra projections . . . . . . . . . . . . . . . . . . . . . 182
7.5.1 Enveloping algebras . . . . . . . . . . . . . . . . . . . . . 182
7.5.2 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . 184
7.5.3 Quantum Weil algebras . . . . . . . . . . . . . . . . . . . 188
8 Applications to reductive Lie algebras . . . . . . . . . . . . . . . . . 191
8.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.2 Harish-Chandra projections . . . . . . . . . . . . . . . . . . . . . 192
8.2.1 Harish-Chandra projection for U(g) . . . . . . . . . . . . . 192
8.2.2 Harish-Chandra projection of the quadratic Casimir . . . . 194
8.2.3 Harish-Chandra projection for Cl(g) . . . . . . . . . . . . 195
8.3 Equal rank subalgebras . . . . . . . . . . . . . . . . . . . . . . . 197
8.4 The kernel of DV . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.5 q-dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.6 The shifted Dirac operator . . . . . . . . . . . . . . . . . . . . . . 208
8.7 Dirac induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.7.1 Central extensions of compact Lie groups . . . . . . . . . . 209
8.7.2 Twistedrepresentations . . . . . . . . . . . . . . . . . . . 211
8.7.3 The ρ-representation of g as a twisted representation of G . 212
8.7.4 Definition of the induction map . . . . . . . . . . . . . . . 213
8.7.5 The kernel of DM . . . . . . . . . . . . . . . . . . . . . . 215
9 D(g, k) as a geometric Dirac operator . . . . . . . . . . . . . . . . . 219
9.1 Differential operators on homogeneous spaces . . . . . . . . . . . 219
9.2 Dirac operators on manifolds . . . . . . . . . . . . . . . . . . . . 222
9.2.1 Linear connections . . . . . . . . . . . . . . . . . . . . . . 222
9.2.2 Principal connections . . . . . . . . . . . . . . . . . . . . 223
9.2.3 Dirac operators . . . . . . . . . . . . . . . . . . . . . . . . 225
9.3 Dirac operators on homogeneous spaces . . . . . . . . . . . . . . 227
10 The Hopf–Koszul–Samelson Theorem . . . . . . . . . . . . . . . . . 231
10.1 Lie algebra cohomology . . . . . . . . . . . . . . . . . . . . . . . 231
10.2 Lie algebra homology . . . . . . . . . . . . . . . . . . . . . . . . 233
10.2.1 Definition and basic properties . . . . . . . . . . . . . . . 233
10.2.2 Schouten bracket . . . . . . . . . . . . . . . . . . . . . . . 235
10.3 Lie algebra homology for reductive Lie algebras . . . . . . . . . . 238
10.3.1 Hopf algebra structure on (∧g)g . . . . . . . . . . . . . . 240
10.4 Primitive elements . . . . . . . . . . . . . . . . . . . . . . . . . . 241
10.5 Hopf–Koszul–Samelson Theorem . . . . . . . . . . . . . . . . . . 242
10.6 Consequences of the Hopf–Koszul–Samelson Theorem . . . . . . 244
10.7 Transgression Theorem . . . . . . . . . . . . . . . . . . . . . . . 245
11 The Clifford algebra of a reductive Lie algebra . . . . . . . . . . . . 249
11.1 Cl(g) and the ρ-representation . . . . . . . . . . . . . . . . . . . 249
11.2 Relationwithextremalprojectors . . . . . . . . . . . . . . . . . . 255
11.3 The isomorphism (Clg)g ∼=Cl(P (g)) . . . . . . . . . . . . . . . . 260
11.4 The ρ-decomposition of elements ξ ∈ g ⊆ Clg . . . . . . . . . . . 262
11.4.1 The space Homg(g,λ(Sg)) . . . . . . . . . . . . . . . . . 262
11.4.2 The space Homg(g,γ (Ug)) . . . . . . . . . . . . . . . . . 265
11.5 The Harish-Chandra projection of q(P(g)) ⊆ Clg . . . . . . . . . 269
11.6 RelationwiththeprincipalTDS . . . . . . . . . . . . . . . . . . . 271
Appendix A Graded and filtered super spaces . . . . . . . . . . . . . . . 275
A.1 Super vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . 275
A.2 Graded super vector spaces . . . . . . . . . . . . . . . . . . . . . 277
A.3 Filtered super vector spaces . . . . . . . . . . . . . . . . . . . . . 279
Appendix B Reductive Lie algebras . . . . . . . . . . . . . . . . . . . . 281
B.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . 281
B.2 Cartan subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 282
B.3 Representation theory of sl(2,C) . . . . . . . . . . . . . . . . . . 283
B.4 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
B.5 Simple roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
B.6 TheWeylgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
B.7 Weyl chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
B.8 Weightsof representations . . . . . . . . . . . . . . . . . . . . . . 293
B.9 Highest weight representations . . . . . . . . . . . . . . . . . . . 295
B.10 Extremalweights . . . . . . . . . . . . . . . . . . . . . . . . . . 298
B.11 Multiplicity computations . . . . . . . . . . . . . . . . . . . . . . 299
Appendix C Background on Lie groups . . . . . . . . . . . . . . . . . . 301
C.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
C.2 Group actionsonmanifolds . . . . . . . . . . . . . . . . . . . . . 302
C.3 The exponential map . . . . . . . . . . . . . . . . . . . . . . . . . 303
C.4 The vector field 12(ξL +ξR) . . . . . . . . . . . . . . . . . . . . . 306
C.5 Maurer–Cartan forms . . . . . . . . . . . . . . . . . . . . . . . . 307
C.6 Quadratic Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . 309
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Source: http://www.springer.com/mathematics/algebra/book/978-3-642-36215-6

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