Clifford Algebras and Lie Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 58)

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. Cover 1 Clifford Algebras and Lie Theory 3 Preface 6 Acknowledgments 9 Conventions 10 Contents 11 List of Symbols 16 Chapter 1: Symmetric bilinear forms 18 1.1 Quadratic vector spaces 18 1.2 Isotropic subspaces 20 1.3 Split bilinear forms 22 1.4 E. Cartan-Dieudonné Theorem 24 1.5 Witt's Theorem 28 1.6 Orthogonal groups for K=R,C 29 1.7 Lagrangian Grassmannians 35 Chapter 2: Clifford algebras 39 2.1 Exterior algebras 39 2.1.1 Definition 39 2.1.2 Universal property, functoriality 40 2.1.3 Derivations 41 2.1.4 Transposition 42 2.1.5 Duality pairings 42 2.2 Clifford algebras 43 2.2.1 Definition and first properties 43 2.2.2 Universal property, functoriality 45 2.2.3 The Clifford algebras Cl(n,m) 46 2.2.4 The Clifford algebras Cl(n) 47 2.2.5 Symbol map and quantization map 48 2.2.6 Transposition 50 2.2.7 Chirality element 51 2.2.8 The trace and the super-trace 52 2.2.9 Lie derivatives and contractions 53 2.2.10 The Lie algebra q(2(V)) 55 2.2.11 A formula for the Clifford product 57 2.3 The Clifford algebra as a quantization 58 2.3.1 Differential operators 58 2.3.2 Graded Poisson algebras 60 2.3.3 Graded super Poisson algebras 61 2.3.4 Poisson structures on (V) 62 Chapter 3: The spin representation 65 3.1 The Clifford group and the spin group 65 3.1.1 The Clifford group 65 3.1.2 The groups Pin(V) and Spin(V) 67 3.2 Clifford modules 70 3.2.1 Basic constructions 70 3.2.2 The spinor module SF 72 3.2.3 The dual spinor module SF 74 3.2.4 Irreducibility of the spinor module 75 3.2.5 Abstract spinor modules 76 3.3 Pure spinors 78 3.4 The canonical bilinear pairing on spinors 81 3.5 The character chi: Gamma(V)F->Kx 85 3.6 Cartan's triality principle 86 3.7 The Clifford algebra Cl(V) 90 3.7.1 The Clifford algebra Cl(V) 90 3.7.2 The groups Spinc(V) and Pinc(V) 91 3.7.3 Spinor modules over Cl(V) 93 3.7.4 Classification of irreducible Cl(V)-modules 95 3.7.5 Spin representation 96 3.7.6 Applications to compact Lie groups 99 Spin(3) 99 Spin(4) 99 Spin(5) 99 Spin(6) 100 Spin(7) 100 Spin(8) 101 Chapter 4: Covariant and contravariant spinors 102 4.1 Pull-backs and push-forwards of spinors 102 4.2 Factorizations 105 4.2.1 The Lie algebra o(V*V) 105 4.2.2 The group SO(V*V) 106 4.2.3 The group Spin(V*V) 107 4.3 The quantization map revisited 109 4.3.1 The symbol map in terms of the spinor module 109 4.3.2 The symbol of elements in the spin group 110 4.3.3 Another factorization 112 4.3.4 The symbol of elements exp(gamma(A)) 114 4.3.5 Clifford exponentials versus exterior algebra exponentials 114 4.3.6 The symbol of elements exp(gamma(A)-i ei taui) 116 4.3.7 The function A->S(A) 118 4.4 Volume forms on conjugacy classes 118 Chapter 5: Enveloping algebras 123 5.1 The universal enveloping algebra 123 5.1.1 Construction 123 5.1.2 Universal property 124 5.1.3 Augmentation map, anti-automorphism 124 5.1.4 Derivations 125 5.1.5 Modules over U(g) 125 5.1.6 Unitary representations 125 5.1.7 Graded or filtered Lie algebras and super Lie algebras 126 5.1.8 Further remarks 126 5.2 The Poincaré-Birkhoff-Witt Theorem 127 5.3 U(g) as left-invariant differential operators 130 5.4 The enveloping algebra as a Hopf algebra 132 5.4.1 Hopf algebras 132 5.4.2 Hopf algebra structure on S(E) 134 5.4.3 Hopf algebra structure on U(g) 135 5.4.4 Primitive elements 137 5.4.5 Coderivations 138 5.4.6 Coderivations of S(E) 139 5.5 Petracci's proof of the Poincaré-Birkhoff-Witt Theorem 140 5.5.1 A g-representation by coderivations 141 5.5.2 The formal vector fields Xzeta(phi) 142 5.5.3 Proof of Petracci's Theorem 144 5.6 The center of the enveloping algebra 145 Chapter 6: Weil algebras 148 6.1 Differential spaces 148 6.2 Symmetric and tensor algebra over differential spaces 150 6.3 Homotopies 150 6.4 Koszul algebras 153 6.5 Symmetrization 154 6.6 g-differential spaces 156 6.7 The g-differential algebra g* 158 6.8 g-homotopies 161 6.9 The Weil algebra 161 6.10 Chern-Weil homomorphisms 164 6.11 The non-commutative Weil algebra Wg 166 6.12 Equivariant cohomology of g-differential spaces 169 6.13 Transgression in the Weil algebra 171 Chapter 7: Quantum Weil algebras 176 7.1 The g-differential algebra Cl(g) 176 7.2 The quantum Weil algebra 180 7.2.1 Poisson structure on the Weil algebra 180 7.2.2 Definition of the quantum Weil algebra 182 7.2.3 The cubic Dirac operator 184 7.2.4 W(g) as a level 1 enveloping algebra 185 7.2.5 Conjugation 186 7.3 Application: Duflo's Theorem 187 7.4 Relative Dirac operators 189 7.5 Harish-Chandra projections 195 7.5.1 Enveloping algebras 195 7.5.2 Clifford algebras 197 7.5.3 Quantum Weil algebras 201 Chapter 8: Applications to reductive Lie algebras 204 8.1 Notation 204 8.2 Harish-Chandra projections 205 8.2.1 Harish-Chandra projection for U(g) 205 8.2.2 Harish-Chandra projection of the quadratic Casimir 207 8.2.3 Harish-Chandra projection for Cl(g) 208 8.3 Equal rank subalgebras 210 8.4 The kernel of DV 216 8.5 q-dimensions 219 8.6 The shifted Dirac operator 221 8.7 Dirac induction 222 8.7.1 Central extensions of compact Lie groups 222 8.7.2 Twisted representations 224 8.7.3 The rho-representation of g as a twisted representation of G 225 8.7.4 Definition of the induction map 226 8.7.5 The kernel of DM 228 Chapter 9: D(g,k) as a geometric Dirac operator 231 9.1 Differential operators on homogeneous spaces 231 9.2 Dirac operators on manifolds 234 9.2.1 Linear connections 234 9.2.2 Principal connections 235 9.2.3 Dirac operators 237 9.3 Dirac operators on homogeneous spaces 239 Chapter 10: The Hopf-Koszul-Samelson Theorem 242 10.1 Lie algebra cohomology 242 10.2 Lie algebra homology 244 10.2.1 Definition and basic properties 244 10.2.2 Schouten bracket 246 10.3 Lie algebra homology for reductive Lie algebras 249 10.3.1 Hopf algebra structure on (g)g 251 10.4 Primitive elements 252 10.5 Hopf-Koszul-Samelson Theorem 253 10.6 Consequences of the Hopf-Koszul-Samelson Theorem 255 10.7 Transgression Theorem 256 Chapter 11: The Clifford algebra of a reductive Lie algebra 260 11.1 Cl(g) and the rho-representation 260 11.2 Relation with extremal projectors 266 11.3 The isomorphism (Clg)g=Cl(P(g)) 271 11.4 The rho-decomposition of elements xigClg 273 11.4.1 The space Homg(g,lambda(Sg)) 273 11.4.2 The space Homg(g,gamma(Ug)) 276 11.5 The Harish-Chandra projection of q(P(g))Clg 280 11.6 Relation with the principal TDS 282 Appendix A: Graded and filtered super spaces 286 A.1 Super vector spaces 286 A.2 Graded super vector spaces 288 A.3 Filtered super vector spaces 290 Appendix B: Reductive Lie algebras 292 B.1 Definitions and basic properties 292 B.2 Cartan subalgebras 293 B.3 Representation theory of sl(2,C) 294 B.4 Roots 295 B.5 Simple roots 298 B.6 The Weyl group 299 B.7 Weyl chambers 302 B.8 Weights of representations 304 B.9 Highest weight representations 306 B.10 Extremal weights 309 B.11 Multiplicity computations 310 Appendix C: Background on Lie groups 312 C.1 Preliminaries 312 C.2 Group actions on manifolds 313 C.3 The exponential map 314 C.4 The vector field 12(xiL+xiR) 317 C.5 Maurer-Cartan forms 318 C.6 Quadratic Lie groups 320 References 322 Index 327 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