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Clifford Algebras and Lie Groups Lecture Notes, University of Toronto Clifford algebras and Lie groups Eckhard Meinrenken Lecture Notes, University of Toronto, Fall 2009. Last revision: 12/2011 Contents Chapter 1. Symmetric bilinear forms 7 1. Quadratic vector spaces 7 2. Isotropic subspaces 9 3. Split bilinear forms 10 4. E.Cartan-Dieudonn´e'sTheorem 13 5. Witt's Theorem 16 6. Orthogonal groups for K = R; C 17 7. Lagrangian Grassmannians 23 Chapter 2. Clifford algebras 27 1. Exterior algebras 27 1.1. Definition 27 1.2. Universal property, functoriality 28 2. Clifford algebras 29 2.1. Definition and first properties 30 2.2. Universal property, functoriality 31 2.3. The Clifford algebras Cl(n; m) 32 2.4. The Clifford algebras Cl(n) 33 2.5. Symbol map and quantization map 34 2.6. Transposition 35 2.7. Chirality element 36 2.8. The trace and the super-trace 37 2.9. Extension of the bilinear form 38 2.10. Lie derivatives and contractions 38 2.11. The Lie algebra q(^2(V )) 40 2.12. A formula for the Clifford product 41 3. The Clifford algebra as a quantization of the exterior algebra 42 3.1. Differential operators 42 3.2. Graded Poisson algebras 44 3.3. Graded super Poisson algebras 45 3.4. Poisson structures on ^(V ) 46 Chapter 3. The spin representation 49 1. The Clifford group and the spin group 49 1.1. The Clifford group 49 1.2. The groups Pin(V ) and Spin(V ) 51 2. Clifford modules 54 2.1. Basic constructions 54 2.2. The spinor module SF 56 2.3. The dual spinor module SF 57 3 2.4. Irreducibility of the spinor module 58 2.5. Abstract spinor modules 59 3. Pure spinors 61 4. The canonical bilinear pairing on spinors 64 × 5. The character χ: Γ(V )F ! K 67 6. Cartan's Triality Principle 69 7. The Clifford algebra Cl(V ) 72 7.1. The Clifford algebra Cl(V ) 72 7.2. The groups Spinc(V ) and Pinc(V ) 73 7.3. Spinor modules over Cl(V ) 75 7.4. Classification of irreducible Cl(V )-modules 77 7.5. Spin representation 78 7.6. Applications to compact Lie groups 81 Chapter 4. Covariant and contravariant spinors 85 1. Pull-backs and push-forwards of spinors 85 2. Factorizations 88 2.1. The Lie algebra o(V ∗ ⊕ V ) 88 2.2. The group SO(V ∗ ⊕ V ) 89 2.3. The group Spin(V ∗ ⊕ V ) 90 3. The quantization map revisited 91 3.1. The symbol map in terms of the spinor module 92 3.2. The symbol of elements in the spin group 92 3.3. Another factorization 94 3.4. The symbol of elements exp(γ(A)) 95 3.5. Clifford exponentials versus exterior algebra exponentials 96 P i 3.6. The symbol of elements exp(γ(A) − i eiτ ) 97 3.7. The function A 7! S(A). 99 4. Volume forms on conjugacy classes 100 Chapter 5. Enveloping algebras 105 1. The universal enveloping algebra 105 1.1. Construction 105 1.2. Universal property 105 1.3. Augmentation map, anti-automorphism 106 1.4. Anti-automorphism 106 1.5. Derivations 106 1.6. Modules over U(g) 107 1.7. Unitary representations 107 1.8. Graded or filtered Lie (super) algebras 107 1.9. Further remarks 108 2. The Poincar´e-Birkhoff-Witt theorem 108 3. U(g) as left-invariant differential operators 111 4. The enveloping algebra as a Hopf algebra 112 4.1. Hopf algebras 112 4.2. Hopf algebra structure on S(E) 115 4.3. Hopf algebra structure on U(g) 115 4.4. Primitive elements 117 4.5. Coderivations 118 4 CHAPTER 0. CONTENTS 4.6. Coderivations of S(E) 119 5. Petracci's proof of the Poincar´e-Birkhoff-Witttheorem 120 5.1. A g-representation by coderivations 121 5.2. The formal vector fields Xζ (φ) 122 5.3. Proof of Petracci's theorem 124 6. The center of the enveloping algebra 124 Chapter 6. Weil algebras 129 1. Differential spaces 129 2. Symmetric and tensor algebra over differential spaces 130 3. Homotopies 131 4. Koszul algebras 133 5. Symmetrization 135 6. g-differential spaces 136 7. The g-differential algebra ^g∗ 139 8. g-homotopies 141 9. The Weil algebra 142 10. Chern-Weil homomorphisms 144 11. The non-commutative Weil algebra W~ g 146 12. Equivariant cohomology of g-differential spaces 148 13. Transgression in the Weil algebra 150 Chapter 7. Quantum Weil algebras 151 1. The g-differential algebra Cl(g) 151 2. The quantum Weil algebra 155 2.1. Poisson structure on the Weil algebra 155 2.2. Definition of the quantum Weil algebra 157 2.3. The cubic Dirac operator 158 2.4. Wg as a level 1 enveloping algebra 159 3. Application: Duflo’s theorem 160 4. Relative Dirac operators 162 5. Harish-Chandra projections 168 5.1. Harish-Chandra projections for enveloping algebras 168 5.2. Harish-Chandra projections for Clifford algebras 170 5.3. Harish-Chandra projections for quantum Weil algebras 173 Chapter 8. Applications to reductive Lie algebras 175 1. Notation 175 2. Harish-Chandra projections 175 3. The ρ-representation and the representation theory of Cl(g) 177 4. Equal rank subalgebras 181 5. The kernel of DV 186 6. q-dimensions 189 7. The shifted Dirac operator 191 8. Dirac induction 192 8.1. Central extensions of compact Lie groups 192 8.2. Twisted representations 194 8.3. The ρ-representation of g as a twisted representation of G 195 8.4. Definition of the induction map 196 5 8.5. The kernel of DM 198 Chapter 9. D(g; k) as a geometric Dirac operator 201 1. Differential operators on homogeneous spaces 201 2. Geometric Dirac operators 203 3. Dirac operators over G 204 4. Dirac operators over G=K 206 Appendix A. Graded and filtered super spaces 209 1. Super vector spaces 209 2. Graded super vector spaces 211 3. Filtered super vector spaces 213 Appendix B. Reductive Lie algebras 215 1. Definitions and basic properties 215 2. Cartan subalgebras 216 3. Representation theory of sl(2; C). 217 4. Roots 218 5. Simple roots 221 6. The Weyl group 221 7. Weyl chambers 224 8. Weights of representations 225 9. Highest weight representations 227 10. Extremal weights 230 Appendix C. Background on Lie groups 231 1. Preliminaries 231 2. Group actions on manifolds 232 3. The exponential map 233 1 L R 4. The vector field 2 (ξ + ξ ) 235 5. Maurer-Cartan forms 236 6. Quadratic Lie groups 238 Bibliography 241 6 CHAPTER 1 Symmetric bilinear forms In this section, we will describe the foundations of the theory of non- degenerate symmetric bilinear forms on finite-dimensional vector spaces, and their orthogonal groups. Among the highlights of this Chapter are the Cartan-Dieudonn´etheorem, which states that any orthogonal transfor- mation is a finite product of reflections, and Witt's theorem giving a partial normal form for quadratic forms. The theory of split symmetric bilinear forms is found to have many parallels to the theory of symplectic forms, and we will give a discussion of the Lagrangian Grassmannian for this case. Throughout, K will denote a ground field of characteristic 6= 2. We are mainly interested in the cases K = R or C, and sometimes specialize to those two cases. 1. Quadratic vector spaces Suppose V is a finite-dimensional vector space over K. For any bilinear form B : V × V ! K, define a linear map B[ : V ! V ∗; v 7! B(v; ·): The bilinear form B is called symmetric if it satisfies B(v1; v2) = B(v2; v1) [ ∗ [ for all v1; v2 2 V . Since dim V < 1 this is equivalent to (B ) = B . The symmetric bilinear form B is uniquely determined by the associated quadratic form, QB(v) = B(v; v) by the polarization identity, 1 (1) B(v; w) = 2 QB(v + w) − QB(v) − QB(w) : The kernel (also called radical) of B is the subspace ker(B) = fv 2 V j B(v; v1) = 0 for all v1 2 V g; i.e. the kernel of the linear map B[. The bilinear form B is called non- degenerate if ker(B) = 0, i.e. if and only if B[ is an isomorphism. A vector space V together with a non-degenerate symmetric bilinear form B will be referred to as a quadratic vector space. Assume for the rest of this chapter that (V; B) is a quadratic vector space. Definition 1.1. A vector v 2 V is called isotropic if B(v; v) = 0, and non-isotropic if B(v; v) 6= 0. n For instance, if V = C over K = C, with the standard bilinear form Pn B(z; w) = i=1 ziwi, then v = (1; i; 0;:::; 0) is an isotropic vector. If 7 1. QUADRATIC VECTOR SPACES 2 V = R over K = R, with bilinear form B(x; y) = x1y1 − x2y2, then the set of isotropic vectors x = (x1; x2) is given by the `light cone' x1 = ±x2. The orthogonal group O(V ) is the group (2) O(V ) = fA 2 GL(V )j B(Av; Aw) = B(v; w) for all v; w 2 V g: The subgroup of orthogonal transformations of determinant 1 is denoted SO(V ), and is called the special orthogonal group. For any subspace F ⊂ V , the orthogonal or perpendicular subspace is defined as ? F = fv 2 V j B(v; v1) = 0 for all v1 2 F g: The image of B[(F ?) ⊂ V ∗ is the annihilator of F .
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