Lectures in Mathematical Physics Lectures on Quantum Groups Pavel Etingof and Olivier Schiffinann Second Edition International Press * s. c *''.. \ir.ik,!.'.....
Contents Introduction ix 1 Poisson algebras and quantization 1 1.1 Modules over rings of power series 1 1.1.1 Topologically free if-modules 1 1.1.2 Completion of if-modules 2 1.2 Poisson algebras 2 1.2.1 Definition 2 1.2.2 Examples of Poisson algebras 3 1.3 Quantization of Poisson algebras 4 1.3.1 Deformations 4 1.3.2 Quantization 4 1.3.3 Examples of quantization 5 1.3.4 Loss of symmetry in quantization 7 1.4 Poisson manifolds and quantization 8 1.4.1 Definition 8 1.4.2. Symplectic leaves of a Poisson manifold 9 1.4.3 Quantization of Poisson manifolds 10 1.4.4 Example of quantization of a Poisson manifold (Geometric quantization) '. - 10 1.5 Rational forms of a quantization 11 1.6 Physical meaning of quantization. 12 2 Poisson-Lie groups 15 2.1 Poisson-Lie groups 15 2.1.1 Definition 15 2.2 Lie bialgebras 16 2.2.1 Definition 16 2.2.2 Examples of Lie bialgebras 18 2.2.3 Duality 19 2.3 Poisson-Lie theory 19 2.3.1 Main theorem of Poisson-Lie theory 19 2.3.2 Dual Poisson-Lie group 21 2.3.3 Examples of dual Lie bialgebras and dual Poisson-Lie groups 22 3 Coboundary Lie bialgebras 25 3.1 Some Lie algebra cohomology 25 3.2 Coboundary Lie bialgebras 26 3.3 The classical Yang-Baxter map 26 3.4 Triangular Lie bialgebras and the classical Yang-Baxter equation 28 iii
iv CONTENTS 3.5 Classification of triangular structures 29 3.6 Quasitriangular Lie bialgebras 31 3.7 Examples of coboundary, triangular and quasitriangular Lie bialgebras 32 4 Drinfeld's double construction 34 4.1 Manin triples 34 4.2 Drinfeld's double 35 4.3 Examples 38 4.4 Standard Lie bialgebra structure on simple Lie algebras 38 4.4.1 Notations 38 4.4.2 Standard structure 39 5 Belavin-Drinfeld classification (I) 41 5.1 Coboundary structure on simple Lie bialgebras 41 5.2 Skew-symmetric r-matrices 42 5.3 Non skew-symmetric r-matrices 42 5.4 Proof of the classification theorem 44 5.4.1 The Cayley transform 44 5.4.2 Proof of part 1) 45 5.4.3 Proof of part 2) 47 6 Infinite dimensional Lie bialgebras. 49 6.1 Infinite Manin triples 49 6.2 Examples 50 6.2.1 The standard structure on Kac-Moody algebras.... 50 6.3 The CYBE with spectral parameter \.... 51 6.3.1 An example: the Yangian and its dual 51 6.3.2 The CYBE with spectral parameter 53 6.3.3 Construction of a Lie bialgebra from an r-matrix 54 6.4 Solutions of the CYBE with spectral parameters. 55 6.5 Affine Lie algebras 57 6.5.1 Definition 57 6.5.2 Lie bialgebra structure 59 7 Belavin-Drinfeld classification (II) 60 7.1 Properties of nondegenerate solutions 60 7.2 Meromorphic continuation of r{z) to C 62 7.3 Proof of the classification theorem 64 7.3.1 Myberg's theorem 64 7.3.2 Elliptic solutions 65 7.3.3 Rational and trigonometric r-matrices 65
CONTENTS v 8 Hopf algebras 68 8.1 Definition of Hopf algebras 68 8.1.1 Finite groups revisited 68~ 8.1.2 Coalgebras 69 8.1.3 Hopf algebras 69 8.2 Pictorial representation 71 8.3 Examples of Hopf algebras 73 8.4 Duality in Hopf algebras 76 8.5 Deformation Hopf algebras 76 9 Quantized universal enveloping algebras 78 9.1 Quantized enveloping algebras 78 9.2 The quantization theorem 80 9.3 Examples 80 9.4 Coboundary, quasitriangular, triangular Hopf algebras 82 9.4.1 Coboundary Hopf algebras 82 9.4.2 (Quasi)triangular Hopf algebras 83 9.4.3 Modifications of the quantization theorem 86 9.5 Quantization by twists 87 10 Formal groups and /i-formal groups 90 10.1 Definition 90 10.2 Duality 91 10.3 R-matrices and R-forms 93 10.3.1 Comodules 93 10.3.2 Universal R-forms (coquasitriangular structures) 94 11 Infinite dimensional quantum groups 96 11.1 The RTT formalism and h-formal groups 96 11.1.1 Formal groups revisited 96 11.1.2 The RTT formalism 97 11.1.3 Examples 98 11.2 RTT formalism and quantum groups 99 11.3 Examples 101 11.3.1 TheYangian 101 11.3.2 The dual Yangian 101 11.3.3 Quantum elliptic algebra 102 11.3.4 Quantized affine Lie algebra 102 12 The quantum double 104 12.1 The quantum double 104 12.2 The quantum double for quantized universal enveloping algebras 110 12.3 Quasitriangular structure on Uh(g) HI
vi CONTENTS 13 Tensor categories and quasi-hopf algebras 114 13.1 Semigroup categories 114 13.1.1 Definition 114 13.1.2 Examples 116 13.1.3 Tensor functors - 117 13.2 Monoidal categories 119 13.2.1 Units in semigroup categories 119 13.2.2 MacLane's theorem 121 13.3 Quasi-bialgebras and quasi-hopf algebras 121 13.3.1 Definition 121 13.3.2 Equivalence of quasi-bialgebras and twists 123 13.3.3 "Nonabelian cohomology" 124 14 Braided tensor categories 126 14.1 Braided monoidal categories 126 14.1.1 Motivation 126 14.1.2 The braid group 127 14.1.3 Braided tensor functors 131 14.1.4 Braid group representations 131 14.1.5 Symmetric categories 131 14.2 Quasitriangular Quasi-Hopf algebras 131 14.2.1 Equivalence of quasitriangular quasi-hopf algebras... 133 15 KZ equations and the Drinfeld Category 134 15.1 The KZ equations: 134 15.1:1 Definition 134 15.1.2"Link with the CYBE 135 15.2 Monodromy of the KZ equations,... 136 15.2.1 The KZ associator L... 136 15.2.2 Quasi-Hopf structure 138 15.2.3 Braided (quasitriangular) structure 141 15.3 The Drinfeld category 144 15.4 Braid group representation 145 16 Quasi-Hopf quantized enveloping algebras 146 16.1 Quasi-Hopf quantized enveloping algebras 146 16.1.1 Definition 146 16.1.2 Examples.147 16.1.3 Twists 147 16.2 Lie quasibialgebras 147 16.2.1 Definition. 148 16.2.2 Quantization of Lie quasibialgebras 148 16.2.3 Quasitriangular Lie quasibialgebras -149 16.2.4 Twists 149 16.3 Associators 150 16.3.1 Definition 150
CONTENTS vii 16.3.2 Action of twists on Ass(g, ft) 151 16.4 Classification of quasitriangular quasi-hopf QUE algebras... 151 16.5 The Drinfeld-Kohno theorem...152 16.6 Geometric interpretation of Lie quasibialgebras 153 17 Lie associators 157 17.1 Lie associators 157 17.1.1 Definition 157 17.1.2 The space of Lie associators 159 17.2 The Grothendieck-Teichmuller group 159 17.2.1 Definition 159 17.2.2 The action of GTi(k) on completed braid groups 163 17.2.3 Drinfeld's conjecture 164 18 Fiber functors and Tannaka-Krein duality 165 18.1 Tensor categories 165 18.2 Fiber functor 165 18.2.1 First example 165 18.2.2 Tannaka-Krein duality 167 18.2.3 Tannaka-Krein duality for bialgebras 168 19 Quantization of finite dimensional Lie bialgebras, 170 19.1 Quantization of the Drinfeld double 170 19.1.1 The Drinfeld category 170 19.1.2 The forgetful functor - 171 19.1.3 The Verma modules 171 19.1.4 Tensor structure on the forgetful functor 172 19.1.5 Quantization of g. 176 19.2 Quantization of finite-dimensional Lie bialgebras. 177 19.3 Quasitriangular quantization 183 19.4 Quantization of r-matrices 183 20 Universal constructions 185 20.1 Cyclic categories 185 20.1.1 Definition 185 20.1.2 Basic notions related to cyclic categories 185 20.1.3 Linear algebraic structures 186 20.2 Universal constructions 188 20.2.1 Acyclic tensor calculus 190 21 Universal quantization 192 21.1 Statement of the theorem 192 21.2 Quantization of finite-dimensional Lie bialgebras revisited... 193 21.3 Categorical Drinfeld double 198 21.4 The Drinfeld category 200 21.5 Quantization of g+ 201
viii CONTENTS 21.6 Quantization of Poisson-Lie groups 203 22 DEQUANTIZATION AND THE EQUIVALENCE... 205 22.1 THE QUANTUM DOUBLE IN A SYMMETRIC 205 22.2 Dequantization 208 1 KZ ASSOCIATOR AND MULTIPLE... 209 1.1 The multiple zeta function 209 1.2 Multiple zeta values and the KZ equation 210 1.3 The relations between multiple zeta values 212 2 Solutions to Problems and Exercises 214