**Mathematical Surveys and Monographs**

Volume: 58;
1998;
198 pp;
Hardcover

MSC: Primary 81;
Secondary 17

**Print ISBN: 978-0-8218-0496-4
Product Code: SURV/58**

List Price: $67.00

AMS Member Price: $53.60

MAA Member Price: $60.30

**Electronic ISBN: 978-1-4704-1285-2
Product Code: SURV/58.E**

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# Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations

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*Pavel I. Etingof; Igor B. Frenkel; Alexander A. Kirillov, Jr.*

This book is devoted to mathematical structures arising in conformal field theory and the \(q\)-deformations. The authors give a self-contained exposition of the theory of Knizhnik-Zamolodchikov equations and related topics. No previous knowledge of physics is required. The text is suitable for a one-semester graduate course.

#### Readership

Graduate students and research mathematicians interested in mathematical physics.

#### Table of Contents

# Table of Contents

## Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations

- Contents ix10 free
- Preface xiii14 free
- Lecture 1. Introduction 116 free
- 1.1. Simple Lie algebras and Lie groups and their generalizations 116
- 1.2. Affine Lie algebras 116
- 1.3. Quantum groups 318
- 1.4. Knizhnik-Zamolodchikov equations 621
- 1.5. Quantum affine algebras and quantum Knizhnik-Zamolodchikov equations 823
- 1.6. Further generalizations of affine Lie algebras and quantum groups 1126
- 1.7. Contents of the book 1227

- Lecture 2. Representations of finite-dimensional and affine Lie algebras 1530
- 2.1. Simple Lie algebras 1530
- 2.2. Cartan matrices of simple Lie algebras 1631
- 2.3. Highest-weight modules over simple Lie algebras and contravariant forms 1732
- 2.4. Finite-dimensional representations and irreducibility of Verma modules 1833
- 2.5. The maximal root, the Coxeter numbers, and the Casimir operator 1934
- 2.6. Affine Lie algebras 2035
- 2.7. Verma modules and Weyl modules for affine Lie algebras 2237
- 2.8. Integrable representations of affine Lie algebras 2439
- 2.9. The Virasoro algebra and its action on g-modules 2540
- 2.10. Generating functions and currents 2641

- Lecture 3. Knizhnik-Zamolodchikov equations 2944
- 3.1. Classification of intertwining operators 2944
- 3.2. Operator KZ equation 3045
- 3.3. Gauge invariance of the intertwining operators 3348
- 3.4. KZ equations for correlation functions 3348
- 3.5. Consistency and g-invariance of the KZ equations 3651
- 3.6. Analyticity of the correlation functions 3752
- 3.7. Correlation functions span the space of solutions of the KZ equations 3954
- 3.8. Trigonometric form of the KZ equations 4257
- 3.9. Consistent systems of differential equations and the classical Yang-Baxter equation 4459

- Lecture 4. Solutions of the Knizhnik- Zamolodchikov equations 4964
- 4.1. The simplest solution of the KZ equations for g=sl[sub(2)] 4964
- 4.2. Simplest level one solution and Gauss hypergeometric function 5166
- 4.3. Integral formulas for level one solutions 5368
- 4.4. Solutions of the KZ equations for sl[sub(2)]: arbitrary level 5671
- 4.5. Solutions of the KZ equations for a general simple Lie algebra 6075

- Lecture 5. Free field realization 6378
- 5.1. Fock modules and vertex operators 6378
- 5.2. Matrix elements of products of vertex operators 6681
- 5.3. Interpretation of the rational part of solutions of the KZ equations in terms of creation and annihilation operators 6782
- 5.4. Factorization of solutions of the KZ equations 6984
- 5.5. Free field realization of Verma modules over sl[sub(2)] 6984
- 5.6. Intertwining operators in the free field realization: level zero 7388
- 5.7. Intertwining operators in the free field realization: positive level 7590
- 5.8. Calculation of the correlation functions 7792

- Lecture 6. Quantum groups 7994
- 6.1. Hopf algebras and their representations 7994
- 6.2. Definition of quantum groups 8196
- 6.3. Quasitriangular structure and braided tensor categories 8499
- 6.4. Quantum Yang-Baxter equation and representations of braid groups 87102
- 6.5. Quantum double construction 88103
- 6.6. Quantum double construction for U[sub(q)](ge) 89104
- 6.7. Quantum Casimir element 92107
- 6.8. Intertwining operators and their commutation relations 93108

- Lecture 7. Local systems and configuration spaces 97112
- 7.1. Local systems 97112
- 7.2. Cohomology and homology with coefficients in local systems 99114
- 7.3. Configuration spaces and Orlik-Solomon algebra 101116
- 7.4. Cohomology of configuration spaces with coefficients in local systems associated with the KZ equations for sl[sub(2)] 103118
- 7.5. Gauss-Manin connection 105120
- 7.6. Relative homology 106121
- 7.7. The case of arbitrary g 110125

- Lecture 8. Monodromy of Knizhnik-Zamolodchikov equations 113128
- 8.1. Monodromy of KZ equations and the braid group 113128
- 8.2. Asymptotics of solutions of the KZ equations 115130
- 8.3. Asymptotics of the correlation functions 118133
- 8.4. Monodromy with respect to an infinite base point 119134
- 8.5. Commutation relations for intertwining operators 122137
- 8.6. Equivalence of categories and Drinfeld-Kohno theorem 124139
- 8.7. Geometric approach to equivalence of categories 127142

- Lecture 9. Quantum affine algebras 131146
- 9.1. Definition of quantum affine algebras 131146
- 9.2. Evaluation representations of quantum affine algebras 132147
- 9.3. Intertwining operators 135150
- 9.4. Quasitriangular structure in quantum affine algebras 136151
- 9.5. Factorization of the R-matrix 137152
- 9.6. Evaluation representations and R-matrix for u[sub(q)](sl[sub(2)]) 139154
- 9.7. Quantum currents 144159
- 9.8. Quantum Sugawara construction in degree zero 145160

- Lecture 10. Quantum Knizhnik-Zamolodchikov equations 147162
- 10.1. Operator quantum KZ equation 147162
- 10.2. Quantum correlation functions 150165
- 10.3. Quantum KZ equations for correlation functions 151166
- 10.4. A fundamental set of solutions of the quantum KZ equations 151166
- 10.5. Holonomic systems of difference equations 152167
- 10.6. Analyticity of the fundamental solution of the quantum KZ equations 153168
- 10.7. The noncommutative product formula for the fundamental solution 155170
- 10.8. Classical limit of the quantum KZ equations 155170
- 10.9. Modified quantum KZ equations 156171
- 10.10. Another proof of the quantum KZ equations 157172

- Lecture 11. Solutions of the quantum Knizhnik-Zamolodchikov equations for sl[sub(2)] 161176
- 11.1. q-analogues of classical special functions 161176
- 11.2. Jackson integral 163178
- 11.3. The q-hypergeometric function 164179
- 11.4. Some second order difference equations 165180
- 11.5. The simplest solutions of the quantum KZ equations and the q-hypergeometric function 166181
- 11.6. Integral formulas for solutions 168183

- Lecture 12. Connection matrices for the quantum Knizhnik-Zamolodchikov equations and elliptic functions 171186
- 12.1. Linear difference equations for functions of one complex variable 171186
- 12.2. Connection relation for the g-hypergeometric equation 174189
- 12.3. The connection matrix for the quantum KZ equations in the simplest case 175190
- 12.4. The connection matrix and the exchange matrix for intertwining operators 176191

- Lecture 13. Current developments and future perspectives 179194
- 13.1. KZ equations: quantum versus classical 179194
- 13.2. Monodromy of the KZ equations, tensor categories, and quantum groups 180195
- 13.3. Vertex operator algebras, conformal field theory and their q-deformations 182197
- 13.4. Elliptic KZ equations and special values of the central charge 185200
- 13.5. Double loop algebras and quantum affine algebras 186201
- 13.6. Quantum KZ equations and physical models 187202

- References 189204
- Index 197212 free