Lectures on representation theory and Knizhnik-Zamolodchikov equations:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[1998]
|
| Series: | Mathematical surveys and monographs
Volume 58 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 198 Seiten Diagramme |
| ISBN: | 0821804960 9780821804964 |
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MARC
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| 245 | 1 | 0 | |a Lectures on representation theory and Knizhnik-Zamolodchikov equations |c Pavel I. Etingof ; Igor B. Frenkel ; Alexander A. Kirillov Jr. |
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Record in the Search Index
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| adam_text | Contents
Preface xiii
Lecture 1. Introduction 1
1.1. Simple Lie algebras and Lie groups and their generalizations 1
1.2. Affine Lie algebras 1
1.3. Quantum groups 3
1.4. Knizhnik Zamolodchikov equations 6
1.5. Quantum affine algebras and quantum Knizhnik Zamolodchikov
equations 8
1.6. Further generalizations of affine Lie algebras and quantum groups. 11
1.7. Contents of the book 12
Lecture 2. Representations of finite dimensional and affine Lie algebras 15
2.1. Simple Lie algebras 15
2.2. Cartan matrices of simple Lie algebras 16
2.3. Highest weight modules over simple Lie algebras and contravariant
forms 17
2.4. Finite dimensional representations and irreducibility of Verma mod¬
ules 18
2.5. The maximal root, the Coxeter numbers, and the Casimir operator 19
2.6. Affine Lie algebras 20
2.7. Verma modules and Weyl modules for affine Lie algebras 22
2.8. Integrable representations of affine Lie algebras 24
2.9. The Virasoro algebra and its action on g modules 25
2.10. Generating functions and currents 26
Lecture 3. Knizhnik Zamolodchikov equations 29
3.1. Classification of intertwining operators 29
3.2. Operator KZ equation 30
3.3. Gauge invariance of the intertwining operators 33
3.4. KZ equations for correlation functions 33
3.5. Consistency and g invariance of the KZ equations 36
3.6. Analyticity of the correlation functions 37
3.7. Correlation functions span the space of solutions of the KZ equa¬
tions 39
3.8. Trigonometric form of the KZ equations 42
3.9. Consistent systems of differential equations and the classical Yang
Baxter equation 44
ix
x CONTENTS
Lecture 4. Solutions of the Knizhnik Zamolodchikov equations 49
4.1. The simplest solution of the KZ equations for g = 3(2 49
4.2. Simplest level one solution and Gauss hypergeometric function 51
4.3. Integral formulas for level one solutions 53
4.4. Solutions of the KZ equations for 5(2: arbitrary level 56
4.5. Solutions of the KZ equations for a general simple Lie algebra 60
Lecture 5. Free field realization 63
5.1. Fock modules and vertex operators 63
5.2. Matrix elements of products of vertex operators 66
5.3. Interpretation of the rational part of solutions of the KZ equations
in terms of creation and annihilation operators. 67
5.4. Factorization of solutions of the KZ equations 69
5.5. Free field realization of Verma modules over 3(2 69
5.6. Intertwining operators in the free field realization: level zero 73
5.7. Intertwining operators in the free field realization: positive level 75
5.8. Calculation of the correlation functions 77
Lecture 6. Quantum groups 79
6.1. Hopf algebras and their representations 79
6.2. Definition of quantum groups 81
6.3. Quasitriangular structure and braided tensor categories 84
6.4. Quantum Yang Baxter equation and representations of braid
groups 87
6.5. Quantum double construction 88
6.6. Quantum double construction for Uq(Qe) 89
6.7. Quantum Casimir element 92
6.8. Intertwining operators and their commutation relations 93
Lecture 7. Local systems and configuration spaces 97
7.1. Local systems 97
7.2. Cohomology and homology with coefficients in local systems 99
7.3. Configuration spaces and Orlik Solomon algebra 101
7.4. Cohomology of configuration spaces with coefficients in local sys¬
tems associated with the KZ equations for 5(2 103
7.5. Gauss Manin connection 105
7.6. Relative homology 106
7.7. The case of arbitrary g 110
Lecture 8. Monodromy of Knizhnik Zamolodchikov equations 113
8.1. Monodromy of KZ equations and the braid group 113
8.2. Asymptotics of solutions of the KZ equations 115
8.3. Asymptotics of the correlation functions 118
8.4. Monodromy with respect to an infinite base point 119
8.5. Commutation relations for intertwining operators 122
8.6. Equivalence of categories and Drinfeld Kohno theorem 124
8.7. Geometric approach to equivalence of categories 127
Lecture 9. Quantum affine algebras 131
9.1. Definition of quantum affine algebras 131
9.2. Evaluation representations of quantum affine algebras 132
CONTENTS xi
9.3. Intertwining operators 135
9.4. Quasitriangular structure in quantum affine algebras 136
9.5. Factorization of the .R matrix 137
9.6. Evaluation representations and .R matrix for W,(sb) 139
9.7. Quantum currents 144
9.8. Quantum Sugawara construction in degree zero 145
Lecture 10. Quantum Knizhnik Zamolodchikov equations 147
10.1. Operator quantum KZ equation 147
10.2. Quantum correlation functions 150
10.3. Quantum KZ equations for correlation functions 151
10.4. A fundamental set of solutions of the quantum KZ equations 151
10.5. Holonomic systems of difference equations 152
10.6. Analyticity of the fundamental solution of the quantum KZ equa¬
tions 153
10.7. The noncommutative product formula for the fundamental solution 155
10.8. Classical limit of the quantum KZ equations 155
10.9. Modified quantum KZ equations 156
10.10 Another proof of the quantum KZ equations 157
Lecture 11. Solutions of the quantum Knizhnik Zamolodchikov equations for
sl2 161
11.1. q analogues of classical special functions 161
11.2. Jackson integral 163
11.3. The q hypergeometric function 164
11.4. Some second order difference equations 165
11.5. The simplest solutions of the quantum KZ equations and the q
hypergeometric function 166
11.6. Integral formulas for solutions 168
Lecture 12. Connection matrices for the quantum Knizhnik Zamolodchikov
equations and elliptic functions 171
12.1. Linear difference equations for functions of one complex variable 171
12.2. Connection relation for the g hypergeometric equation 174
12.3. The connection matrix for the quantum KZ equations in the sim¬
plest case 175
12.4. The connection matrix and the exchange matrix for intertwining
operators 176
Lecture 13. Current developments and future perspectives 179
13.1. KZ equations: quantum versus classical 179
13.2. Monodromy of the KZ equations, tensor categories, and quantum
groups 180
13.3. Vertex operator algebras, conformal field theory and their q defor
mations 182
13.4. Elliptic KZ equations and special values of the central charge 185
13.5. Double loop algebras and quantum affine algebras 186
13.6. Quantum KZ equations and physical models 187
References 189
Index 197
|
| any_adam_object | 1 |
| author | Etingof, Pavel 1969- Frenkel, Igor B. 1952- Kirillov, Alexander Jr. 1967- |
| author_GND | (DE-588)133258858 (DE-588)172080584 (DE-588)135930243 |
| author_facet | Etingof, Pavel 1969- Frenkel, Igor B. 1952- Kirillov, Alexander Jr. 1967- |
| author_role | aut aut aut |
| author_sort | Etingof, Pavel 1969- |
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| dewey-ones | 530 - Physics |
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| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV012373506 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:26:27Z |
| institution | BVB |
| isbn | 0821804960 9780821804964 |
| language | English |
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| physical | xiv, 198 Seiten Diagramme |
| publishDate | 1998 |
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| series2 | Mathematical surveys and monographs |
| spelling | Etingof, Pavel 1969- Verfasser (DE-588)133258858 aut Lectures on representation theory and Knizhnik-Zamolodchikov equations Pavel I. Etingof ; Igor B. Frenkel ; Alexander A. Kirillov Jr. Providence, Rhode Island American Mathematical Society [1998] © 1998 xiv, 198 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs Volume 58 Deformation <Mathematik> - Analytische Funktion - Quantengruppe Symmetriebrechung (DE-588)4184200-5 gnd rswk-swf Quantengruppe (DE-588)4252437-4 gnd rswk-swf Symmetriebrechung (DE-588)4184200-5 s Quantengruppe (DE-588)4252437-4 s DE-604 Frenkel, Igor B. 1952- Verfasser (DE-588)172080584 aut Kirillov, Alexander Jr. 1967- Verfasser (DE-588)135930243 aut Erscheint auch als Online-Ausgabe 978-1-4704-1285-2 Mathematical surveys and monographs Volume 58 (DE-604)BV000018014 58 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008391780&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Etingof, Pavel 1969- Frenkel, Igor B. 1952- Kirillov, Alexander Jr. 1967- Lectures on representation theory and Knizhnik-Zamolodchikov equations Mathematical surveys and monographs Deformation <Mathematik> - Analytische Funktion - Quantengruppe Symmetriebrechung (DE-588)4184200-5 gnd Quantengruppe (DE-588)4252437-4 gnd |
| subject_GND | (DE-588)4184200-5 (DE-588)4252437-4 |
| title | Lectures on representation theory and Knizhnik-Zamolodchikov equations |
| title_auth | Lectures on representation theory and Knizhnik-Zamolodchikov equations |
| title_exact_search | Lectures on representation theory and Knizhnik-Zamolodchikov equations |
| title_full | Lectures on representation theory and Knizhnik-Zamolodchikov equations Pavel I. Etingof ; Igor B. Frenkel ; Alexander A. Kirillov Jr. |
| title_fullStr | Lectures on representation theory and Knizhnik-Zamolodchikov equations Pavel I. Etingof ; Igor B. Frenkel ; Alexander A. Kirillov Jr. |
| title_full_unstemmed | Lectures on representation theory and Knizhnik-Zamolodchikov equations Pavel I. Etingof ; Igor B. Frenkel ; Alexander A. Kirillov Jr. |
| title_short | Lectures on representation theory and Knizhnik-Zamolodchikov equations |
| title_sort | lectures on representation theory and knizhnik zamolodchikov equations |
| topic | Deformation <Mathematik> - Analytische Funktion - Quantengruppe Symmetriebrechung (DE-588)4184200-5 gnd Quantengruppe (DE-588)4252437-4 gnd |
| topic_facet | Deformation <Mathematik> - Analytische Funktion - Quantengruppe Symmetriebrechung Quantengruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008391780&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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