Quantum groups from coalgebras to Drinfeld algebras: a guided tour
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Internat. Press
1993
|
| Series: | Graduate texts in mathematical physics
2 |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019641705&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | 496 S. |
| ISBN: | 1571460004 |
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| 245 | 1 | 0 | |a Quantum groups from coalgebras to Drinfeld algebras |b a guided tour |c Steven Shnider and Shlomo Sternberg |
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Record in the Search Index
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|---|---|
| adam_text | Contents
1 Coalgebras and addition laws 1
1.1 Addition laws 1
1.2 Matrix comultiplication 4
1.3 Units and counits 6
2 Comodules and categories 9
2.1 Comodules and their morphisms 9
2.2 Categories 10
2.3 Functors and morphisms 11
2.4 The forgetful functor 12
2.5 Coalgebras as representatives 15
2.6 Opmatrix comultiplication as minimal example 18
2.7 Reconstruction 21
3 Bialgebras and Hopf spaces 23
3.1 Bialgebras 23
3.2 The cohomology of Hopf spaces 26
3.3 Graded bialgebras over a field 28
3.4 Antipodes and Hopf algebras 32
3.5 Self adjoint bialgebras 34
3.6 Antipodes of self adjoint bialgebras 35
3.7 The universal enveloping algebra 38
3.8 The shuffle algebra 39
3.9 Chen s iterated integrals 46
4 Representations of Sn 51
4.1 PSH algebras 51
4.2 PSH algebras with one generator 54
CONTENTS
4.3 Relations between x,y and z 61
4.4 Irreducible elements 65
4.5 Actions on basis elements 66
4.6 The Littlewood Richardson rule 71
4.7 Clebsch Gordan decompositions 74
4.8 Symmetric functions 77
4.9 Representations of Gl(n, q) 79
4.10 The compatibility of m and A 81
5 The Stasheff polyhedra 85
5.1 Associativity laws 85
5.2 Loop spaces 88
5.3 The polyhedron K5 90
5.4 Closed paths 92
5.5 Free magmas 94
6 Mac Lane s coherence 101
6.1 Monoidal categories 101
6.2 Braided categories 109
6.3 The category of braids 114
6.4 Duality and crossing 120
6.5 Constructions on graphs 124
6.6 Cyclically ordered graphs 128
6.7 Restricted moves on rooted trees 131
7 Lattice statistical mechanics 137
7.1 Ising models 137
7.2 The Peierls argument 139
7.3 The Kramers - Wannier computation 143
7.4 Honeycomb-triangular duality 146
7.5 The star triangle relation 149
7.6 Star triangle and commutation 152
7.7 Vertex models 156
7.8 Yang-Baxter and the Mac Lane hexagons 162
7.9 Electrical star triangle 165
8 Drinfeld algebras and Majid algebras 169
8.1 Drinfeld algebras 169
8.2 Monoidal reconstruction 171
CONTENTS
8.3 Twisting 180
8.4 Modules over a monoidal category 181
8.5 The Dual Category and Drinfeld Double 188
9 Gerstenhaber s Deformation theory 193
9.1 Deformation of coalgebras 196
9.2 The cobar complex 198
9.3 Deformation of algebras 200
9.4 Deformation of bialgebras 203
9.5 Deformation cohomology of bialgebras 206
9.6 Formal bialgebra deformations 210
9.7 Bialgebra to Drinfeld algebra deformations 215
9.8 Poisson algebras and Gerstenhaber algebras 218
10 Homological tools 223
10.1 Double complexes 224
10.2 B-4 modules 231
10.3 B-4 complexes 236
10.4 Projective and injective resolutions 241
10.5 Relative projectives and injectives 248
10.6 The Koszul complexes 250
10.7 B-4 properties of di, and 6C 254
10.8 Acyclicity of the Koszul complexes 256
10.9 The main reduction 258
10.lOHopf modules 262
11 Deformations of U{Q) 269
11.1 The normalized bar and cobar resolutions 270
11.2 Units, counits, antipodes under deformation 273
11.3 Normalized forms of deformations of U(Q) 276
11.4 The first uniqueness theorem 281
11.5 Uniqueness of Drinfeld algebra deformations 288
11.6 Quasitriangular structures 291
11.7 Coboundary deformations 299
11.8 Lie bialgebras and Poisson-Lie groups 308
11.9 Quantum 5/(2) 314
CONTENTS
12 The Knizhnik Zamolodchikov equations 321
12.1 Introduction 321
12.2 The Knizhnik Zamolodchikov equations 323
12.3 KZE3 and the associativity 326
12.4 KZE4 and the pentagon constraint 330
12.5 KZE2 and the R matrix 336
12.6 Braiding and the hexagons 338
12.7 The Drinfeld-Yang-Baxter equations 342
12.8 Solution by Chen integrals 344
13 Deformation of Drinfeld algebras 349
13.1 The modified cobar complex 351
13.2 Deformation theory 360
13.3 Nonassociative tensor algebra of a bimodule 365
13.4 Comodules over a Drinfeld algebra 369
13.5 Interpolation operators Mjt 373
13.6 Homotopy operators Dk 379
13.7 Relation of the Dk to deformation theory 384
Bibliography 391
|
| any_adam_object | 1 |
| author | Shnider, Steven Sternberg, Shlomo 1936- |
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| discipline | Physik Mathematik |
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| id | DE-604.BV024974127 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T14:19:30Z |
| institution | BVB |
| isbn | 1571460004 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-019641705 |
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| physical | 496 S. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | Internat. Press |
| record_format | marc |
| series | Graduate texts in mathematical physics |
| series2 | Graduate texts in mathematical physics |
| spellingShingle | Shnider, Steven Sternberg, Shlomo 1936- Quantum groups from coalgebras to Drinfeld algebras a guided tour Graduate texts in mathematical physics |
| title | Quantum groups from coalgebras to Drinfeld algebras a guided tour |
| title_auth | Quantum groups from coalgebras to Drinfeld algebras a guided tour |
| title_exact_search | Quantum groups from coalgebras to Drinfeld algebras a guided tour |
| title_full | Quantum groups from coalgebras to Drinfeld algebras a guided tour Steven Shnider and Shlomo Sternberg |
| title_fullStr | Quantum groups from coalgebras to Drinfeld algebras a guided tour Steven Shnider and Shlomo Sternberg |
| title_full_unstemmed | Quantum groups from coalgebras to Drinfeld algebras a guided tour Steven Shnider and Shlomo Sternberg |
| title_short | Quantum groups from coalgebras to Drinfeld algebras |
| title_sort | quantum groups from coalgebras to drinfeld algebras a guided tour |
| title_sub | a guided tour |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019641705&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV024974126 |
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