Introduction to the quantum Yang-Baxter equation and quantum groups: an algebraic approach
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| Main Authors: | , |
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| Format: | Book |
| Language: | Undetermined |
| Published: |
Dordrecht u.a.
Dordrecht u.a.
1997
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 293 S. |
| ISBN: | 0792347218 |
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Record in the Search Index
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| adam_text | INTRODUCTION TO THE QUANTUM YANG-BAXTER EQUATION AND QUANTUM GROUPS: AN
ALGEBRAIC APPROACH BY LARRY A. LAMBE CAIP, RUTGERS UNIVERSITY,
PISCATAWAY, NJ, U.S.A. AND UNIVERSITY OF WALES, BANGOR BANGOR, GWYNEDD,
U.K. AND DAVID E. RADFORD UNIVERSITY OF ILLINOIS AT CHICAGO, CHICAGO,
ILLINOIS, U.S.A. KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
CONTENTS FOREWORD XI PREFACE XV ACKNOWLEDGMENTS XVII INTRODUCTION XIX 1.
ALGEBRAIC PRELIMINARIES N 1 1.1 COALGEBRAS 1 1.2 THE ALGEBRA C* 10 1.3
THE COALGEBRA A 0 17 1.3.1 THE CONSTRUCTION AND CHARACTERIZATIONS O A
17 1.3.2 DOUBLE DUALS 20 1.3.3 THE FUNDAMENTAL THEOREM OF COALGEBRAS 21
1.4 RATIONAL MODULES AND COMODULES 23 1.4.1, - RATIONAL MODULES 23 1.4.2
COMODULES 24 1.4.3 M R ANDM R 27 1.4.4 M R CHARACTERIZED IN TERMS OF
ANNIHILATORS 27 1.4.5 ANOTHER PROOF OF THE FUNDAMENTAL THEOREM OF
COALGEBRAS 29 1.5 BIALGEBRAS . 32 1.6 HOPF ALGEBRAS 39 1.6.1 THE
CONVOLUTION ALGEBRA 40 1.6.2 DEFINITION OF HOPF ALGEBRA AND ANTIPODE 41
1.7 THE CORADICAL AND THE CORADICAL FILTRATION 46 1.8 POINTED HOPF
ALGEBRAS V 53 1.9 (CO)MODULE (CO)ALGEBRAS 54 1.9.1 I/-MODULE ALGEBRAS
AND COALGEBRAS 55 1.9.2 #-COMODULE ALGEBRAS AND COALGEBRAS 59 VIII
INTRODUCTION TO THE QYBE 2. THE QUANTUM YANG-BAXTER EQUATION (QYBE) 65
2.1 THE CONSTANT FORM OF THE QYBE 66 2.1.1 THE CONSTANT FORM OF THE QYBE
IN H-S NOTATION 67 2.1.2 THE CONSTANT FORM OF THE QYBE IN COORDINATES 67
2.2 THE BRAID EQUATION 68 2.3 SYMMETRIES 70 2.4 THE ONE-PARAMETER FORM
OF THE QYBE 72 2.5 THE TWO-PARAMETER FORM OF THE QYBE 74 2.6 A SYSTEM OF
POLYNOMIAL EQUATIONS (THE QYB VARIETY) 74 2.7 THE BIALGEBRA ASSOCIATED
TO THE QYBE 76 2.7.1 A MODULE ACTION ASSOCIATED TO A QYBE SOLUTION 76
2.7.2 COMODULE COACTION 77 2.8 FACTORING A QYBE SOLUTION OVER A
BIALGEBRA 78 2.9 COMPATIBILITY CONDITIONS IN THE CONSTANT CASE 79 2.9.1
THE FUNDAMENTAL COMPATIBILITY CONDITION IN COORDINATES 79 2.9.2 THE
(CO)COMMUTATIVE COMPATIBILITY CONDITION 80 2.9.3 COMPATIBILITY
CONDITIONS IN H-S NOTATION 81 2.10 COMPATIBILITY CONDITIONS IN THE
PARAMETERIZED CASES 81 2.11 REDUCING THE DEGREE OF THE QYB VARIETY 83
2.11.1 FROM CUBIC TO QUADRATIC TO LINEAR 83 2.11.2 A CURIOUS EXAMPLE 83
3. CATEGORIES OF QUANTUM YANG-BAXTER MODULES 87 3.1 VARIOUS CATEGORIES
87 3.1.1 LEFT QYB A-MODULES 88 3.1.2 CQYB A-MODULES 89 3.1.3 RIGHT QYB
A-MODULES 90 3.1.4 WEAK QYB A-MODULES 91 3.2 CONGRUENCE IN A QYB 93 3.3
RECOLLECTIONS OF VARIOUS MODULE AND COMODULE STRUCTURES 94 3.4 GENERAL
CONSTRUCTIONS IN A QYB 96 3.4.1 SUB-OBJECTS, QUOTIENT OBJECTS OF A QYB
96 3.4.2 DIRECT SUMS IN A QYB 96 3.4.3 DUALS OF OBJECTS OF A QYB 96
3.4.4 STRUCTURE INDUCED FROM OBJECTS OF A QYB 99 3.5 CONSTRUCTIONS IN H
QYB WHEN H P HAS AN ANTIPODE 99 3.5.1 EQUIVALENT FORMULATIONS OF
COMPATIBILITY 99 3.5.2 THE RATIONAL PART OF A LEFT H, #*-MODULE 101
3.5.3 DIRECT PRODUCTS IN A QYB 102 3.5.4 SUB-OBJECTS OF OBJECTS OF H QYB
WHEN H OP HAS AN ANTIPODE 103 3.6 THE RELATIONSHIP BETWEEN QYBE
SOLUTIONS R AND R T 104 3.7 QYB STRUCTURES ON H WHEN H P IS A HOPF
ALGEBRA 105 CONTENTS IX 3.7.1 GENERALIZED COADJOINT ACTION 106 3.7.2
GENERALIZED ADJOINT ACTION 109 3.8 TENSOR PRODUCT IN A QYB 110 3.8.1 THE
TENSOR ALGEBRA 113 3.8.2 EOM(M,N) AND QUANTUM YANG-BAXTER SUBMODULES 113
3.9 TENSOR PRODUCT OF PARAMETERIZED QYBE SOLUTIONS 114 3.10 ALGEBRAS OF
H QYB 115 3.11 COALGEBRAS, BIALGEBRAS, AND HOPF ALGEBRAS OF H QYB 116
3.12 SMASH BIPRODUCTS ASSOCIATED TO H H QYB 117 4. MORE ON THE BIALGEBRA
ASSOCIATED TO THE QYBE 121 4.1 MODULE-COMODULE COMPATIBILITY REVISITED
121 4.2 A BASIS-FREE DESCRIPTION OF THE FRT CONSTRUCTION 128 4.3 A(R)P,
A(R) COP , AND A{R) OP COP AS FRT CONSTRUCTIONS 131 4.4 CONDITIONS FOR
A(R) TO BE A POINTED BIALGEBRA 138 5. THE FUNDAMENTAL EXAMPLE OF A
QUANTUM GROUP 143 5.1 REVIEW OF SL (2, K) 143 5.1.1 THE COORDINATE RING
OF SL(2, K) 144 5.1.2 THE LIE ALGEBRA SL(2,FC) 146 5.1.3 IRREDUCIBLE
REPRESENTATIONS OF SI(2, K) 148 5.2 DERIVATIONS AND (CO)ALGEBRA ACTIONS
REVISITED 149 5.3 A HOPF ALGEBRA CLOSELY RELATED TO FC[SL(2, K)} 150 5.4
GROUPLIKES AND SKEW PRIMITIVES OF K[SL Q (2,K)] 151 5.5 EMBEDDING
W(SL(2, K)) INTO K[SL(2,K)] 153 5.6 QUANTUM ANALOGS OF U(S (2,K)) 155
6. QUASITRIANGULAR STRUCTURES AND THE DOUBLE 161 6.1 QUASITRIANGULAR
ALGEBRAS 161 6.2 QUASITRIANGULAR STRUCTURES ARISING FROM INTEGRALS 162
6.3 QUASITRIANGULAR BIALGEBRAS AND QUASITRIANGULAR HOPF ALGEBRAS 164 6.4
THE QUANTUM DOUBLE 175 6.5 SOME FUNDAMENTAL EXAMPLES OF POINTED HOPF
ALGEBRAS 181 6.5.1 Q-BINOMIAL COEFFICIENTS 182 6.5.2 CONSTRUCTION OF THE
EXAMPLES 184 6.6 A FAMILY OF QT HOPF ALGEBRAS AND ASSOCIATED DOUBLES 186
6.6.1 CONSTRUCTION AND PROPERTIES OF H( N ,V,U) 187 6.6.2 CONSTRUCTION
AND PROPERTIES OF U(M,U,UI) 1 91 7. COQUASITRIANGULAR STRUCTURES 197 7.1
FURTHER PROPERTIES OF A(R) 197 X INTRODUCTION TO THE QYBE 7.2
COQUASITRIANGULAR COALGEBRAS 199 7.3 COQUASITRIANGULAR BIALGEBRAS AND
HOPF ALGEBRAS 203 7.4 THE FREE COQUASITRIANGULAR BIALGEBRA 209 7.5
ONE-PARAMETER QYBE, COQUASITRIANGULARITY, AND TENSOR PRODUCT 213 7.5.1
I?-COMMUTATIVE SPECTRAL PARAMETER 214 7.5.2 CONSTRUCTIONS WHEN X IS A
GROUP 215 7.5.3 TENSOR PRODUCT OF ONE-PARAMETER QYBE SOLUTIONS 218 8.
SOME CLASSES OF SOLUTIONS 219 8.1 SOME CONSEQUENCES OF M-REDUCTION 220
8.2 WHEN A(R) IS GENERATED BY GROUPLIKE ELEMENTS 222 8.3 SOLUTIONS WHEN
DIMM = 2 AND ^L(-R) IS POINTED 226 8.4 PATCHING AND SOLUTIONS IN HIGHER
DIMENSION 232 8.5 A CLASS OF WEAK QYB MODULES 233 8.6 SOME ONE-PARAMETER
SOLUTIONS 244 8.6.1 SOME SPECIFIC SOLUTIONS 244 8.6.2 A P-PERTURBATION
EXAMPLE 247 9. CATEGORICAL CONSTRUCTIONS 249 9.1 COENDS 249 9.2
QUASI-SYMMETRIC MONOIDAL CATEGORIES 250 9.3 RIGID MONOIDAL CATEGORIES
AND HOPF ALGEBRAS 254 9.4 CATEGORIES AND COQUASITRIANGULAR HOPF ALGEBRAS
258 9.5 THE QYBE IN OTHER CATEGORIES 258 9.6 THE CATEGORY OF GRADED
MODULES 259 APPENDICES 261 A-PREREQUISITES 261 A.1 THE GROUND RING K AND
BASIC FE-LINEAR MAPS 261 A.2 ALGEBRAS, COALGEBRAS, AND THEIR
REPRESENTATIONS 262 A.3 VARIOUS NOTATIONS RELATED TO THE QYBE 263 A.3.1
STRUCTURE CONSTANTS 263 A.3.2 HEYNEMAN-SWEEDLER AND H-S NOTATIONS 267
A.3.3 CATEGORICAL NOTATION 268 A.4 SOME RESULTS FROM LINEAR ALGEBRA 269
A.4.1 RANK OF TENSORS AND ENDOMORPHISMS 269 A.4.2 CLOSED SUBSPACES OF U*
272 A.4.3 COFINITE SUBSPACES AND CONTINUOUS LINEAR MAPS 277 REFERENCES
281 INDEX 291
|
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| indexdate | 2024-07-09T18:46:51Z |
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| spelling | Lambe, Larry A. Verfasser aut Introduction to the quantum Yang-Baxter equation and quantum groups an algebraic approach by Larry A. Lambe and David E. Radford Dordrecht u.a. Dordrecht u.a. 1997 XX, 293 S. txt rdacontent n rdamedia nc rdacarrier Quantengruppe (DE-588)4252437-4 gnd rswk-swf Yang-Baxter-Gleichung (DE-588)4291478-4 gnd rswk-swf Yang-Baxter-Gleichung (DE-588)4291478-4 s Quantengruppe (DE-588)4252437-4 s 1\p DE-604 Radford, David E. 1943- Verfasser (DE-588)1020225114 aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009213045&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lambe, Larry A. Radford, David E. 1943- Introduction to the quantum Yang-Baxter equation and quantum groups an algebraic approach Quantengruppe (DE-588)4252437-4 gnd Yang-Baxter-Gleichung (DE-588)4291478-4 gnd |
| subject_GND | (DE-588)4252437-4 (DE-588)4291478-4 |
| title | Introduction to the quantum Yang-Baxter equation and quantum groups an algebraic approach |
| title_auth | Introduction to the quantum Yang-Baxter equation and quantum groups an algebraic approach |
| title_exact_search | Introduction to the quantum Yang-Baxter equation and quantum groups an algebraic approach |
| title_full | Introduction to the quantum Yang-Baxter equation and quantum groups an algebraic approach by Larry A. Lambe and David E. Radford |
| title_fullStr | Introduction to the quantum Yang-Baxter equation and quantum groups an algebraic approach by Larry A. Lambe and David E. Radford |
| title_full_unstemmed | Introduction to the quantum Yang-Baxter equation and quantum groups an algebraic approach by Larry A. Lambe and David E. Radford |
| title_short | Introduction to the quantum Yang-Baxter equation and quantum groups |
| title_sort | introduction to the quantum yang baxter equation and quantum groups an algebraic approach |
| title_sub | an algebraic approach |
| topic | Quantengruppe (DE-588)4252437-4 gnd Yang-Baxter-Gleichung (DE-588)4291478-4 gnd |
| topic_facet | Quantengruppe Yang-Baxter-Gleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009213045&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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