Quantum groups:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1995
|
| Series: | Graduate texts in mathematics
155 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 531 S. |
| ISBN: | 3540943706 0387943706 |
Staff View
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Record in the Search Index
| _version_ | 1804124179639828480 |
|---|---|
| adam_text | CHRISTIAN KASSEL
QUANTU
M GROUP
S
WITH 88 ILLUSTRATIONS
SPRINGER-VERLAG
NEW YOR
K BERLIN
HEIDELBERG LONDON
PARI
S
TOKY
O HON
G KON
G
BARCELONA BUDAPEST
CONTENT
S
PREFAC
E
V
PAR
T ONE QUANTU
M
SL(2)
1
I
PRELIMINARIE
S 3
1 ALGEBRAS AND MODULES 3
2 FREE ALGEBRAS 7
3 THE AFFINE LINE AND PLANE 8
4 MATRIX MULTIPLICATION 10
5 DETERMINANTS AND INVERTIBLE MATRICES 10
6 GRADED AND FILTERED ALGEBRAS 12
7 ORE EXTENSIONS 14
8 NOETHERIAN RINGS 18
9 EXERCISES 20
10 NOTES 22
I
I
TENSOR PRODUCT
S 2
3
1 TENSOR PRODUCTS OF VECTOR SPACES 23
2 TENSOR PRODUCTS OF LINEAR MAPS 26
3 DUALITY AND TRACES 29
4 TENSOR PRODUCTS OF ALGEBRAS 32
5 TENSOR AND SYMMETRIC ALGEBRAS 34
6 EXERCISES 36
7 NOTES 38
CONTENTS
LANGUAGE OF HOP
F ALGEBRA
S 39
COALGEBRAS 39
BIALGEBRAS 45
HOPF ALGEBRAS 49
RELATIONSHIP WITH CHAPTE
R I. TH
E HOPF ALGEBRAS
GL{2)
ANDL(2
) 57
5 MODULES OVER A HOPF ALGEBRA 57
6 COMODULES 61
7 COMODULE-ALGEBRAS. COACTION OF
SL(2)
ON TH
E AFFINE PLAN
E . 64
8 EXERCISES 66
9 NOTES 70
TH
E QUANTU
M PLAN
E AN
D IT
S SYMMETRIE
S 72
1 THE QUANTU
M PLAN
E 72
2 GAUSS POLYNOMIALS AN
D TH
E G-BINOMIAL FORMULA 74
3 THE ALGEBRA
M
Q
(2) 77
4 RING-THEORETICAL PROPERTIE
S OF
M
Q
{2)
81
5 BIALGEBRA STRUCTUR
E ON
M
Q
(2)
82
6 THE HOPF ALGEBRAS
GL
Q
(2) AND
SL
Q
(2)
83
7 COACTION ON TH
E QUANTU
M PLAN
E 85
8 HOPF *-ALGEBRAS 86
9 EXERCISES 88
10 NOTES 90
V
TH
E LIE ALGEBR
A O
F
SL(2)
9
3
1 LIE ALGEBRAS 93
2 ENVELOPING ALGEBRAS 94
3 THE LIE ALGEBRA SL(2) 99
4 REPRESENTATIONS OF SL(2) 101
5 TH
E CLEBSCH-GORDAN FORMULA 105
6 MODULE-ALGEBRA OVER A BIALGEBRA. ACTION OF ST(2) ON TH
E
AFFINE PLANE 107
7 DUALITY BETWEEN TH
E HOPF ALGEBRAS
U(SL{2))
AND
SL{2)
..
. 109
8 EXERCISES 117
9 NOTES 119
V
I
TH
E QUANTU
M ENVELOPIN
G ALGEBR
A O
F
SL(2)
121
1 THE ALGEBRA
U
Q
(SL(2))
121
2 RELATIONSHIP WIT
H TH
E ENVELOPING ALGEBRA OF
SL(2)
125
3 REPRESENTATIONS OF
U
Q
127
4 THE HARISH-CHANDRA HOMOMORPHISM AND TH
E CENTRE OF
U
Q
.
130
CONTENTS IX
5 CASE WHEN
Q
IS A ROOT OF UNITY 134
6 EXERCISES 138
7 NOTES 138
VI
I
A HOP
F ALGEBR
A STRUCTUR
E O
N
U
Q
(SL{2))
14
0
1 COMULTIPLICATION 140
2 SEMISIMPLICITY 143
3 ACTION OF
U
Q
(SI(2))
ON TH
E QUANTUM PLANE 146
4 DUALITY BETWEEN TH
E HOPF ALGEBRAS
U
Q
(SI{2))
AND
SL
Q
(2) . .
150
5 DUALITY BETWEEN [/
Q
(SL(2))-MODULES AND 5L
Q
(2)-COMODULES . 154
6 SCALAR PRODUCT
S ON
U
Q
{SL(2))-MODU ES
155
7 QUANTUM CLEBSCH-GORDAN 157
8 EXERCISES 162
9 NOTES 163
PAR
T TWO UNIVERSAL ^-MATRICE
S 165
VII
I
TH
E YANG-BAXTE
R EQUATIO
N AN
D (CO)BRAIDE
D BIALGEBRA
S 16
7
1 TH
E YANG-BAXTER EQUATION 167
2 BRAIDED BIALGEBRAS 172
3 HOW A BRAIDED BIALGEBRA GENERATES I?-MATRICES 178
4 TH
E SQUARE OF TH
E ANTIPODE IN A BRAIDED HOPF ALGEBRA ..
. 179
5 A DUAL CONCEPT: COBRAIDED BIALGEBRAS 184
6 THE FRT CONSTRUCTION 188
7 APPLICATION T
O
GL
Q
(2)
AND
SL
Q
{2)
194
8 EXERCISES 196
9 NOTES 198
I
X
DRINFELD
S QUANTU
M DOUBL
E 19
9
1 BICROSSED PRODUCT
S OF GROUPS 199
2 BICROSSED PRODUCT
S OF BIALGEBRAS
,
202
3 VARIATIONS ON TH
E ADJOINT REPRESENTATION 207
4 DRINFELD S QUANTU
M DOUBLE 213
5 REPRESENTATION-THEORETIC INTERPRETATIO
N OF TH
E
QUANTU
M DOUBLE 220
6 APPLICATION T
O
U
Q
(SI(2))
223
7 .R-MATRICES FOR
U
Q
230
8 EXERCISES 236
9 NOTES 238
M
CONTENT
S
JP*JART THREE LOW-DIMENSIONA
L TOPOLOGY AN
D 23
9
- TENSOR CATEGORIE
S
X
KNOTS
, LINKS, TANGLES
, AND BRAID
S 241
!)S F
1
KNOTS AND LINKS 242
01 S 2 CLASSIFICATION OF LINKS UP T
O ISOTOPY 244
&?YY I 3 LINK DIAGRAMS 246
J$M 4 THE JONES-CONWAY POLYNOMIAL 252
TEL
5 TANGLES 257
J*I 6 BRAIDS 262
$R.F 7 EXERCISES 269
TGI
8 NOTES 270
%
$.1 9 APPENDIX. THE FUNDAMENTAL GROUP 273
M
YY
XI
1 ENSOR
CATEGORIE
S 27
5
%
THE LANGUAGE OF CATEGORIES AND FUNCTORS 275
;I
2 TENSOR CATEGORIES 281
F&*§
EXAMPLES OF TENSOR CATEGORIES 284
4 TENSOR FUNCTORS 287
^,
6 TURNING TENSOR CATEGORIES INTO STRICT ONES 288
;*
YY EXERCISES 291
F
T 7 NOTES 293
^
]
TH
E TANGLE
CATEGOR
Y 29
4
T
I PRESENTATION OF A STRICT TENSOR CATEGORY 294
2 THE CATEGORY OF TANGLES 299
3 THE CATEGORY OF TANGLE DIAGRAMS 302
4 REPRESENTATIONS OF TH
E CATEGORY OF TANGLES 305
5 EXISTENCE PROOF FOR JONES-CONWAY POLYNOMIAL 311
6 EXERCISES 313
7 NOTES 313
XII
I
BRAIDING
S 31
4
1 BRAIDE
D TENSOR CATEGORIE
S 314
2 TH
E BRAI
D CATEGOR
Y 321
3 UNIVERSALITY OF TH
E BRAI
D CATEGOR
Y 322
4 TH
E CENTR
E CONSTRUCTIO
N 330
5 A CATEGORICA
L INTERPRETATIO
N OF TH
E QUANTU
M DOUBL
E ...
. 333
6 EXERCISES 337
7 NOTE
S 338
CONTENTS XI
XI
V
DUALIT
Y IN TENSOR CATEGORIE
S 339
1 REPRESENTING MORPHISMS IN A TENSOR CATEGORY 339
2 DUALITY 342
3 RIBBON CATEGORIES 348
4 QUANTUM TRACE AND DIMENSION 354
5 EXAMPLES OF RIBBON CATEGORIES 358
6 RIBBON ALGEBRAS 361
7 EXERCISES 365
8 NOTES 366
X
V
QUASI-BIALGEBRAS 368
1 QUASI-BIALGEBRAS 368
2 BRAIDED QUASI-BIALGEBRAS 371
3 GAUGE TRANSFORMATIONS 372
4 BRAID GROUP REPRESENTATIONS 377
5 QUASI-HOPF ALGEBRAS 379
6 EXERCISES 381
7 NOTES 381
PART FOUR QUANTU
M GROUPS AND MONODROM
Y 38
3
XV
I
GENERALITIE
S O
N QUANTU
M ENVELOPIN
G ALGEBRA
S 385
1 THE RING OF FORMAL SERIES AND /I-ADIC TOPOLOGY 385
2 TOPOLOGICALLY FREE MODULES 388
3 TOPOLOGICAL TENSOR PRODUCT 390
4 TOPOLOGICAL ALGEBRAS 392
5 QUANTUM ENVELOPING ALGEBRAS 395
6 SYMMETRIZING THE UNIVERSAL I?-MATRIX 398
7 EXERCISES 400
8 NOTES 401
9 APPENDIX. INVERSE LIMITS 401
XVI
I
DRINFELD AND JIMBO
S QUANTU
M ENVELOPIN
G ALGEBRA
S 40
3
1 SEMISIMPLE LIE ALGEBRAS 403
2 DRINFELD-JIMBO ALGEBRAS 406
3 QUANTUM GROUP INVARIANTS OF LINKS 410
4 THE CASE OF L(2) 412
5 EXERCISES 418
6 NOTES 418
XII CONTENTS
XVII
I
COHOMOLOG
Y AN
D RIGIDIT
Y THEOREM
S 42
0
1 COHOMOLOGY OF LIE ALGEBRA
S 420
2 RIGIDIT
Y FOR LI
E ALGEBRA
S 424
3 VANISHING RESULT
S FOR SEMISIMPLE LIE ALGEBRA
S 427
4 APPLICATIO
N T
O DRINFELD-JIMB
O QUANTU
M ENVELOPIN
G ALGEBRA
S 430
5 COHOMOLOGY OF COALGEBRA
S 431
6 ACTIO
N OF A SEMISIMPL
E LIE ALGEBR
A O
N TH
E COBA
R COMPLEX . 434
7 COMPUTATION
S FOR SYMMETRI
C COALGEBRA
S . 435
8 UNIQUENESS THEORE
M FOR QUANTU
M ENVELOPIN
G ALGEBRA
S ..
. 442
9 EXERCISES 446
10 NOTE
S 446
11 APPENDIX
. COMPLEXE
S AN
D RESOLUTION
S 447
XI
X
MONODROM
Y O
F TH
E KNIZHNIK-ZAMOLODCHIKO
V EQUATION
S 44
9
1
CONNECTIONS 449
2 BRAID GROUP REPRESENTATIONS FROM MONODROMY 451
3 THE KNIZHNIK-ZAMOLODCHIKOV EQUATIONS 455
4 THE DRINFELD-KOHNO THEOREM 458
5 EQUIVALENCE OF
U
H
(G)
AND
A
Q T
461
6 DRINFELD S ASSOCIATOR 463
7 CONSTRUCTION OF TH
E TOPOLOGICAL BRAIDED QUASI-BIALGEBRA
A
&T
468
8 VERIFICATION OF TH
E AXIOMS 471
9 EXERCISES 479
10 NOTES 479
11 APPENDIX. ITERATE
D INTEGRALS 480
X
X
POSTLUDE
. A UNIVERSA
L KNO
T INVARIANT 48
4
1 KNOT INVARIANTS OF FINITE TYPE 484
2 CHORD DIAGRAMS AND KONTSEVICH S THEOREM 486
3 ALGEBRA STRUCTURES ON CHORD DIAGRAMS 491
4 INFINITESIMAL SYMMETRIC CATEGORIES 494
5 A UNIVERSAL CATEGORY FOR INFINITESIMAL BRAIDINGS 496
6 FORMAL INTEGRATION OF INFINITESIMAL SYMMETRIC CATEGORIES . . 498
7 CONSTRUCTION OF KONTSEVICH S UNIVERSAL INVARIANT 499
8 RECOVERING QUANTUM GROUP INVARIANTS 502
9 EXERCISES 505
10 NOTES 505
REFERENCE
S 50
6
INDE
X 52
3
|
| any_adam_object | 1 |
| author | Kassel, Christian |
| author_facet | Kassel, Christian |
| author_role | aut |
| author_sort | Kassel, Christian |
| author_variant | c k ck |
| building | Verbundindex |
| bvnumber | BV009825010 |
| classification_rvk | SK 230 SK 340 UK 3000 |
| classification_tum | MAT 160f MAT 200f |
| ctrlnum | (OCoLC)246648976 (DE-599)BVBBV009825010 |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV009825010 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:41:33Z |
| institution | BVB |
| isbn | 3540943706 0387943706 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006504558 |
| oclc_num | 246648976 |
| open_access_boolean | |
| owner | DE-384 DE-12 DE-703 DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-824 DE-355 DE-BY-UBR DE-29T DE-739 DE-19 DE-BY-UBM DE-634 DE-83 DE-20 |
| owner_facet | DE-384 DE-12 DE-703 DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-824 DE-355 DE-BY-UBR DE-29T DE-739 DE-19 DE-BY-UBM DE-634 DE-83 DE-20 |
| physical | XII, 531 S. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Kassel, Christian Verfasser aut Quantum groups Christian Kassel New York [u.a.] Springer 1995 XII, 531 S. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 155 Quantengruppe (DE-588)4252437-4 gnd rswk-swf Hopf-Algebra (DE-588)4160646-2 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Lie-Algebra (DE-588)4130355-6 gnd rswk-swf Quantengruppe (DE-588)4252437-4 s DE-604 Lie-Algebra (DE-588)4130355-6 s 1\p DE-604 Topologie (DE-588)4060425-1 s 2\p DE-604 Hopf-Algebra (DE-588)4160646-2 s 3\p DE-604 Graduate texts in mathematics 155 (DE-604)BV000000067 155 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006504558&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 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Kassel, Christian Quantum groups Graduate texts in mathematics Quantengruppe (DE-588)4252437-4 gnd Hopf-Algebra (DE-588)4160646-2 gnd Topologie (DE-588)4060425-1 gnd Lie-Algebra (DE-588)4130355-6 gnd |
| subject_GND | (DE-588)4252437-4 (DE-588)4160646-2 (DE-588)4060425-1 (DE-588)4130355-6 |
| title | Quantum groups |
| title_auth | Quantum groups |
| title_exact_search | Quantum groups |
| title_full | Quantum groups Christian Kassel |
| title_fullStr | Quantum groups Christian Kassel |
| title_full_unstemmed | Quantum groups Christian Kassel |
| title_short | Quantum groups |
| title_sort | quantum groups |
| topic | Quantengruppe (DE-588)4252437-4 gnd Hopf-Algebra (DE-588)4160646-2 gnd Topologie (DE-588)4060425-1 gnd Lie-Algebra (DE-588)4130355-6 gnd |
| topic_facet | Quantengruppe Hopf-Algebra Topologie Lie-Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006504558&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT kasselchristian quantumgroups |