Introduction to quantum groups:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
1993
|
| Schriftenreihe: | Progress in mathematics
110 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 340 S. |
| ISBN: | 0817637125 3764337125 |
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| 100 | 1 | |a Lusztig, George |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introduction to quantum groups |c George Lusztig |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 1993 | |
| 300 | |a XII, 340 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Progress in mathematics |v 110 | |
| 650 | 7 | |a Canonische analyse |2 gtt | |
| 650 | 7 | |a Hopf-algebra's |2 gtt | |
| 650 | 7 | |a Kwantumgroepen |2 gtt | |
| 650 | 7 | |a Physique mathématique |2 ram | |
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Mathematical physics | |
| 650 | 4 | |a Quantum groups | |
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Datensatz im Suchindex
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CONTENTS
PART I. THE DRINFELD-JIMBO ALGEBRA U. 1
CHAPTER 1. THE ALGEBRA F . 2
1.1. CARTAN DATUM. 2
1.2. THE ALGEBRAS 'F AND F. 2
1.3. PRELIMINARIES ON GAUSSIAN BINOMIAL COEFFICIENTS. 9
1.4. THE QUANTUM SERRE RELATIONS.10
CHAPTER 2. WEYL GROUP, ROOT DATUM.14
2.1. THE WEYL GROUP.14
2.2. ROOT DATUM.15
2.3. COROOTS.18
CHAPTER 3. THE ALGEBRA U.19
3.1. THE ALGEBRAS 'U AND U .19
3.2. TRIANGULAR DECOMPOSITION FOR 'TJ AND U .25
3.3. ANTIPODE.28
3.4. THE CATEGORY
C
.30
3.5. INTEGRABLE OBJECTS OF
C
.31
CHAPTER 4. THE QUASI-7-MATRIX .34
4.1. THE ELEMENT .34
4.2. SOME IDENTITIES FOR .37
CHAPTER 5. THE SYMMETRIES
T
'E,
T"E
OF AN INTEGRABLE
U-MODULE .40
5.1. THE CATEGORY C'.40
5.2. FIRST PROPERTIES OF
T'
E,
T"E
.42
5.3. THE OPERATORS
L'^L"
.45
CHAPTER 6. COMPLETE REDUCIBILITY THEOREMS.48
6.1. THE QUANTUM CASIMIR OPERATOR.48
6.2. COMPLETE REDUCIBILITY IN
CHX DC'
.51
6.3. AFFINE OR FINITE CARTAN DATA .53
CHAPTER 7. HIGHER ORDER QUANTUM SERRE RELATIONS . 55
NOTES ON PART I .59
PART II. GEOMETRIC REALIZATION OF F .61
CHAPTER 8. REVIEW OF THE THEORY OF PERVERSE SHEAVES . 63
BIBLIOGRAFISCHE INFORMATIONEN
HTTP://D-NB.INFO/940025167
VI
CONTENTS
CHAPTER 9. QUIVERS AND PERVERSE SHEAVES
9.1. THE COMPLEXES
LV
.
9.2. THE FUNCTORS IND AND RES.
9.3. THE CATEGORIES
VWX, I
AND PV;I';7 .
CHAPTER 10. FOURIER-DELIGNE TRANSFORM , . .
10.1. FOURIER-DELIGNE TRANSFORM AND RESTRICTION
10.2. FOURIER-DELIGNE TRANSFORM AND INDUCTION
10.3. A KEY INDUCTIVE STEP.
CHAPTER 11. PERIODIC FUNCTORS.
CHAPTER 12. QUIVERS WITH AUTOMORPHISMS . .
12.1. THE GROUP /C(QV) .
12.2. INNER PRODUCT .
12.3. PROPERTIES OF
LYY
.
12.4. VERDIER DUALITY.
12.5. SELF-DUAL ELEMENTS .
12.6.
LYY
AS ADDITIVE GENERATORS.
CHAPTER 13. THE ALGEBRAS O K AND K.
13.1. THE ALGEBRA
0'K .
13.2. THE ALGEBRA K .
CHAPTER 14. THE SIGNED BASIS OF F.
14.1. CARTAN DATA AND GRAPHS WITH AUTOMORPHISMS
14.2. THE SIGNED BASIS
B
.
14.3. THE SUBSETS Z?I|RL OF
B
.
14.4. THE CANONICAL BASIS B OF F .
14.5. EXAMPLES .
NOTES ON PART II.
68
68
71
77
81
81
84
87
89
92
92
95
95
99
100
102
106
106
110
113
113
118
120
122
125
127
PART III. KASHIWARA'S OPERATORS AND APPLICATIONS
129
CHAPTER 15. THE ALGEBRA IT .
CHAPTER 16. KASHIWARA'S OPERATORS IN RANK 1
16.1. DEFINITION OF THE OPERATORS
FA, U
AND
FI, E,
16.2. ADMISSIBLE FORMS.
16.3. ADAPTED BASES.
CHAPTER 17. APPLICATIONS .
17.1. FIRST APPLICATION TO TENSOR PRODUCTS . .
17.2. SECOND APPLICATION TO TENSOR PRODUCTS
17.3. THE OPERATORS
FA, :
F F .
130
132
132
133
139
142
142
148
150
CONTENTS
VII
CHAPTER 18. STUDY OF THE OPERATORS
EI
ON A A. 152
18.1. PRELIMINARIES. 152
18.2. A GENERAL HYPOTHESIS AND SOME CONSEQUENCES. 154
18.3. FURTHER CONSEQUENCES OF THE GENERAL HYPOTHESIS . 159
CHAPTER 19. INNER PRODUCT ON A. 164
19.1. FIRST PROPERTIES OF THE INNER PRODUCT . 164
19.2. NORMALIZATION OF SIGNS. 167
19.3. FURTHER PROPERTIES OF THE INNER PRODUCT. 170
CHAPTER 20. BASES AT OO . 173
20.1. THE BASIS AT OO OF AX. 173
20.2. BASIS AT OO IN A TENSOR PRODUCT. 175
CHAPTER 21. CARTAN DATA OF FINITE TYPE. 177
CHAPTER 22. POSITIVITY OF THE ACTION OF
FI,EI
IN THE SYMMETRIC CASE . 179
NOTES ON PART III. 182
PART 4. CANONICAL BASIS OF U. 183
CHAPTER 23. THE ALGEBRA U . 185
23.1. DEFINITION AND FIRST PROPERTIES OF TJ. 185
23.2. TRIANGULAR DECOMPOSITION, A-FORM FOR U. 188
23.3. U AND TENSOR PRODUCTS . 189
CHAPTER 24. CANONICAL BASES IN CERTAIN TENSOR PRODUCTS 192
24.1. INTEGRALITY PROPERTIES OF THE QUASI-7-MATRIX. 192
24.2. A LEMMA ON SYSTEMS OF (SEMI)-LINEAR EQUATIONS . 194
24.3. THE CANONICAL BASIS OF "'A*
8
AV. 195
CHAPTER 25. THE CANONICAL BASIS B OF IJ. 198
25.1. STABILITY PROPERTIES. 198
25.2. DEFINITION OF THE BASIS B OF U . 202
25.3. EXAMPLE (RANK 1) 205
25.4. STRUCTURE CONSTANTS. 206
CHAPTER 26. INNER PRODUCT ON U. 208
26.1. FIRST DEFINITION OF THE INNER PRODUCT. 208
26.2. DEFINITION OF THE INNER PRODUCT AS A LIMIT. 211
26.3. A CHARACTERIZATION OF B U (-B) . 212
CHAPTER 27. BASED MODULES . 214
27.1. ISOTYPICAL COMPONENTS . 214
27.2. THE SUBSETS B[A]. 217
27.3. TENSOR PRODUCT OF BASED MODULES. 219
VIII
CONTENTS
CHAPTER 28. BASES FOR COINVARIANTS AND CYCLIC PERMUTATIONS 224
28.1. MONOMIALS. 224
28.2. THE ISOMORPHISM P. 226
CHAPTER 29. A REFINEMENT OF THE PETER-WEYL THEOREM 230
29.1. THE SUBSETS B[A] OF B. 230
29.2. THE FINITE DIMENSIONAL ALGEBRAS U/U[P] . 231
29.3. THE REFINED PETER-WEYL THEOREM. 233
29.4. CELLS .^ ^ | 235
29.5. THE QUANTUM COORDINATE ALGEBRA. 237
CHAPTER 30. THE CANONICAL TOPOLOGICAL BASIS OF (U- U+)T 238
30.1. THE DEFINITION OF THE CANONICAL TOPOLOGICAL BASIS
. 238
30.2. ON THE COEFFICIENTS P6LI6/.62I!, . 240
NOTES ON PART IV. 242
PART V. CHANGE OF RINGS. 244
CHAPTER 31. THE ALGEBRA 245
31.1. DEFINITION OF /JU. 245
31.2. INTEGRABLE U-MODULES. 249
31.3. HIGHEST WEIGHT MODULES. 251
CHAPTER 32. COMMUTATIVITY ISOMORPHISM . 252
32.1. THE ISOMORPHISM JTZM,M' . 252
32.2. THE HEXAGON PROPERTY. 255
CHAPTER 33. RELATION WITH KAC-MOODY LIE ALGEBRAS . . . 258
33.1. THE SPECIALIZATION V = L. 258
33.2. THE QUASI-CLASSICAL CASE. 260
CHAPTER 34. GAUSSIAN BINOMIAL COEFFICIENTS AT ROOTS OF 1 265
CHAPTER 35. THE QUANTUM FROBENIUS HOMOMORPHISM . . 269
35.1. STATEMENTS OF RESULTS. 269
35.2. PROOF OF THEOREM 35.1.8 . 271
35.3. THE STRUCTURE OF CERTAIN HIGHEST WEIGHT MODULES OF *U . 273
35.4. A TENSOR PRODUCT DECOMPOSITION OF R{. 276
35.5. PROOF OF THEOREM 35.1.7 . 278
CHAPTER 36. THE ALGEBRAS FTF, RU. 280
36.1. THE ALGEBRA
RF. 280
36.2. THE ALGEBRAS RU, RU. 282
NOTES ON PART V. 285
IX
PART VI. BRAID GROUP ACTION. 286
CHAPTER 37. THE SYMMETRIES T/
E, T"E OF U. 287
37.1. DEFINITION OF THE SYMMETRIES. 287
37.2. CALCULATIONS IN RANK 2. 288
37.3. RELATION OF THE SYMMETRIES WITH COMULTIPLICATION . 293
CHAPTER 38. SYMMETRIES AND INNER PRODUCT ON F. 294
38.1. THE ALGEBRAS F[I],
FFF[I]. 294
38.2. A COMPUTATION OF INNER PRODUCTS. 300
CHAPTER 39. BRAID GROUP RELATIONS. 304
39.1. PREPARATORY RESULTS. 304
39.2. BRAID GROUP RELATIONS FOR U IN RANK 2 . 305
39.3. THE QUANTUM VERMA IDENTITIES. 311
39.4. PROOF OF THE BRAID GROUP RELATIONS. 314
CHAPTER 40. SYMMETRIES AND U+. 318
40.1. PREPARATORY RESULTS. 318
40.2. THE SUBSPACE U +(W,E) OF U+ . 321
CHAPTER 41. INTEGRALITY PROPERTIES OF THE SYMMETRIES . . 324
41.1. BRAID GROUP ACTION ON U . 324
41.2. BRAID GROUP ACTION ON INTEGRABLE /JU-MODULES. 326
CHAPTER 42. THE ADE CASE . 328
42.1. COMBINATORIAL DESCRIPTION OF THE LEFT COLORED GRAPH . . 328
42.2. REMARKS ON THE PIECEWISE LINEAR BISECTIONS : NN - N" 334
NOTES ON PART VI. 338
INDEX OF NOTATION . 339
INDEX OF TERMINOLOGY . 341 |
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| indexdate | 2024-10-09T18:17:00Z |
| institution | BVB |
| isbn | 0817637125 3764337125 |
| language | English |
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| physical | XII, 340 S. |
| publishDate | 1993 |
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| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spelling | Lusztig, George Verfasser aut Introduction to quantum groups George Lusztig Boston [u.a.] Birkhäuser 1993 XII, 340 S. txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 110 Canonische analyse gtt Hopf-algebra's gtt Kwantumgroepen gtt Physique mathématique ram Mathematische Physik Mathematical physics Quantum groups Quantengruppe (DE-588)4252437-4 gnd rswk-swf Quantengruppe (DE-588)4252437-4 s DE-604 Progress in mathematics 110 (DE-604)BV000004120 110 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005408784&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lusztig, George Introduction to quantum groups Progress in mathematics Canonische analyse gtt Hopf-algebra's gtt Kwantumgroepen gtt Physique mathématique ram Mathematische Physik Mathematical physics Quantum groups Quantengruppe (DE-588)4252437-4 gnd |
| subject_GND | (DE-588)4252437-4 |
| title | Introduction to quantum groups |
| title_auth | Introduction to quantum groups |
| title_exact_search | Introduction to quantum groups |
| title_full | Introduction to quantum groups George Lusztig |
| title_fullStr | Introduction to quantum groups George Lusztig |
| title_full_unstemmed | Introduction to quantum groups George Lusztig |
| title_short | Introduction to quantum groups |
| title_sort | introduction to quantum groups |
| topic | Canonische analyse gtt Hopf-algebra's gtt Kwantumgroepen gtt Physique mathématique ram Mathematische Physik Mathematical physics Quantum groups Quantengruppe (DE-588)4252437-4 gnd |
| topic_facet | Canonische analyse Hopf-algebra's Kwantumgroepen Physique mathématique Mathematische Physik Mathematical physics Quantum groups Quantengruppe |
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