Gespeichert in:
| Titel: | Multidimensional hypergeometric functions and representation theory of lie algebras and quantum groups |
|---|---|
| Von: |
A. Varchenko
|
| Person: |
Varčenko, Aleksandr N.
1949- Verfasser aut |
| Hauptverfasser: | |
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Singapore [u.a.]
World Scientific Publ.
1995
|
| Schriftenreihe: | Advanced series in mathematical physics
21 |
| Schlagworte: | |
| Online-Zugang: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006998593&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | IX, 371 S. graph. Darst. |
| ISBN: | 981021880X |
Internformat
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| 100 | 1 | |a Varčenko, Aleksandr N. |d 1949- |e Verfasser |0 (DE-588)115203257 |4 aut | |
| 245 | 1 | 0 | |a Multidimensional hypergeometric functions and representation theory of lie algebras and quantum groups |c A. Varchenko |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific Publ. |c 1995 | |
| 300 | |a IX, 371 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Advanced series in mathematical physics |v 21 | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-006998593 | |
Datensatz im Suchindex
| DE-BY-UBR_call_number | 80/SK 340 V289 |
|---|---|
| DE-BY-UBR_katkey | 2201724 |
| DE-BY-UBR_location | UB Lesesaal Mathematik |
| DE-BY-UBR_media_number | 069008141927 |
| _version_ | 1835086569610936321 |
| adam_text | Contents
1. Introduction 1
1.1 Example. Three points on the line 2
1.2 Brief description of the contents 9
1.3 Acknowledgements 14
2. Construction of complexes calculating homology of the
complement of a configuration 15
2.1 Configuration in a real space 15
2.2 Configuration in a complex space 18
2.3 Homology 19
2.4 Compatible orientation of cells of complexes Q, Q /Q , X 22
2.5 Local system S*(a) and distinguished sections over cells 22
2.6 Quantum bilinear form of a configuration 25
2.7 Homomorphism of the projection on the real part 29
2.8 Action of a group preserving configuration 33
2.9 Abstract complexes of a real configuration 35
3. Construction of homology complexes for discriminantal
configuration 39
3.1 Discriminantal configuration 39
3.2 Facets of a discriminantal configuration 40
3.3 Centers of top dimensional facets 41
3.4 Basic polytopes 42
3.5 Centers of facets of arbitrary codimension 42
3.6 Cells of the construction of Sec. 2 are convex polytopes 43
3.7 Description of basic polytopes by inequalities 45
3.8 Admissible monomials 46
3.9 Equalities and inequalities defining cells 48
3.10 Distinguished coorientation of a discriminantal configuration 50
3.11 Weights of Cn k(zi,. ..,zn) and the reduced quantum bilinear
form for n 1 53
3.12 Complexes C.(Q /Q ,S*(a)), C.{X,S*{a)) for a
discriminantal configuration Cn * 54
vi Contents
3.13 Action of the permutation group £jt on a discriminantal
configuration CUtk (z) 58
3.14 Complexes X.(Cn,k) and Q.(Cn,k) 61
3.15 Homology of Q.(Cn,k) 62
3.16 Action of the permutation group on X.(CU:k), Q (Cn,k) 62
4. Algebraic interpretation of chain complexes of a discriminantal
configuration 67
4.1 Quantum groups 67
4.2 Contravariant form 69
4.3 Coalgebra structure on f/,n_, algebra structure on ((7,n_)* 75
4.4 Hochschild homology 79
4.5 Two sided Hochschild complexes connected with a
discriminantal configuration 80
4.6 Algebraic interpretation of the abstract complexes X.(Cn k)
and Q.(CUik) of a discriminantal configuration 82
4.7 Geometric interpretation of complexes C.(+Uqn _; M; 2; +A)a
and C.*(+t/,n _; M; 2; fi)x defined in (4.5) 85
4.8 Symmetrization 88
4.9 Proof of Theorem (4.7.5) 91
5. Quasiisomorphism of two sided Hochschild complexes to suitable
one sided Hochschild complexes 93
5.1 One sided Hochschild complexes connected with a
discriminantal configuration 93
5.2 Complexes (5.1.1) and (5.1.24) as subcomplexes of
complexes (4.5.5) 100
5.3 Construction of a monomorphism p : C.(+t/,n_; M; +A)a
»C.(+£/,n _;M;2;+A)x 106
5.4 Theorem. The monomorphisnup : C?(+Uqti ;M;i/,)
— C*(+Uqn __;M; 2;/j)j is a quasiisomorphism 115
5.5 Theorem. The monomorphism (p : C(+C/,n_;M; +A)
— C.(+Uqn _;M; 2;+A)j is a quasiisomorphism 118
5.6 Filtration in C.(+I/,n _;M;+A) 119
5.7 Degree 120
5.8 Proof of Theorem 5.6.12 121
5.9 Proof of Theorem 5.6.11 123
5.10 Remark 126
Contents vii
5.11 Geometric interpretation of Theorems 5.3.43, 5.4,
and 5.5 126
6. Bundle properties of a discriminantal configuration 145
6.1 Subordinated monomials 145
6.2 Leaves 145
6.3 Properties of leaves 147
6.4 Proof of Theorem 6.2.4 151
7. R matrix for the two sided Hochschild complexes 155
7.1 Bistructures on (+t/,n )®n, (+[/,n_)*®n 155
7.2 t/,n_ bimodule structure on (C/,n_)* 158
7.3 (f/,n_)* bimodule structure on C/,n_ 158
7.4 R matrix 159
7.5 Symmetrization and R matrix 165
7.6 R matrix for Verma modules 166
7.7 Connection of R matrices for two and one sided Hochschild
complexes 168
8. Monodromy 171
8.1 Gauss Manin connection 171
8.2 Chain complexes over real points of the base 173
8.3 Parallel translations along special curves 176
8.4 Isotopy of the real line 178
8.5 Factorization properties of cells 180
8.6 Involution 183
8.7 Bundle property 183
8.8 Construction of Tt 185
8.9 Lemma. The deformation Tt defined on each cell
separately is compatible on intersections of cells 186
8.10 Example 188
8.11 Computation of the action 7 : C.(Q /Q ,S*(A), z°)
C.(Q IQ ,S*(A), z°) for the isotropy Tt constructed
in (8.8) 189
8.12 Proof of Theorem 8.3.4 for Ttu 190
8.13 Geometric interpretation of the R matrix operators acting
on the two sided Hochschild complexes constructed in Sec. 4 191
8.14 Geometric interpretation of the R matrix operators on the
complexes C.(+[/,n_;M6; +A)V and C.*(+t/,n_; Aff;^)A,
constructed in Sec. 5. 194
viii Contents
9. R matrix operator as the canonical element, quantum doubles 197
9.1 Quantum double 197
9.2 Quantum doubles £ ((£/,b_) ) and T (Uqb+) 198
9.3 The action of the quantum doubles on Verma modules
and their duals 202
9.4 The quotient complex C.(+l/,n_; Ma; +A)/ker5 207
9.5 Quantum groups corresponding to Kac Moody algebras 210
10. Hypergeometric integrals 215
10.1 Orlik Solomon algebra and flags 215
10.2 Framings and bases 216
10.3 Quasiclassical contravariant form of a configuration 218
10.4 Relative complexes 220
10.5 Integrable connection on the horizontal complexes 223
10.6 Hypergeometric differential forms 228
10.7 Hypergeometric integrals 230
10.8 Resolution of singularities of a configuration of hyperplanes 233
10.9 Integration pairings and symmetric frames 236
10.10 Hypergeometric differential equations 239
11. Kac Moody Lie algebras without Serre s relations
and their doubles 247
11.1 Kac Moody Lie algebras without Serre s relations 247
11.2 Complexes 250
11.3 The double 252
11.4 Homology in degree zero 261
11.5 Knizhnik Zamolodchikov differential equation 261
12. Hypergeometric integrals of a discriminantal configuration 265
12.1 Complexes of a discriminantal configuration 265
12.2 Hypergeometric pairings 271
12.3 Hypergeometric integrals and the Knizhnik Zamolodchikov
connection 278
12.4 Resonances 285
12.5 Nondegeneracy of the hypergeometric pairing
J{z) : Ho(C.(n_;L)A) ® H0(C.(+Uqn^L(q) 6)x) C
for almost all k. Asymptotics for k + oo. 290
Contents ix
13. Resonances at infinity 295
13.1 Projective transformations of the complex line and
discriminantal configurations 295
13.2 Complementary weight 296
13.3 The inversion isomorphism for one Verma module 298
13.4 An inversion isomorphism for n Verma modules 303
13.5 Generic sets 309
13.6 Transformations of flag forms 310
13.7 The kernel of the hypergeometric pairing for s(2 312
13.8 Remarks on the representation theory of the quantum
double of the subalgebra C/?b_ c Uqs£2 317
14. Degenerations of discriminantal configurations 329
14.1 Composition of singular vectors 329
14.2 Asymptotics of the hypergeometric pairing 337
14.3 Rank of the hypergeometric pairing 341
14.4 Remarks on the kernel of the hypergeometric pairing 342
14.5 The Selberg formula 343
14.6 The hypergeometric pairing for g = s£2 and two modules 344
15. Remarks on homology groups of a configuration with coefficients
in local systems more general than complex one dimensional 347
15.1 Complexified real configuration 347
15.2 Universal quantum group 349
15.3 Discriminantal configuration 352
15.4 Remarks on homology groups of braid groups 357
15.5 Local systems of rank greater than 1 359
References 367
|
| any_adam_object | 1 |
| author | Varčenko, Aleksandr N. 1949- |
| author_GND | (DE-588)115203257 |
| author_facet | Varčenko, Aleksandr N. 1949- |
| author_role | aut |
| author_sort | Varčenko, Aleksandr N. 1949- |
| author_variant | a n v an anv |
| building | Verbundindex |
| bvnumber | BV010501917 |
| classification_rvk | SK 340 |
| ctrlnum | (OCoLC)246752724 (DE-599)BVBBV010501917 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV010501917 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T09:55:17Z |
| institution | BVB |
| isbn | 981021880X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006998593 |
| oclc_num | 246752724 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-12 DE-29T DE-11 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-12 DE-29T DE-11 DE-188 |
| physical | IX, 371 S. graph. Darst. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | World Scientific Publ. |
| record_format | marc |
| series | Advanced series in mathematical physics |
| series2 | Advanced series in mathematical physics |
| spellingShingle | Varčenko, Aleksandr N. 1949- Multidimensional hypergeometric functions and representation theory of lie algebras and quantum groups Advanced series in mathematical physics Quantengruppe (DE-588)4252437-4 gnd Lie-Algebra (DE-588)4130355-6 gnd Darstellungstheorie (DE-588)4148816-7 gnd Hypergeometrische Reihe (DE-588)4161061-1 gnd Kac-Moody-Algebra (DE-588)4223399-9 gnd |
| subject_GND | (DE-588)4252437-4 (DE-588)4130355-6 (DE-588)4148816-7 (DE-588)4161061-1 (DE-588)4223399-9 |
| title | Multidimensional hypergeometric functions and representation theory of lie algebras and quantum groups |
| title_auth | Multidimensional hypergeometric functions and representation theory of lie algebras and quantum groups |
| title_exact_search | Multidimensional hypergeometric functions and representation theory of lie algebras and quantum groups |
| title_full | Multidimensional hypergeometric functions and representation theory of lie algebras and quantum groups A. Varchenko |
| title_fullStr | Multidimensional hypergeometric functions and representation theory of lie algebras and quantum groups A. Varchenko |
| title_full_unstemmed | Multidimensional hypergeometric functions and representation theory of lie algebras and quantum groups A. Varchenko |
| title_short | Multidimensional hypergeometric functions and representation theory of lie algebras and quantum groups |
| title_sort | multidimensional hypergeometric functions and representation theory of lie algebras and quantum groups |
| topic | Quantengruppe (DE-588)4252437-4 gnd Lie-Algebra (DE-588)4130355-6 gnd Darstellungstheorie (DE-588)4148816-7 gnd Hypergeometrische Reihe (DE-588)4161061-1 gnd Kac-Moody-Algebra (DE-588)4223399-9 gnd |
| topic_facet | Quantengruppe Lie-Algebra Darstellungstheorie Hypergeometrische Reihe Kac-Moody-Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006998593&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000900258 |
| work_keys_str_mv | AT varcenkoaleksandrn multidimensionalhypergeometricfunctionsandrepresentationtheoryofliealgebrasandquantumgroups |
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