Tensor categories:
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2015]
|
| Series: | Mathematical surveys and monographs
volume 205 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverzeichnis Seite 325 - 337 |
| Physical Description: | xvi, 343 Seiten |
| ISBN: | 9781470420246 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents
Preface xi
Chapter 1. Abelian categories 1
1.1. Categorical prerequisites and notation 1
1.2. Additive categories 1
1.3. Definition of abelian category 2
1.4. Exact sequences 4
1.5. Length of objects and the Jordan-Holder theorem 5
1.6. Projective and injective objects 6
1.7. Higher Ext groups and group cohomology 7
1.8. Locally finite (artinian) and finite abelian categories 9
1.9. Coalgebras 12
1.10. The Coend construction 14
1.11. Deligne’s tensor product of locally finite abelian categories 15
1.12. The finite dual of an algebra 16
1.13. Pointed coalgebras and the coradical filtration 16
1.14. Bibliographical notes 19
Chapter 2. Monoidal categories 21
2.1. Definition of a monoidal category 21
2.2. Basic properties of unit objects 22
2.3. First examples of monoidal categories 26
2.4. Monoidal functors and their morphisms 30
2.5. Examples of monoidal functors 32
2.6. Monoidal functors between categories of graded vector spaces 33
2.7. Group actions on categories and equivariantization 35
2.8. The Mac Lane strictness theorem 36
2.9. The coherence theorem 39
2.10. Rigid monoidal categories 40
2.11. Invertible objects and Gr-categories 43
2.12. 2-categories 45
2.13. Bibliographical notes 46
Chapter 3. Z+-rings 49
3.1. Definition of a Z^-ring 49
3.2. The Frobenius-Perron theorem 51
3.3. The Frobenius-Perron dimensions 52
3.4. Z+-modules 55
3.5. Graded based rings 5g
3.6. The adjoint based subring and universal grading 60
vii
viii CONTENTS
3.7. Complexified Z+-rings and *-algebras 62
3.8. Weak based rings 63
3.9. Bibliographical notes 63
Chapter 4. Tensor categories 65
4.L Tensor and multitensor categories 65
4.2. Exactness of the tensor product 66
4.3. Semisimplicity of the unit object 69
4.4. Absence of self-extensions of the unit object 70
4.5. Grothendieck ring and Frobenius-Perron dimension 71
4.6. Deligne’s tensor product of tensor categories 73
4.7. Quantum traces, pivotal and spherical categories 73
4.8. Semisimple multitensor categories 76
4.9. Grothendieck rings of semisimple tensor categories 76
4.10. Categorification of based rings 78
4.11. Tensor subcategories 80
4.12. Chevalley property of tensor categories 81
4.13. Groupoids 82
4.14. The adjoint subcategory and universal grading 83
4.15. Equivariantization of tensor categories 86
4.16. Multitensor categories over arbitrary fields 87
4.17. Bibliographical notes 88
Chapter 5. Representation categories of Hopf algebras 91
5.1. Fiber functors 91
5.2. Bialgebras 91
5.3. Hopf algebras 93
5.4. Reconstruction theory in the infinite setting 97
5.5. More examples of Hopf algebras 99
5.6. The quantum group Uq(sla) 101
5.7. The quantum group Uq(g) 103
5.8. Representations of quantum groups and quantum function algebras 104
5.9. Absence of primitive elements 106
5.10. The Cartier-Gabriel-Kostant theorem 106
5.11. Pointed tensor categories and Hopf algebras 108
5.12. Quasi-bialgebras 110
5.13. Quasi-bialgebras with an antipode and quasi-Hopf algebras 112
5.14. Twists for bialgebras and Hopf algebras 114
5.15. Bibliographical notes 115
5.16. Other results 117
Chapter 6. Finite tensor categories 119
6.1. Properties of projective objects 119
6.2. Categorical freeness 122
6.3. Injective and surjective tensor functors 124
6.4. The distinguished invertible object 126
6.5. Integrals in quasi-Hopf algebras and unimodular categories 127
6.6. Degeneracy of the Cartan matrix 129
6.7. Bibliographical notes 129
CONTENTS
ix
Chapter 7. Module categories 131
7.1. The definition of a module category 131
7.2. Module functors 134
7.3. Module categories over multitensor categories 135
7.4. Examples of module categories 136
7.5. Exact module categories over finite tensor categories 138
7.6. First properties of exact module categories 139
7.7. Module categories and Z+-modules 140
7.8. Algebras in multitensor categories 141
7.9. Internal Homs in module categories 147
7.10. Characterization of module categories in terms of algebras 150
7.11. Categories of module functors 154
7.12. Dual tensor categories and categorical Morita equivalence 155
7.13. The center construction 162
7.14. The quantum double construction for Hopf algebras 163
7.15. Yetter-Drinfeld modules 166
7.16. Invariants of categorical Morita equivalence 166
7.17. Duality for tensor functors and Lagrange’s Theorem 170
7.18. Hopf bimodules and the Fundamental Theorem 172
7.19. Radford’s isomorphism for the fourth dual 176
7.20. The canonical Frobenius algebra of a unimodular category 178
7.21. Categorical dimension of a multifusion category 179
7.22. Davydov-Yetter cohomology and deformations of tensor categories 183
7.23. Weak Hopf algebras 186
7.24. Bibliographical notes 189
7.25. Other results 192
Chapter 8. Braided categories 195
8.1. Definition of a braided category 195
8.2. First examples of braided categories and functors 197
8.3. Quasitriangular Hopf algebras 198
8.4. Pre֊metric groups and pointed braided fusion categories 203
8.5. The center as a braided category 207
8.6. Factorizable braided tensor categories 208
8.7. Module categories over braided tensor categories 209
8.8. Commutative algebras and central functors 210
8.9. The Drinfeld morphism 214
8.10. Ribbon monoidal categories 216
8.11. Ribbon Hopf algebras 221
8.12. Characterization of Morita equivalence 221
8.13. The 5-matrix of a pre-modular category 224
8.14. Modular categories 227
8.15. Gauss sums and the central charge 229
8.16. Representation of the modular group 230
8.17. Modular data 232
8.18. The Anderson-Moore-Vafa property and Verlinde categories 233
8.19. A non-spherical generalization of the 5-matrix 237
8.20. Centralizers and non-degeneracy 239
8.21. Dimensions of centralizers 244
X
CONTENTS
8.22. Projective centralizers 247
8.23. De-equivariantization 249
8.24. Braided G-crossed categories 253
8.25. Braided Hopf algebras, Nichols algebras, pointed Hopf algebras 256
8.26. Bibliographical notes 261
8.27. Other results 262
Chapter 9. Fusion categories 275
9.1. Ocneanu rigidity (absence of deformations) 275
9.2. Induction to the center 278
9.3. Duality for fusion categories 279
9.4. Pseudo-unitary fusion categories 282
9.5. Canonical spherical structure 284
9.6. Integral and weakly integral fusion categories 285
9.7. Group-theoretical fusion categories 287
9.8. Weakly group-theoretical fusion categories 289
9.9. Symmetric and Tannakian fusion categories 291
9.10. Existence of a fiber functor 298
9.11. Deligne’s theorem for infinite categories 301
9.12. The Deligne categories Rep(St), Rep(GLt), Rep(Ot), Rep(Sp2t) 305
9.13. Recognizing group-theoretical fusion categories 311
9.14. Fusion categories of prime power dimension 314
9.15. Burnside s theorem for fusion categories 317
9.16. Lifting theory 320
9.17. Bibliographical notes 321
9.18. Other results 322
Bibliography 325
Index 339
|
| any_adam_object | 1 |
| author | Etingof, Pavel 1969- Gelaki, Shlomo 1964- Nikshych, Dmitri 1973- Ostrik, Victor |
| author_GND | (DE-588)133258858 (DE-588)1183593279 (DE-588)1183593651 |
| author_facet | Etingof, Pavel 1969- Gelaki, Shlomo 1964- Nikshych, Dmitri 1973- Ostrik, Victor |
| author_role | aut aut aut aut |
| author_sort | Etingof, Pavel 1969- |
| author_variant | p e pe s g sg d n dn v o vo |
| building | Verbundindex |
| bvnumber | BV042814776 |
| classification_rvk | SK 230 SK 300 SK 320 |
| ctrlnum | (OCoLC)924601171 (DE-599)OBVAC12385536 |
| dewey-full | 512/.57 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.57 |
| dewey-search | 512/.57 |
| dewey-sort | 3512 257 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV042814776 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:10:15Z |
| institution | BVB |
| isbn | 9781470420246 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028244207 |
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| physical | xvi, 343 Seiten |
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| publisher | American Mathematical Society |
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| series | Mathematical surveys and monographs |
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| spelling | Etingof, Pavel 1969- Verfasser (DE-588)133258858 aut Tensor categories Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik Providence, Rhode Island American Mathematical Society [2015] © 2015 xvi, 343 Seiten txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs volume 205 Literaturverzeichnis Seite 325 - 337 Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Monoidale Kategorie (DE-588)4170466-6 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Monoidale Kategorie (DE-588)4170466-6 s Darstellungstheorie (DE-588)4148816-7 s Quantenmechanik (DE-588)4047989-4 s DE-604 Geometrie (DE-588)4020236-7 s Funktionalanalysis (DE-588)4018916-8 s Gelaki, Shlomo 1964- Verfasser (DE-588)1183593279 aut Nikshych, Dmitri 1973- Verfasser (DE-588)1183593651 aut Ostrik, Victor Verfasser aut Erscheint auch als Online-Ausgabe 978-1-4704-2349-0 Mathematical surveys and monographs volume 205 (DE-604)BV000018014 205 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028244207&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Etingof, Pavel 1969- Gelaki, Shlomo 1964- Nikshych, Dmitri 1973- Ostrik, Victor Tensor categories Mathematical surveys and monographs Funktionalanalysis (DE-588)4018916-8 gnd Darstellungstheorie (DE-588)4148816-7 gnd Monoidale Kategorie (DE-588)4170466-6 gnd Geometrie (DE-588)4020236-7 gnd Quantenmechanik (DE-588)4047989-4 gnd |
| subject_GND | (DE-588)4018916-8 (DE-588)4148816-7 (DE-588)4170466-6 (DE-588)4020236-7 (DE-588)4047989-4 |
| title | Tensor categories |
| title_auth | Tensor categories |
| title_exact_search | Tensor categories |
| title_full | Tensor categories Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik |
| title_fullStr | Tensor categories Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik |
| title_full_unstemmed | Tensor categories Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik |
| title_short | Tensor categories |
| title_sort | tensor categories |
| topic | Funktionalanalysis (DE-588)4018916-8 gnd Darstellungstheorie (DE-588)4148816-7 gnd Monoidale Kategorie (DE-588)4170466-6 gnd Geometrie (DE-588)4020236-7 gnd Quantenmechanik (DE-588)4047989-4 gnd |
| topic_facet | Funktionalanalysis Darstellungstheorie Monoidale Kategorie Geometrie Quantenmechanik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028244207&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018014 |
| work_keys_str_mv | AT etingofpavel tensorcategories AT gelakishlomo tensorcategories AT nikshychdmitri tensorcategories AT ostrikvictor tensorcategories |