Introduction to representation theory:
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| Main Authors: | , , , , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2011]
|
| Series: | Student mathematical library
volume 59 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | vii, 228 Seiten Illustrationen, Diagramme |
| ISBN: | 9780821853511 |
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Record in the Search Index
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| adam_text |
Titel: Introduction to representation theory
Autor: Etingof, Pavel
Jahr: 2011
Contents
Chapter 1. Introduction 1
Chapter 2. Basic notions of representation theory 5
§2.1. What is representation theory? 5
§2.2. Algebras 8
§2.3. Representations 9
§2.4. Ideals 15
§2.5. Quotients 15
§2.6. Algebras defined by generators and relations 16
§2.7. Examples of algebras 17
§2.8. Quivers 19
§2.9. Lie algebras 22
§2.10. Historical interlude: Sophus Lie's trials and
transformations 26
§2.11. Tensor products 30
§2.12. The tensor algebra 35
§2.13. Hilbert's third problem 36
§2.14. Tensor products and duals of representations of Lie
algebras 36
§2.15. Representations of sl(2) 37
§2.16. Problems on Lie algebras 39
Chapter 3. General results of representation theory 41
§3.1. Subrepresentations in semisimple representations 41
§3.2. The density theorem 43
§3.3. Representations of direct sums of matrix algebras 44
§3.4. Filtrations 45
§3.5. Finite dimensional algebras 46
§3.6. Characters of representations 48
§3.7. The Jordan-Holder theorem 50
§3.8. The Krull-Schmidt theorem 51
§3.9. Problems 53
§3.10. Representations of tensor products 56
Chapter 4. Representations of finite groups: Basic results 59
§4.1. Maschke's theorem 59
§4.2. Characters 61
§4.3. Examples 62
§4.4. Duals and tensor products of representations 65
§4.5. Orthogonality of characters 65
§4.6. Unitary representations. Another proof of Maschke's
theorem for complex representations 68
§4.7. Orthogonality of matrix elements 70
§4.8. Character tables, examples 71
§4.9. Computing tensor product multiplicities using character
tables 74
§4.10. Frobenius determinant 75
§4.11. Historical interlude: Georg Frobenius's "Principle of
Horse Trade" 77
§4.12. Problems 81
§4.13. Historical interlude: William Rowan Hamilton's
quaternion of geometry, algebra, metaphysics, and
poetry 86
Chapter 5. Representations of finite groups: Further results 91
§5.1. Frobenius-Schur indicator 91
§5.2. Algebraic numbers and algebraic integers 93
§5.3. Frobenius divisibility 96
§5.4. Burnside's theorem 98
§5.5. Historical interlude: William Burnside and intellectual
harmony in mathematics 100
§5.6. Representations of products 104
§5.7. Virtual representations 105
§5.8. Induced representations 105
§5.9. The Frobenius formula for the character of an induced
representation 106
§5.10. Frobenius reciprocity 107
§5.11. Examples 110
§5.12. Representations of Sn 110
§5.13. Proof of the classification theorem for representations
of Sn 112
§5.14. Induced representations for Sn 114
§5.15. The Frobenius character formula 115
§5.16. Problems 118
§5.17. The hook length formula 118
§5.18. Schur-Weyl duality for al(K) 119
§5.19. Schur-Weyl duality for GL(V) 122
§5.20. Historical interlude: Hermann Weyl at the intersection
of limitation and freedom 122
§5.21. Schur polynomials 128
§5.22. The characters of La 129
§5.23. Algebraic representations of GL(V) 130
§5.24. Problems 131
§5.25. Representations of GL2(¥q) 132
§5.26. Artin's theorem 141
§5.27. Representations of semidirect products 142
Chapter 6. Quiver representations 145
§6.1. Problems 145
§6.2. Indecomposable representations of the quivers
AUA2,A3 150
§6.3. Indecomposable representations of the quiver D± 154
§6.4. Roots 160
§6.5. Gabriel's theorem 163
§6.6. Reflection functors 164
§6.7. Coxeter elements 169
§6.8. Proof of Gabriel's theorem 170
§6.9. Problems 173
Chapter 7. Introduction to categories 177
§7.1. The definition of a category 177
§7.2. Functors 179
§7.3. Morphisms of functors 181
§7.4. Equivalence of categories 182
§7.5. Representable functors 183
§7.6. Adjoint functors 184
§7.7. Abelian categories 186
§7.8. Complexes and cohomology 187
§7.9. Exact functors 190
§7.10. Historical interlude: Eilenberg, Mac Lane, and
"general abstract nonsense" 192
Chapter 8. Homological algebra 201
§8.1. Projective and injective modules 201
§8.2. Tor and Ext functors 203
Chapter 9. Structure of finite dimensional algebras 209
§9.1. Lifting of idempotents 209
§9.2. Projective covers 210
§9.3. The Cartan matrix of a finite dimensional algebra 211
§9.4. Homological dimension 212
§9.5. Blocks 213
§9.6. Finite abelian categories 214
§9.7. Morita equivalence 215
References for historical interludes 217
Mathematical references 223
Index 225 |
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| spelling | Etingof, Pavel 1969- Verfasser (DE-588)133258858 aut Introduction to representation theory Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina ; with historical interludes by Slava Gerovitch Providence, Rhode Island American Mathematical Society [2011] © 2011 vii, 228 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Student mathematical library volume 59 Representations of algebras Representations of groups Associative rings and algebras -- Representation theory of rings and algebras -- Representation theory of rings and algebras msc Group theory and generalizations -- Linear algebraic groups and related topics -- Linear algebraic groups and related topics msc Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 s DE-604 Golberg, Oleg Verfasser aut Hensel, Sebastian Wolfgang 1986- Verfasser (DE-588)174020414 aut Liu, Tiankai Verfasser aut Schwendner, Alex Verfasser aut Vaintrob, Dmitry Verfasser aut Yudovina, Elena Verfasser aut Gerovič, Slava 1963- (DE-588)1051691524 wat Erscheint auch als Online-Ausgabe 978-1-4704-1222-7 Student mathematical library volume 59 (DE-604)BV013184751 59 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024410465&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Etingof, Pavel 1969- Golberg, Oleg Hensel, Sebastian Wolfgang 1986- Liu, Tiankai Schwendner, Alex Vaintrob, Dmitry Yudovina, Elena Introduction to representation theory Student mathematical library Representations of algebras Representations of groups Associative rings and algebras -- Representation theory of rings and algebras -- Representation theory of rings and algebras msc Group theory and generalizations -- Linear algebraic groups and related topics -- Linear algebraic groups and related topics msc Darstellungstheorie (DE-588)4148816-7 gnd |
| subject_GND | (DE-588)4148816-7 |
| title | Introduction to representation theory |
| title_auth | Introduction to representation theory |
| title_exact_search | Introduction to representation theory |
| title_full | Introduction to representation theory Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina ; with historical interludes by Slava Gerovitch |
| title_fullStr | Introduction to representation theory Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina ; with historical interludes by Slava Gerovitch |
| title_full_unstemmed | Introduction to representation theory Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina ; with historical interludes by Slava Gerovitch |
| title_short | Introduction to representation theory |
| title_sort | introduction to representation theory |
| topic | Representations of algebras Representations of groups Associative rings and algebras -- Representation theory of rings and algebras -- Representation theory of rings and algebras msc Group theory and generalizations -- Linear algebraic groups and related topics -- Linear algebraic groups and related topics msc Darstellungstheorie (DE-588)4148816-7 gnd |
| topic_facet | Representations of algebras Representations of groups Associative rings and algebras -- Representation theory of rings and algebras -- Representation theory of rings and algebras Group theory and generalizations -- Linear algebraic groups and related topics -- Linear algebraic groups and related topics Darstellungstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024410465&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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