Verfügbarkeit: Representations of semisimple Lie algebras in the BGG category O

Representations of semisimple Lie algebras in the BGG category O:

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Bibliographische Detailangaben
1. Verfasser:	Humphreys, James E. 1939-2020 (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Providence, R.I. Amer. Math. Soc. 2008
Schriftenreihe:	Graduate studies in mathematics 94
Schlagworte:	Representations of Lie algebras Categories (Mathematics) Modulkategorie Halbeinfache Lie-Algebra
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XVI, 289 S. 27 cm
ISBN:	9780821846780 0821846787

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Datensatz im Suchindex

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adam_text	Contents Preface xv Chapter 0. Review of Semisimple Lie Algebras 1 §0.1. Cartan Decomposition 1 §0.2. Root Systems 3 §0.3. Weyl Groups 4 §0.4. Chevalley-Bruhat Ordering of W 5 §0.5. Universal Enveloping Algebras 6 §0.6. Integral Weights 7 §0.7. Representations 8 §0.8. Finite Dimensional Modules 9 §0.9. Simple Modules for sl(2,C) 9 Part I. Highest Weight Modules Chapter 1. Category O: Basics 13 §1.1. Axioms and Consequences 13 §1.2. Highest Weight Modules 15 §1.3. Verma Modules and Simple Modules 17 §1.4. Maximal Vectors in Verma Modules 18 §1.5. Example: sl(2, C) 20 §1.6. Finite Dimensional Modules 20 §1.7. Action of the Center 22 §1.8. Central Characters and Linked Weights 24 vii §1.9. Harish-Chandra Homomorphism 25 §1.10. Harish-Chandra s Theorem 26 §1.11. Category O is Artinian 28 §1.12. Subcategories Ox 30 §1.13. Blocks 30 §1.14. Formal Characters of Finite Dimensional Modules 32 §1.15. Formal Characters of Modules in O 33 §1.16. Formal Characters of Verma Modules 34 Notes 35 Chapter 2. Characters of Finite Dimensional Modules 37 §2.1. Summary of Prerequisites 37 §2.2. Formal Characters Revisited 38 §2.3. The Functions p and q 38 §2.4. Formulas of Weyl and Kostant 40 §2.5. Dimension Formula 42 §2.6. Maximal Submodule of M(A), A E A+ 43 §2.7. Related Topics 45 Notes 46 Chapter 3. Category O: Methods 47 §3.1. Horn and Ext 47 §3.2. Duality in O 49 §3.3. Duals of Highest Weight Modules 51 §3.4. The Reflection Group WyX] 52 §3.5. Dominant and Antidominant Weights 54 §3.6. Tensoring Verma Modules with Finite Dimensional Modules 56 §3.7. Standard Filtrations 58 §3.8. Projectives in O 60 §3.9. Indecomposable Projectives 62 §3.10. Standard Filtrations of Projectives 64 §3.11. BGG Reciprocity 65 §3.12. Example: sl(2,C) 66 §3.13. Projective Generators and Finite Dimensional Algebras 68 §3.14. Contravariant Forms 68 §3.15. Universal Construction 70 Notes 71 Chapter 4. Highest Weight Modules I 73 §4.1. Simple Submodules of Verma Modules 74 §4.2. Homomorphisms between Verma Modules 75 §4.3. Special Case: Dominant Integral Weights 76 §4.4. Simplicity Criterion: Integral Case 77 §4.5. Existence of Embeddings: Preliminaries 78 §4.6. Existence of Embeddings: Integral Case 79 §4.7. Existence of Embeddings: General Case 81 §4.8. Simplicity Criterion: General Case 82 §4.9. Blocks of O Revisited 83 §4.10. Example: Antidominant Projectives 84 §4.11. Application to st(3, C) 85 §4.12. Shapovalov Elements 86 §4.13. Proof of Shapovalov s Theorem 88 §4.14. A Look Back at Verma s Thesis 90 Notes 91 Chapter 5. Highest Weight Modules II 93 §5.1. BGG Theorem 93 §5.2. Bruhat Ordering 94 §5.3. Jantzen Filtration 95 §5.4. Example: sI(3,C) 97 §5.5. Application to BGG Theorem 98 §5.6. Key Lemma 98 §5.7. Proof of Jantzen s Theorem 100 §5.8. Determinant Formula 102 §5.9. Details of Shapovalov s Proof 103 Notes 106 Chapter 6. Extensions and Resolutions 107 §6.1. BGG Resolution of a Finite Dimensional Module 108 §6.2. Weak BGG Resolution 109 §6.3. Exactness of the Sequence 110 §6.4. Weights of the Exterior Powers 111 §6.5. Extensions of Verma Modules 113 §6.6. Application: Bott s Theorem 115 §6.7. Squares 116 §6.8. Maps in a BGG Resolution 118 §6.9. Homological Dimension 120 §6.10. Higher Ext Groups 122 §6.11. Vanishing Criteria for Extn 123 §6.12. Computation of Extg(M(/x),M(A)v) 124 §6.13. Ext Criterion for Standard Filtrations 125 §6.14. Characters in Terms of Ext^ 126 §6.15. Comparison of Ext£ and Lie Algebra Cohomology 127 Notes 128 Chapter 7. Translation Functors 129 §7.1. Translation Functors 130 §7.2. Adjoint Functor Property 131 §7.3. Weyl Group Geometry 132 §7.4. Nonintegral Weights 134 §7.5. Key Lemma 135 §7.6. Translation Functors and Verma Modules 137 §7.7. Translation Functors and Simple Modules 138 §7.8. Application: Category Equivalences 138 §7.9. Translation to Upper Closures 140 §7.10. Character Formulas 142 §7.11. Translation Functors and Projective Modules 143 §7.12. Translation from a Facet Closure 144 §7.13. Example 145 §7.14. Translation from a Wall 146 §7.15. Wall-Crossing Functors 148 §7.16. Self-Dual Projectives 149 Notes 152 Chapter 8. Kazhdan-Lusztig Theory 153 §8.1. The Multiplicity Problem for Verma Modules 154 §8.2. Hecke Algebras and Kazhdan-Lusztig Polynomials 156 §8.3. Examples 157 §8.4. Kazhdan-Lusztig Conjecture 159 §8.5. Schubert Varieties and KL Polynomials 160 §8.6. Example: W of Type C3 161 §8.7. Jantzen s Multiplicity One Criterion 162 §8.8. Proof of the KL Conjecture 165 §8.9. Outline of the Proof 166 §8.10. Ext Functors and Vogan s Conjecture 168 §8.11. KLV Polynomials 169 §8.12. The Jantzen Conjecture and the KL Conjecture 171 §8.13. Weight Filtrations and Jantzen Filtrations 172 §8.14. Review of Loewy Filtrations 173 §8.15. Loewy Filtrations and KL Polynomials 174 §8.16. Some Details 177 Part II. Further Developments Chapter 9. Parabolic Versions of Category O 181 §9.1. Standard Parabolic Subalgebras 182 §9.2. Modules for Levi Subalgebras 183 §9.3. The Category C 3 184 §9.4. Parabolic Verma Modules 186 §9.5. Example: sl(3,C) 188 §9.6. Formal Characters and Composition Factors 189 §9.7. Relative Kazhdan-Lusztig Theory 190 §9.8. Projectives and BGG Reciprocity in Op 191 §9.9. Structure of Parabolic Verma Modules 192 §9.10. Maps between Parabolic Verma Modules 193 §9.11. Parabolic Verma Modules of Scalar Type 195 §9.12. Simplicity of Parabolic Verma Modules 196 §9.13. Jantzen s Simplicity Criterion 198 §9.14. Socles and Self-Dual Projectives 199 §9.15. Blocks of Op 200 §9.16. Analogue of the BGG Resolution 201 §9.17. Filtrations and Rigidity 203 §9.18. Special Case: Maximal Parabolic Subalgebras 204 Notes 206 Chapter 10. Projective Functors and Principal Series 207 §10.1. Functors on Category O 208 §10.2. Tensoring With a Dominant Verma Module 210 §10.3. Proof of the Theorem 211 §10.4. Module Categories 212 §10.5. Projective Functors 213 §10.6. Annihilator of a Verma Module 215 §10.7. Comparison of Horn Spaces 216 §10.8. Classification Theorem 218 §10.9. Harish-Chandra Modules 219 §10.10. Principal Series Modules and Category O 221 Notes 222 Chapter 11. Tilting Modules 223 §11.1. Tilting Modules 224 §11.2. Indecomposable Tilting Modules 225 §11.3. Translation Functors and Tilting Modules 227 §11.4. Grothendieck Groups 229 §11.5. Subgroups of K 230 §11.6. Fusion Rules 231 §11.7. Formal Characters 232 §11.8. The Parabolic Case 234 Chapter 12. Twisting and Completion Functors 235 §12.1. Shuffling Functors 236 §12.2. Shuffled Verma Modules 237 §12.3. Families of Twisted Verma Modules 239 §12.4. Uniqueness of a Family of Twisted Verma Modules 240 §12.5. Existence of Twisted Verma Modules 242 §12.6. Twisting Functors 242 §12.7. Arkhipov s Construction of Twisting Functors 243 §12.8. Twisted Versions of Standard Filtrations 244 §12.9. Complete Modules 245 §12.10. Enright s Completions 247 §12.11. Completion Functors 248 §12.12. Comparison of Functors 249 Chapter 13. Complements 251 §13.1. Primitive Ideals in U(g) 252 §13.2. Classification of Primitive Ideals 253 §13.3. Structure of a Fiber 254 §13.4. Kostant s Problem 255 §13.5. Kac-Moody Algebras 256 §13.6. Category O for Kac-Moody Algebras 258 §13.7. Highest Weight Categories 259 §13.8. Blocks and Finite Dimensional Algebras 260 §13.9. Quiver Attached to a Block 261 §13.10. Representation Type of a Block 263 §13.11. Soergel s Functor V 264 §13.12. Coinvariant Algebra of W 265 §13.13. Application: Category Equivalence 266 §13.14. Endomorphisms and Socles of Projectives 267 §13.15. Koszul Duality 268 Bibliography 271 Frequently Used Symbols 283 Index 287
adam_txt	Contents Preface xv Chapter 0. Review of Semisimple Lie Algebras 1 §0.1. Cartan Decomposition 1 §0.2. Root Systems 3 §0.3. Weyl Groups 4 §0.4. Chevalley-Bruhat Ordering of W 5 §0.5. Universal Enveloping Algebras 6 §0.6. Integral Weights 7 §0.7. Representations 8 §0.8. Finite Dimensional Modules 9 §0.9. Simple Modules for sl(2,C) 9 Part I. Highest Weight Modules Chapter 1. Category O: Basics 13 §1.1. Axioms and Consequences 13 §1.2. Highest Weight Modules 15 §1.3. Verma Modules and Simple Modules 17 §1.4. Maximal Vectors in Verma Modules 18 §1.5. Example: sl(2, C) 20 §1.6. Finite Dimensional Modules 20 §1.7. Action of the Center 22 §1.8. Central Characters and Linked Weights 24 vii §1.9. Harish-Chandra Homomorphism 25 §1.10. Harish-Chandra's Theorem 26 §1.11. Category O is Artinian 28 §1.12. Subcategories Ox 30 §1.13. Blocks 30 §1.14. Formal Characters of Finite Dimensional Modules 32 §1.15. Formal Characters of Modules in O 33 §1.16. Formal Characters of Verma Modules 34 Notes 35 Chapter 2. Characters of Finite Dimensional Modules 37 §2.1. Summary of Prerequisites 37 §2.2. Formal Characters Revisited 38 §2.3. The Functions p and q 38 §2.4. Formulas of Weyl and Kostant 40 §2.5. Dimension Formula 42 §2.6. Maximal Submodule of M(A), A E A+ 43 §2.7. Related Topics 45 Notes 46 Chapter 3. Category O: Methods 47 §3.1. Horn and Ext 47 §3.2. Duality in O 49 §3.3. Duals of Highest Weight Modules 51 §3.4. The Reflection Group WyX] 52 §3.5. Dominant and Antidominant Weights 54 §3.6. Tensoring Verma Modules with Finite Dimensional Modules 56 §3.7. Standard Filtrations 58 §3.8. Projectives in O 60 §3.9. Indecomposable Projectives 62 §3.10. Standard Filtrations of Projectives 64 §3.11. BGG Reciprocity 65 §3.12. Example: sl(2,C) 66 §3.13. Projective Generators and Finite Dimensional Algebras 68 §3.14. Contravariant Forms 68 §3.15. Universal Construction 70 Notes 71 Chapter 4. Highest Weight Modules I 73 §4.1. Simple Submodules of Verma Modules 74 §4.2. Homomorphisms between Verma Modules 75 §4.3. Special Case: Dominant Integral Weights 76 §4.4. Simplicity Criterion: Integral Case 77 §4.5. Existence of Embeddings: Preliminaries 78 §4.6. Existence of Embeddings: Integral Case 79 §4.7. Existence of Embeddings: General Case 81 §4.8. Simplicity Criterion: General Case 82 §4.9. Blocks of O Revisited 83 §4.10. Example: Antidominant Projectives 84 §4.11. Application to st(3, C) 85 §4.12. Shapovalov Elements 86 §4.13. Proof of Shapovalov's Theorem 88 §4.14. A Look Back at Verma's Thesis 90 Notes 91 Chapter 5. Highest Weight Modules II 93 §5.1. BGG Theorem 93 §5.2. Bruhat Ordering 94 §5.3. Jantzen Filtration 95 §5.4. Example: sI(3,C) 97 §5.5. Application to BGG Theorem 98 §5.6. Key Lemma 98 §5.7. Proof of Jantzen's Theorem 100 §5.8. Determinant Formula 102 §5.9. Details of Shapovalov's Proof 103 Notes 106 Chapter 6. Extensions and Resolutions 107 §6.1. BGG Resolution of a Finite Dimensional Module 108 §6.2. Weak BGG Resolution 109 §6.3. Exactness of the Sequence 110 §6.4. Weights of the Exterior Powers 111 §6.5. Extensions of Verma Modules 113 §6.6. Application: Bott's Theorem 115 §6.7. Squares 116 §6.8. Maps in a BGG Resolution 118 §6.9. Homological Dimension 120 §6.10. Higher Ext Groups 122 §6.11. Vanishing Criteria for Extn 123 §6.12. Computation of Extg(M(/x),M(A)v) 124 §6.13. Ext Criterion for Standard Filtrations 125 §6.14. Characters in Terms of Ext^ 126 §6.15. Comparison of Ext£ and Lie Algebra Cohomology 127 Notes 128 Chapter 7. Translation Functors 129 §7.1. Translation Functors 130 §7.2. Adjoint Functor Property 131 §7.3. Weyl Group Geometry 132 §7.4. Nonintegral Weights 134 §7.5. Key Lemma 135 §7.6. Translation Functors and Verma Modules 137 §7.7. Translation Functors and Simple Modules 138 §7.8. Application: Category Equivalences 138 §7.9. Translation to Upper Closures 140 §7.10. Character Formulas 142 §7.11. Translation Functors and Projective Modules 143 §7.12. Translation from a Facet Closure 144 §7.13. Example 145 §7.14. Translation from a Wall 146 §7.15. Wall-Crossing Functors 148 §7.16. Self-Dual Projectives 149 Notes 152 Chapter 8. Kazhdan-Lusztig Theory 153 §8.1. The Multiplicity Problem for Verma Modules 154 §8.2. Hecke Algebras and Kazhdan-Lusztig Polynomials 156 §8.3. Examples 157 §8.4. Kazhdan-Lusztig Conjecture 159 §8.5. Schubert Varieties and KL Polynomials 160 §8.6. Example: W of Type C3 161 §8.7. Jantzen's Multiplicity One Criterion 162 §8.8. Proof of the KL Conjecture 165 §8.9. Outline of the Proof 166 §8.10. Ext Functors and Vogan's Conjecture 168 §8.11. KLV Polynomials 169 §8.12. The Jantzen Conjecture and the KL Conjecture 171 §8.13. Weight Filtrations and Jantzen Filtrations 172 §8.14. Review of Loewy Filtrations 173 §8.15. Loewy Filtrations and KL Polynomials 174 §8.16. Some Details 177 Part II. Further Developments Chapter 9. Parabolic Versions of Category O 181 §9.1. Standard Parabolic Subalgebras 182 §9.2. Modules for Levi Subalgebras 183 §9.3. The Category C"3 184 §9.4. Parabolic Verma Modules 186 §9.5. Example: sl(3,C) 188 §9.6. Formal Characters and Composition Factors 189 §9.7. Relative Kazhdan-Lusztig Theory 190 §9.8. Projectives and BGG Reciprocity in Op 191 §9.9. Structure of Parabolic Verma Modules 192 §9.10. Maps between Parabolic Verma Modules 193 §9.11. Parabolic Verma Modules of Scalar Type 195 §9.12. Simplicity of Parabolic Verma Modules 196 §9.13. Jantzen's Simplicity Criterion 198 §9.14. Socles and Self-Dual Projectives 199 §9.15. Blocks of Op 200 §9.16. Analogue of the BGG Resolution 201 §9.17. Filtrations and Rigidity 203 §9.18. Special Case: Maximal Parabolic Subalgebras 204 Notes 206 Chapter 10. Projective Functors and Principal Series 207 §10.1. Functors on Category O 208 §10.2. Tensoring With a Dominant Verma Module 210 §10.3. Proof of the Theorem 211 §10.4. Module Categories 212 §10.5. Projective Functors 213 §10.6. Annihilator of a Verma Module 215 §10.7. Comparison of Horn Spaces 216 §10.8. Classification Theorem 218 §10.9. Harish-Chandra Modules 219 §10.10. Principal Series Modules and Category O 221 Notes 222 Chapter 11. Tilting Modules 223 §11.1. Tilting Modules 224 §11.2. Indecomposable Tilting Modules 225 §11.3. Translation Functors and Tilting Modules 227 §11.4. Grothendieck Groups 229 §11.5. Subgroups of K 230 §11.6. Fusion Rules 231 §11.7. Formal Characters 232 §11.8. The Parabolic Case 234 Chapter 12. Twisting and Completion Functors 235 §12.1. Shuffling Functors 236 §12.2. Shuffled Verma Modules 237 §12.3. Families of Twisted Verma Modules 239 §12.4. Uniqueness of a Family of Twisted Verma Modules 240 §12.5. Existence of Twisted Verma Modules 242 §12.6. Twisting Functors 242 §12.7. Arkhipov's Construction of Twisting Functors 243 §12.8. Twisted Versions of Standard Filtrations 244 §12.9. Complete Modules 245 §12.10. Enright's Completions 247 §12.11. Completion Functors 248 §12.12. Comparison of Functors 249 Chapter 13. Complements 251 §13.1. Primitive Ideals in U(g) 252 §13.2. Classification of Primitive Ideals 253 §13.3. Structure of a Fiber 254 §13.4. Kostant's Problem 255 §13.5. Kac-Moody Algebras 256 §13.6. Category O for Kac-Moody Algebras 258 §13.7. Highest Weight Categories 259 §13.8. Blocks and Finite Dimensional Algebras 260 §13.9. Quiver Attached to a Block 261 §13.10. Representation Type of a Block 263 §13.11. Soergel's Functor V 264 §13.12. Coinvariant Algebra of W 265 §13.13. Application: Category Equivalence 266 §13.14. Endomorphisms and Socles of Projectives 267 §13.15. Koszul Duality 268 Bibliography 271 Frequently Used Symbols 283 Index 287
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id	DE-604.BV035103477
illustrated	Not Illustrated
index_date	2024-07-02T22:15:02Z
indexdate	2024-07-09T21:22:17Z
institution	BVB
isbn	9780821846780 0821846787
language	English
lccn	2008012667
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-016771418
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owner	DE-384 DE-83 DE-91G DE-BY-TUM DE-11 DE-29T DE-19 DE-BY-UBM
owner_facet	DE-384 DE-83 DE-91G DE-BY-TUM DE-11 DE-29T DE-19 DE-BY-UBM
physical	XVI, 289 S. 27 cm
publishDate	2008
publishDateSearch	2008
publishDateSort	2008
publisher	Amer. Math. Soc.
record_format	marc
series	Graduate studies in mathematics
series2	Graduate studies in mathematics
spelling	Humphreys, James E. 1939-2020 Verfasser (DE-588)108120848 aut Representations of semisimple Lie algebras in the BGG category O James E. Humphreys Providence, R.I. Amer. Math. Soc. 2008 XVI, 289 S. 27 cm txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 94 Representations of Lie algebras Categories (Mathematics) Modulkategorie (DE-588)4170335-2 gnd rswk-swf Halbeinfache Lie-Algebra (DE-588)4193986-4 gnd rswk-swf Halbeinfache Lie-Algebra (DE-588)4193986-4 s Modulkategorie (DE-588)4170335-2 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-2120-5 Graduate studies in mathematics 94 (DE-604)BV009739289 94 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016771418&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Humphreys, James E. 1939-2020 Representations of semisimple Lie algebras in the BGG category O Graduate studies in mathematics Representations of Lie algebras Categories (Mathematics) Modulkategorie (DE-588)4170335-2 gnd Halbeinfache Lie-Algebra (DE-588)4193986-4 gnd
subject_GND	(DE-588)4170335-2 (DE-588)4193986-4
title	Representations of semisimple Lie algebras in the BGG category O
title_auth	Representations of semisimple Lie algebras in the BGG category O
title_exact_search	Representations of semisimple Lie algebras in the BGG category O
title_exact_search_txtP	Representations of semisimple Lie algebras in the BGG category O
title_full	Representations of semisimple Lie algebras in the BGG category O James E. Humphreys
title_fullStr	Representations of semisimple Lie algebras in the BGG category O James E. Humphreys
title_full_unstemmed	Representations of semisimple Lie algebras in the BGG category O James E. Humphreys
title_short	Representations of semisimple Lie algebras in the BGG category O
title_sort	representations of semisimple lie algebras in the bgg category o
topic	Representations of Lie algebras Categories (Mathematics) Modulkategorie (DE-588)4170335-2 gnd Halbeinfache Lie-Algebra (DE-588)4193986-4 gnd
topic_facet	Representations of Lie algebras Categories (Mathematics) Modulkategorie Halbeinfache Lie-Algebra
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016771418&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV009739289
work_keys_str_mv	AT humphreysjamese representationsofsemisimpleliealgebrasinthebggcategoryo

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