Lie superalgebras and enveloping algebras:
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| 245 | 1 | 0 | |a Lie superalgebras and enveloping algebras |c Ian M. Musson |
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| 650 | 7 | |a Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Automorphisms, derivations, other operators |2 msc | |
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| _version_ | 1804149044002422784 |
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| adam_text | Titel: Lie superalgebras and enveloping algebras
Autor: Musson, Ian M
Jahr: 2012
Contents
Preface xv
Chapter 1. Introduction 1
§1.1. Basic Definitions 1
§1.2. Simple Lie Superalgebras 3
§1.3. Classification of Classical Simple Lie Superalgebras 8
§1.4. Exercises 9
Chapter 2. The Classical Simple Lie Superalgebras. I 11
§2.1. Introduction 11
§2.2. Lie Superalgebras of Type A(m,n) 12
§2.3. The Orthosymplectic Lie Superalgebras 14
2.3.1. The Lie Superalgebras 0Sp(2m + 1,2ra) 15
2.3.2. The Lie Superalgebras osp(2m, 2n) 16
2.3.3. The Lie Superalgebras 0Sp(2,2n - 2) 16
§2.4. The Strange Lie Superalgebras p(n) and q(n) 17
2.4.1. The Lie Superalgebras p(n) 17
2.4.2. The Lie Superalgebras q(n) 17
§2.5. Rationality Issues 19
§2.6. The Killing Form 19
§2.7. Exercises 20
Chapter 3. Borel Subalgebras and Dynkin-Kac Diagrams 25
§3.1. Introduction 25
§3.2. Cartan Subalgebras and Borel-Penkov-Serganova
Subalgebras 28
§3.3. Flags, Shuffles, and Borel Subalgebras 30
§3.4. Simple Roots and Dynkin-Kac Diagrams 38
3.4.1. Definitions and Low Rank Cases 38
3.4.2. From Borel Subalgebras and Shuffles to Simple
Roots 39
3.4.3. From Simple Roots to Diagrams 41
3.4.4. Back from Diagrams to Shuffles and Simple
Roots 44
3.4.5. Distinguished Simple Roots and Diagrams 47
3.4.6. Cartan Matrices 48
3.4.7. Connections with Representation Theory 50
§3.5. Odd Reflections 51
§3.6. Borel Subalgebras in Types A(l,l), p(n), and q(n) 55
3.6.1. Lie Superalgebras of Type A(1,1) 55
3.6.2. The Lie Superalgebra p(n) 58
3.6.3. The Lie Superalgebra q(n) 63
§3.7. Exercises 64
Chapter 4. The Classical Simple Lie Superalgebras. II 69
§4.1. Introduction and Preliminaries 69
§4.2. The Lie Superalgebras D(2,1; a) 71
§4.3. Alternative Algebras 75
§4.4. Octonions and the Exceptional Lie Superalgebra (2(3) 78
§4.5. Fierz Identities and the Exceptional Lie Superalgebra
F(4) 82
§4.6. Borel Subalgebras versus BPS-subalgebras 88
§4.7. Exercises 88
Chapter 5. Contragredient Lie Superalgebras 95
§5.1. Realizations and the Algebras q(A, t) 96
§5.2. Contragredient Lie Superalgebras: First Results 101
5.2.1. The Center, Root Space Decomposition, and
Antiautomorphism 101
5.2.2. Equivalent Matrices 104
5.2.3. Integrability and Kac-Moody Superalgebras 107
5.2.4. Serre Relations 108
§5.3. Identifying Contragredient Lie Superalgebras 109
5.3.1. The Exceptional Lie Superalgebras 110
5.3.2. The Nonexceptional Lie Superalgebras 111
§5.4. Invariant Bilinear Forms on Contragredient Lie
Superalgebras 112
5.4.1. The Invariant Form 112
§5.5. Automorphisms of Contragredient Lie Superalgebras 115
5.5.1. Semisimple Lie Algebras 115
5.5.2. Automorphisms Preserving Cartan and Borel
Subalgebras 116
5.5.3. Diagram and Diagonal Automorphisms 119
5.5.4. The Structure of H and (Aut g)° 120
5.5.5. More on Diagram Automorphisms 121
5.5.6. Outer Automorphisms 124
5.5.7. Automorphisms of Type A Lie Superalgebras 125
§5.6. Exercises 127
Chapter 6. The PBW Theorem and Filtrations on Enveloping
Algebras 131
§6.1. The Poincare-Birkhoff-Witt Theorem 131
§6.2. Free Lie Superalgebras and Witt s Theorem 135
§6.3. Filtered and Graded Rings 136
§6.4. Supersymmetrization 138
§6.5. The Clifford Filtration 142
§6.6. The Rees Ring and Homogenized Enveloping Algebras 143
§6.7. Exercises 145
Chapter 7. Methods from Ring Theory 147
§7.1. Introduction and Review of Basic Concepts 147
7.1.1. Motivation and Hypothesis 147
7.1.2. Bimodules 148
7.1.3. Prime and Primitive Ideals 149
7.1.4. Localization 150
§7.2. Torsion-Free Bimodules, Composition Series, and Bonds 152
§7.3. Gelfand-Kirillov Dimension 154
§7.4. Restricted Extensions 161
7.4.1. Main Results 161
7.4.2. Applications 165
§7.5. Passing Properties over Bonds 166
§7.6. Prime Ideals in Z2-graded Rings and Finite Ring
Extensions 170
7.6.1. Z2-graded Rings 170
7.6.2. Lying Over and Direct Lying Over 172
7.6.3. Further Results 177
§7.7. Exercises 178
Chapter 8. Enveloping Algebras of Classical Simple Lie
Superalgebras 181
§8.1. Root Space and Triangular Decompositions 181
§8.2. Verma Modules and the Category O 184
8.2.1. Verma Modules 184
8.2.2. Highest Weight Modules in the Type I Case 187
8.2.3. The Category O 188
8.2.4. Central Characters and Blocks 189
8.2.5. Contravariant Forms 190
8.2.6. Base Change 192
8.2.7. Further Properties of the Category O 193
§8.3. Basic Classical Simple Lie Superalgebras and a
Hypothesis 195
8.3.1. Basic Lie Superalgebras 195
8.3.2. A Hypothesis 197
§8.4. Partitions and Characters 199
§8.5. The Casimir Element 201
§8.6. Changing the Borel Subalgebra 204
§8.7. Exercises 205
Chapter 9. Verma Modules. I 207
§9.1. Introduction 207
§9.2. Universal Verma Modules and Sapovalov Elements 208
9.2.1. Basic Results and Hypotheses 208
9.2.2. Universal Verma Modules 210
9.2.3. Sapovalov Elements for Nonisotropic Roots 210
9.2.4. Sapovalov Elements for Isotropic Roots 212
§9.3. Verma Module Embeddings 213
9.3.1. Reductive Lie Algebras 213
9.3.2. Contragredient Lie Superalgebras 214
9.3.3. Typical Verma Modules 217
§9.4. Construction of Sapovalov Elements 218
§9.5. Exercises 221
Chapter 10. Verma Modules. II 223
§10.1. The Sapovalov Determinant 223
§10.2. The Jantzen Filtration 227
10.2.1. The p-adic Valuation of a Certain Determinant 227
10.2.2. The Jantzen Filtration 228
10.2.3. Evaluation of the Sapovalov Determinant 229
§10.3. The Jantzen Sum Formula 232
§10.4. Further Results 233
10.4.1. The Typical Case 233
10.4.2. Reductive Lie Algebras 233
10.4.3. Restriction of Verma Modules to go 234
§10.5. Exercises 235
Chapter 11. Schur-Weyl Duality 239
§11.1. The Double Commutant Theorem 239
§11.2. Schur s Double Centralizer Theorem 240
§11.3. Diagrams, Tableaux, and Representations of Symmetric
Groups 246
§11.4. The Robinson-Schensted-Knuth Correspondence 249
§11.5. The Decomposition of W and a Basis for U 253
§11.6. The Module U as a Highest Weight Module 258
§11.7. The Robinson-Schensted Correspondence 259
§11.8. Exercises 260
Chapter 12. Supersymmetric Polynomials 263
§12.1. Introduction 263
§12.2. The Sergeev-Pragacz Formula 265
§12.3. Super Schur Polynomials and Semistandard Tableaux 272
§12.4. Some Consequences 278
§12.5. Exercises 278
Chapter 13. The Center and Related Topics 281
§13.1. The Harish-Chandra Homomorphism: Introduction 281
§13.2. The Harish-Chandra Homomorphism: Details of the
Proof 284
§13.3. The Chevalley Restriction Theorem 293
§13.4. Supersymmetric Polynomials and Generators for I(h) 298
§13.5. Central Characters 299
13.5.1. Equivalence Relations for Central Characters 299
13.5.2. More on Central Characters 302
§13.6. The Ghost Center 304
§13.7. Duality in the Category O 304
§13.8. Exercises 306
Chapter 14. Finite Dimensional Representations of Classical Lie
Superalgebras 307
§14.1. Introduction 307
§14.2. Conditions for Finite Dimensionality 308
§14.3. The Orthosymplectic Case 310
14.3.1. Statements of the Results 310
14.3.2. A Special Case 311
14.3.3. The General Case 314
§14.4. The Kac-Weyl Character Formula 316
§14.5. Exercises 317
Chapter 15. Prime and Primitive Ideals in Enveloping Algebras 319
§15.1. The Dixmier-Moeglin Equivalence 320
§15.2. Classical Simple Lie Superalgebras 322
15.2.1. A Theorem of Duflo and Its Superalgebra Analog 322
15.2.2. Type I Lie Superalgebras 323
§15.3. Semisimple Lie Algebras 324
15.3.1. Notation 325
15.3.2. The Characteristic Variety 326
15.3.3. Translation Functors on the Category O 329
15.3.4. Translation Maps on Primitive Ideals 332
15.3.5. Primitive Ideals for Type A Lie Algebras 337
15.3.6. The Poset of Primitive Ideals and the Kazhdan-
Lusztig Conjecture 342
15.3.7. The Lie Superalgebra 0Sp(l,2n) 344
§15.4. More on Prime Ideals and Related Topics 346
15.4.1. Strongly Typical Representations, Annihilation,
and Separation Theorems 346
15.4.2. Primeness of U(g) 347
15.4.3. The Unique Minimal Prime 348
15.4.4. The Goldie Rank of U(g) 349
15.4.5. Enveloping Algebras of Nilpotent and Solvable
Lie Superalgebras 349
§15.5. Exercises 350
Chapter 16. Cohomology of Lie Superalgebras 355
§16.1. Introduction and Preliminaries 355
16.1.1. Complexes and Filtrations 355
§16.2. Spectral Sequences 357
16.2.1. The Spectral Sequence Associated to a Filtered
Complex 357
16.2.2. Bounded Filtrations and Convergence 359
§16.3. The Standard Resolution and the Cochain Complex 361
16.3.1. The Standard Resolution 361
16.3.2. The Cochain Complex 363
§16.4. Cohomology in Low Degrees 367
§16.5. The Cup Product 369
16.5.1. Definition and Basic Properties 369
16.5.2. Examples of Cup Products 372
§16.6. The Hochschild-Serre Spectral Sequence 374
§16.7. Exercises 379
Chapter 17. Zero Divisors in Enveloping Algebras 381
§17.1. Introduction 381
§17.2. Derived Functors and Global Dimension 383
§17.3. The Yoneda Product and the Bar Resolution 386
17.3.1. The Yoneda Product 386
17.3.2. The Bar Resolution 387
§17.4. The Lofwall Algebra 389
§17.5. Proof of the Main Results 392
§17.6. Further Homological Results 398
17.6.1. Tor and Homology of Lie Superalgebras 398
17.6.2. The Auslander and Macaulay Conditions 399
§17.7. Exercises 400
Chapter 18. Affine Lie Superalgebras and Number Theory 403
§18.1. Some Identities 403
§18.2. Affine Kac-Moody Lie Superalgebras 405
§18.3. Highest Weight Modules and the Affine Weyl Group 412
§18.4. The Casimir Operator 414
§18.5. Character Formulas 418
§18.6. The Jacobi Triple Product Identity 420
§18.7. Basic Classical Simple Lie Superalgebras 422
§18.8. The Case g= sl(2,1) 425
§18.9. The Case g= osp(3,2) 428
§18.10. Exercises 430
Appendix A. 433
§A.l. Background from Lie Theory 433
A. 1.1. Root Systems 433
A. 1.2. The Weyl Group 434
A. 1.3. Reductive Lie Algebras 435
A.1.4. A Theorem of Harish-Chandra 436
§A.2. Hopf Algebras and Z2-Graded Structures 437
A.2.1. Hopf Algebras 437
A.2.2. Remarks on Z2-Graded Structures: The Rule Of
Signs 440
A.2.3. Some Constructions with U(g)-Modules 443
A.2.4. The Supersymmetric and Superexterior Algebras 446
A.2.5. Actions of the Symmetric Group 447
§A.3. Some Ring Theoretic Background 449
A.3.1. The Diamond Lemma 449
A.3.2. Clifford Algebras 452
A.3.3. Ore Extensions 455
§A.4. Exercises 456
Appendix B. 463
Bibliography 471
Index 485
|
| any_adam_object | 1 |
| author | Musson, Ian M. 1953- |
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| id | DE-604.BV040095710 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:16:46Z |
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| isbn | 9780821868676 |
| language | English |
| lccn | 2011044064 |
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| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Musson, Ian M. 1953- Verfasser (DE-588)173254004 aut Lie superalgebras and enveloping algebras Ian M. Musson Providence, RI American Math. Soc. 2012 XX, 488 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 131 Lie superalgebras Universal enveloping algebras Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Universal enveloping (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Representations, algebraic theory (weights) msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Simple, semisimple, reductive (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Root systems msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Exceptional (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Solvable, nilpotent (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Automorphisms, derivations, other operators msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Homological methods in Lie (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Cohomology of Lie (super)algebras msc Associative rings and algebras -- Rings and algebras arising under various constructions -- Universal enveloping algebras of Lie algebras msc Lie-Superalgebra (DE-588)4304027-5 gnd rswk-swf Universelle Einhüllende (DE-588)4792961-3 gnd rswk-swf Lie-Superalgebra (DE-588)4304027-5 s Universelle Einhüllende (DE-588)4792961-3 s DE-604 Erscheint auch als Online-Ausgabe 978-0-8218-8504-8 Graduate studies in mathematics 131 (DE-604)BV009739289 131 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024952384&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Musson, Ian M. 1953- Lie superalgebras and enveloping algebras Graduate studies in mathematics Lie superalgebras Universal enveloping algebras Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Universal enveloping (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Representations, algebraic theory (weights) msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Simple, semisimple, reductive (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Root systems msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Exceptional (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Solvable, nilpotent (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Automorphisms, derivations, other operators msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Homological methods in Lie (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Cohomology of Lie (super)algebras msc Associative rings and algebras -- Rings and algebras arising under various constructions -- Universal enveloping algebras of Lie algebras msc Lie-Superalgebra (DE-588)4304027-5 gnd Universelle Einhüllende (DE-588)4792961-3 gnd |
| subject_GND | (DE-588)4304027-5 (DE-588)4792961-3 |
| title | Lie superalgebras and enveloping algebras |
| title_auth | Lie superalgebras and enveloping algebras |
| title_exact_search | Lie superalgebras and enveloping algebras |
| title_full | Lie superalgebras and enveloping algebras Ian M. Musson |
| title_fullStr | Lie superalgebras and enveloping algebras Ian M. Musson |
| title_full_unstemmed | Lie superalgebras and enveloping algebras Ian M. Musson |
| title_short | Lie superalgebras and enveloping algebras |
| title_sort | lie superalgebras and enveloping algebras |
| topic | Lie superalgebras Universal enveloping algebras Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Universal enveloping (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Representations, algebraic theory (weights) msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Simple, semisimple, reductive (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Root systems msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Exceptional (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Solvable, nilpotent (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Automorphisms, derivations, other operators msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Homological methods in Lie (super)algebras msc Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Cohomology of Lie (super)algebras msc Associative rings and algebras -- Rings and algebras arising under various constructions -- Universal enveloping algebras of Lie algebras msc Lie-Superalgebra (DE-588)4304027-5 gnd Universelle Einhüllende (DE-588)4792961-3 gnd |
| topic_facet | Lie superalgebras Universal enveloping algebras Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Universal enveloping (super)algebras Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Representations, algebraic theory (weights) Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Simple, semisimple, reductive (super)algebras Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Root systems Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Exceptional (super)algebras Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Solvable, nilpotent (super)algebras Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Automorphisms, derivations, other operators Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Homological methods in Lie (super)algebras Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Cohomology of Lie (super)algebras Associative rings and algebras -- Rings and algebras arising under various constructions -- Universal enveloping algebras of Lie algebras Lie-Superalgebra Universelle Einhüllende |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024952384&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT mussonianm liesuperalgebrasandenvelopingalgebras |