Dualities and representations of Lie superalgebras
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Providence, Rhode Island
American Mathematical Society
2012
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| Schriftenreihe: | Graduate studies in mathematics
144 |
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| 245 | 1 | 0 | |a Dualities and representations of Lie superalgebras |c Shun-Jen Cheng, Weiqiang Wang |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c 2012 | |
| 300 | |a XVII, 302 S. | ||
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| 490 | 1 | |a Graduate studies in mathematics |v 144 | |
| 650 | 4 | |a Lie superalgebras | |
| 650 | 4 | |a Duality theory (Mathematics) | |
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| adam_text | Titel: Dualities and representations of Lie superalgebras
Autor: Cheng, Shun-Jen
Jahr: 2012
Contents Preface xiii Chapter 1. Lie superalgebra ABC 1 §1.1. Lie superalgebras: Definitions and examples 1 1.1.1. Basic definitions 2 1.1.2. The general and special linear Lie superalgebras 4 1.1.3. The ortho-symplectic Lie superalgebras 6 1.1.4. The queer Lie superalgebras 8 1.1.5. The periplectic and exceptional Lie superalgebras 9 1.1.6. The Cartan series 10 1.1.7. The classification theorem 12 §1.2. Structures of classical Lie superalgebras 13 1.2.1. A basic structure theorem 13 1.2.2. Invariant bilinear forms for 0 1 and 05p 16 1.2.3. Root system and Weyl group for gl(m|n) 16 1.2.4. Root system and Weyl group for spo(2m 2n+ 1) 17 1.2.5. Root system and Weyl group for spo (2m |2n) 17 1.2.6. Root system and odd invariant form for q («) 18 §1.3. Non-conjugate positive systems and odd reflections 19 1.3.1. Positive systems and fundamental systems 19 1.3.2. Positive and fundamental systems for gi(m|«) 21 1.3.3. Positive and fundamental systems for spo(2m|2n+1) 22 1.3.4. Positive and fundamental systems for spo(2m|2n) 23 1.3.5. Conjugacy classes of fundamental systems 25 §1.4. Odd and real reflections 26 1.4.1. A fundamental lemma 26 vii
Contents viii 1.4.2. Odd reflections 4/ 1.4.3. Real reflections 28 1.4.4. Reflections and fundamental systems 28 1.4.5. Examples 30 §1.5. Highest weight theory 31 1.5.1. The Poincare-Birkhoff-Witt (PBW) Theorem 31 1.5.2. Representations of solvable Lie superalgebras 32 1.5.3. Highest weight theory for basic Lie superalgebras 33 1.5.4. Highest weight theory for q(n) 35 §1.6. Exercises 37 Notes 40 Chapter 2. Finite-dimensional modules 43 §2.1. Classification of finite-dimensional simple modules 43 2.1.1. Finite-dimensional simple modules of g[(m|n) 43 2.1.2. Finite-dimensional simple modules of spo(2m|2) 45 2.1.3. A virtual character formula 45 2.1.4. Finite-dimensional simple modules of spo(2m(277 + 1) 47 2.1.5. Finite-dimensional simple modules of spo(2m|2/i) 50 2.1.6. Finite-dimensional simple modules of q (77) 53 §2.2. Harish-Chandra homomorphism and linkage 55 2.2.1. Supersymmetrization 55 2.2.2. Central characters 56 2.2.3. Harish-Chandra homomorphism for basic Lie superalgebras 57 2.2.4. Invariant polynomials for gl and osp 59 2.2.5. Image of Harish-Chandra homomorphism for gl and osp 62 2.2.6. Linkage for gl and osp 65 2.2.7. Typical finite-dimensional irreducible characters 68 §2.3. Harish-Chandra homomorphism and linkage for q (77) 69 2.3.1. Central characters for q(/7) 70 2.3.2. Harish-Chandra homomorphism for q (77) 70 2.3.3. Linkage for q (77) 74 2.3.4. Typical finite-dimensional characters of q (77 ) 76 §2.4. Extremal weights of finite-dimensional simple modules 77 2.4.1. Extremal weights for Ql(m n) 11 2.4.2. Extremal weights for spo(2777|2/7 + 1) 80 2.4.3. Extremal weights for spo(2777|2/7) 82 §2.5. Exercises 85 Notes 89 Chapter 3. Schur duality 91
Contents IX §3.1. Generalities for associative superalgebras 91 3.1.1. Classification of simple superalgebras 92 3.1.2. Wedderburn Theorem and Schur’s Lemma 94 3.1.3. Double centralizer property for superalgebras 95 3.1.4. Split conjugacy classes in a finite supergroup 96 §3.2. Schur-Sergeev duality of type A 98 3.2.1. Schur-Sergeev duality, I 98 3.2.2. Schur-Sergeev duality, II 100 3.2.3. The character formula 104 3.2.4. The classical Schur duality 105 3.2.5. Degree of atypicality of X : 106 3.2.6. Category of polynomial modules 108 §3.3. Representation theory of the algebra !K„ 109 3.3.1. A double cover 110 3.3.2. Split conjugacy classes in B n 111 3.3.3. A ring structure on R~ 114 3.3.4. The characteristic map 116 3.3.5. The basic spin module 118 3.3.6. The irreducible characters 119 §3.4. Schur-Sergeev duality for q(n) 121 3.4.1. A double centralizer property 121 3.4.2. The Sergeev duality 123 3.4.3. The irreducible character formula 125 §3.5. Exercises 125 Notes 128 Chapter 4. Classical invariant theory 131 §4.1. FFT for the general linear Lie group 131 4.1.1. General invariant theory 132 4.1.2. Tensor and multilinear FFT for GL(V) 133 4.1.3. Formulation of the polynomial FFT for GL(V) 134 4.1.4. Polarization and restitution 135 §4.2. Polynomial FFT for classical groups 137 4.2.1. A reduction theorem of Weyl 137 4.2.2. The symplectic and orthogonal groups 139 4.2.3. Formulation of the polynomial FFT 140 4.2.4. From basic to general polynomial FFT 141 4.2.5. The basic case 142 §4.3. Tensor and supersymmetric FFT for classical groups 145 4.3.1. Tensor FFT for classical groups 145 4.3.2. From tensor FFT to supersymmetric FFT 147
x Contents §4.4. Exercises 149 Notes 1^0 Chapter 5. Howe duality 151 §5.1. Weyl-Clifford algebra and classical Lie superalgebras 152 5.1.1. Weyl-Clifford algebra 152 5.1.2. A filtration on Weyl-Clifford algebra 154 5.1.3. Relation to classical Lie superalgebras 155 5.1.4. A general duality theorem 157 5.1.5. A duality for Weyl-Clifford algebras 159 §5.2. Howe duality for type A and type Q 160 5.2.1. Howe dual pair (GL(k),Qi(m n)) 160 5.2.2. (GL(L),g[(m|n))-Howe duality 162 5.2.3. Formulas for highest weight vectors 164 5.2.4. (q(m),q(n))-Howe duality 166 §5.3. Howe duality for symplectic and orthogonal groups 169 5.3.1. Howe dual pair (Sp(V), osp(2m|2n)) 170 5.3.2. (Sp(y), osp(2wz|2n))-Howe duality 172 5.3.3. Irreducible modules of O(V) 175 5.3.4. Howe dual pair (O(fc),spo(2m|2n)) 177 5.3.5. (0(V),spo(2m|2n))-Howe duality 178 §5.4. Howe duality for infinite-dimensional Lie algebras 180 5.4.1. Lie algebras a M , c„c, and boo 180 5.4.2. The fermionic Fock space 183 5.4.3. (GL(f),cioo)-Howe duality 184 5.4.4. (Sp(k), c»o)-Howe duality 187 5.4.5. (0( ),Dc«,)-Howe duality 190 §5.5. Character formula for Lie superalgebras 192 5.5.1. Characters for modules of Lie algebras and be» 192 5.5.2. Characters of oscillator osp(2m|2n)-modules 193 5.5.3. Characters for oscillator spo(2m|2«)-modules 195 §5.6. Exercises 197 Notes 201 Chapter 6. Super duality 205 §6.1. Lie superalgebras of classical types 206 6.1.1. Head, tail, and master diagrams 206 6.1.2. The index sets 208 6.1.3. Infinite-rank Lie superalgebras 208 6.1.4. The case of m = 0 211 6.1.5. Finite-dimensional Lie superalgebras 213
Contents xi 6.1.6. Central extensions 213 §6.2. The module categories 214 6.2.1. Category of polynomial modules revisited 215 6.2.2. Parabolic subalgebras and dominant weights 217 6.2.3. The categories 0, 0, and 0 218 6.2.4. The categories 0„, 0„, and 0„ 220 6.2.5. Truncation functors 221 §6.3. The irreducible character formulas 222 6.3.1. Two sequences of Borel subalgebras of g 223 6.3.2. Odd reflections and highest weight modules 225 6.3.3. The functors T and T 228 6.3.4. Character formulas 231 §6.4. Kostant homology and KLV polynomials 232 6.4.1. Homology and cohomology of Lie superalgebras 232 6.4.2. Kostant u _ -homology and u-cohomology 235 6.4.3. Comparison of Kostant homology groups 236 6.4.4. Kazhdan-Lusztig-Vogan (KLV) polynomials 239 6.4.5. Stability of KLV polynomials 240 §6.5. Super duality as an equivalence of categories 241 6.5.1. Extensions a la Baer-Yoneda 241 6.5.2. Relating extensions in 0, 0, and 0 243 6.5.3. Categories 0-^, ((/, and 0^ 247 6.5.4. Lifting highest weight modules 247 6.5.5. Super duality and strategy of proof 248 6.5.6. The proof of super duality 250 §6.6. Exercises 255 Notes 258 Appendix A. Symmetric functions 261 § A. 1. The ring A and Schur functions 261 A. 1.1. The ring A 261 A. 1.2. Schur functions 265 A.1.3. Skew Schur functions 268 A. 1.4. The Frobenius characteristic map 270 §A.2. Supersymmetric polynomials 271 A.2.1. The ring of supersymmetric polynomials 271 A.2.2. Super Schur functions 273 §A.3. The ring T and Schur Q-functions 275 A.3.1. The ring T 275 A.3.2. Schur ^-functions 277 A.3.3. Inner product on T 278
Contents xii A.3.4. A characterization of T 280 A.3.5. Relating A and r 281 §A.4. The Boson-Fermion correspondence 282 A.4.1. The Maya diagrams 282 A.4.2. Partitions 282 A.4.3. Fermions and fermionic Fock space 284 A.4.4. Charge and energy 286 A.4.5. From Bosons to Fermions 287 A.4.6. Fermions and Schur functions 289 A.4.7. Jacobi triple product identity 289 Notes 290 Bibliography 291 Index 299
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| id | DE-604.BV040657445 |
| illustrated | Not Illustrated |
| indexdate | 2025-02-03T17:41:49Z |
| institution | BVB |
| isbn | 9780821891186 |
| language | English |
| lccn | 2012031989 |
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| physical | XVII, 302 S. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
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| publisher | American Mathematical Society |
| record_format | marc |
| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spellingShingle | Cheng, Shun-Jen 1963- Wang, Weiqiang 1970- Dualities and representations of Lie superalgebras Graduate studies in mathematics Lie superalgebras Duality theory (Mathematics) 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 Dualitätstheorie (DE-588)4150801-4 gnd Lie-Superalgebra (DE-588)4304027-5 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4150801-4 (DE-588)4304027-5 (DE-588)4037944-9 |
| title | Dualities and representations of Lie superalgebras |
| title_auth | Dualities and representations of Lie superalgebras |
| title_exact_search | Dualities and representations of Lie superalgebras |
| title_full | Dualities and representations of Lie superalgebras Shun-Jen Cheng, Weiqiang Wang |
| title_fullStr | Dualities and representations of Lie superalgebras Shun-Jen Cheng, Weiqiang Wang |
| title_full_unstemmed | Dualities and representations of Lie superalgebras Shun-Jen Cheng, Weiqiang Wang |
| title_short | Dualities and representations of Lie superalgebras |
| title_sort | dualities and representations of lie superalgebras |
| topic | Lie superalgebras Duality theory (Mathematics) 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 Dualitätstheorie (DE-588)4150801-4 gnd Lie-Superalgebra (DE-588)4304027-5 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Lie superalgebras Duality theory (Mathematics) 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 Dualitätstheorie Lie-Superalgebra Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025484266&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT chengshunjen dualitiesandrepresentationsofliesuperalgebras AT wangweiqiang dualitiesandrepresentationsofliesuperalgebras |