پیشرفت‌های اخیر در نظریه نمایش، گروه‌های کوانتومی، هندسه جبری و موضوعات مرتبط: جلسات ویژهٔ AMS در جنبه‌های هندسی و جبری نظریه نمایش و گروه‌های کوانتومی، و هندسه جبری غیرمتقارن، ۱۳-۱۴ اکتبر

This volume contains the proceedings of two AMS Special Sessions “Geometric and Algebraic Aspects of Representation Theory” and “Quantum Groups and Noncommutative Algebraic Geometry” held October 13–14, 2012, at Tulane University, New Orleans, Louisiana. Included in this volume are original research and some survey articles on various aspects of representations of algebras including Kac–Moody algebras, Lie superalgebras, quantum groups, toroidal algebras, Leibniz algebras and their connections with other areas of mathematics and mathematical physics. Preface 8 A classification of irreducible Wakimoto modules for the affine Lie algebra A1(1) 14 1. Introduction 14 2. Clifford vertex superalgebras 15 3. The vertex superalgebra \cV and its modules 16 4. Schur polynomials and irreducibility of ̃Fχ 18 5. Wakimoto modules 22 References 24 A note on U_{q}(D4(3))-Demazure crystals 26 1. Introduction 26 2. Quantum affine algebras and the perfect crystals 27 3. U_{q}(D4(3))- Perfect crystals 29 4. Demazure Modules and Demazure crystals 31 5. U_{q}(D4(3))-Demazure crystals 33 References 34 Dimensions of imaginary root spaces of hyperbolic Kac–Moody algebras 36 1. Introduction 36 2. Berman-Moody formula and Peterson formula 38 3. Works of Feingold and Frenkel, and Kac, Moody and Wakimoto 40 4. Kang’s generalizations 41 5. Frenkel’s conjecture 44 6. Multiplicities of norm zero root spaces 45 7. Extended and overextended Dynkin diagrams 45 ࠀ⸀ 刀漀漀琀 洀甀氀琀椀瀀氀椀挀椀琀椀攀猀 昀漀爀⃘㗜㠠脠耀 愀渀搀⃘㗜㠠脠 46 ऀ⸀ 䈀漀爀挀栀攀爀搀猠ᤀ猀 挀漀渀猀琀爀甀挀琀椀漀渀 46 10. Asymptotics by the method of Hardy-Ramanujan-Rademacher 48 11. Summary of results on root multiplicities 49 Acknowledgments 50 References 50 On some structures of Leibniz algebras 54 1. Introduction 54 2. Preliminaries 55 3. Solvability 57 4. Nilpotency 59 5. Semisimplicity 62 6. Classification of Low dimensional Leibniz algebras 64 References 66 A geometric construction of generalized q-Schur algebras 68 Introduction 68 1. The algebras C, C 68 2. The algebra L_{d} 72 3. A parametrization 73 Acknowledgements 74 References 74 On the classification of irreducible Gelfand-Tsetlin modules of sl(3) 76 1. Introduction 76 2. Preliminaries. 77 3. Gelfand-Tsetlin modules 78 4. Gelfand-Tsetlin modules of sl(3) 80 5. Modules in GT defined by tableaux 85 6. Localization functors 86 7. Generic blocks of GT and localization 87 8. Example 88 9. Main results 89 References 90 Supersymmetry and the modular double 94 1. Introduction 94 2. Reminder of the modular double. 95 3. Modular double for U_{q}(osp(1|2)) 99 4. R-matrix for U_{q,τ(q)}(osp(1|2)) 103 5. Final remarks 108 References 109 On Weyl modules for quantum and hyper loop algebras 112 Introduction 112 1. The Algebras 114 2. Weyl Modules 124 Acknowledgements 143 References 143 Toroidal Lie superalgebras and free field representations 148 1. Introduction 148 2. Preliminaries 149 3. Loop-like toroidal Lie superalgebras T(X) 152 4. Free field realizations of T(X) 154 5. Appendix 163 References 164 Invariants of (-1)-skew polynomial rings under permutation representations 168 0. Introduction 168 1. Definitions and basic properties 170 2. Upper bound for the algebra generators 174 3. Invariants under the full symmetric group S_{n} 178 4. Invariants under A_{n} 186 5. Converse of Kac-Watanabe-Gordeev Theorem 196 6. Appendix 201 References 204 On total Frobenius-Schur indicators 206 1. Introduction 206 2. Total Frobenius-Schur indicators 208 3. Proof of Theorem B 212 4. Twisted quantum doubles 216 5. Tambara-Yamagami categories 220 References 224 Loop Grassmannians in the framework of local spaces over a curve 228 1. Introduction 228 2. Local spaces 232 3. A generalization of loop Grassmannians 236 4. A topological realization of loop Grassmannians 238 References 238 Decorated geometric crystals and polyhedral realizations of type D_{n} 240 1. Introduction 240 2. Crystals and polyhedral realizations 241 3. Decorated geometric crystals 244 4. Explicit form of the decoration f_{B} of type D_{n} 248 5. Ultra-Discretization and Polyhedral Realizations of type D_{n} 251 References 254 Some Koszul properties of standard and irreducible modules 256 1. Introduction 256 2. Preliminaries 258 3. A review of some earlier results 262 4. Some new properties of Koszul and Q-Koszul algebras 265 5. A graded Ext result for G1T 269 6. Some relative results 271 7. Open questions 273 8. Appendix: Comparison of gradings 274 References 276 On higher order Leibniz identities in TCFT 280 1. Introduction 280 2. Toy story: Leibniz algebras for TVOA 281 3. Homotopy Leibniz algebra for TCFT 283 4. Conjectures 289 Acknowledgments 292 References 292