李理论及其应用

本项目研究数学物理中自然出现的一些代数与超代数以及它们的表示理论。主要研究内容在以下几个方面：(1) 李超代数的范畴 O 及 parabolic 范畴 O，包括 Kazhdan-Lusztig 理论、(parabolic) Verma 模的 Jantzen 滤过、不可约模的特征标与维数公式以及 deformed 范畴 O 理论；(2) 仿射李超代数的正能量表示，即可积最高权表示；(3) 量子群、量子 Yangians 以及超 Yangians；(4) 不变量理论，（超）对称性理论，高阶超 Schur-Weyl 对偶；(5) 李超代数的范畴 O 的 Koszul 对偶性。

In this proposal, we work on several problems on algebras and superalgebras which naturally accur in mathematical physics. The main problems we are concerned with are: (1) Category O and parabolic category O of Lie superalgebras, including Kazhdan-Lusztig theory, Jantzen filtrations of (parabolic) Verma modules, character and dimension formulae for irreducible modules and deformed category O theory; (2) positive energy representations, namely, unitary highest weight representations of affine Lie superalgebras; (3) quantum groups, quantized Yangians, super-Yangians; (4) inveriant theory, theory of (super)symmetry, higher super Schur-Weyl duality; (5) Koszul duality of category O of Lie superalgebras.