仿射超代数与格顶点算子代数的表示及相关问题

The structure and representation theory of affine superalgebras,lattice vertex operator algebras and related algebras is one of the most important fields studied in Lie theory. All these algebras are related to infinite-dimensional Lie algebras, especially with the Virasoro Algebra. Vertex operator algebras are graded and so are the most important infinite-dimensional Lie (super)algebras. We will focus on studying the representation theory on several classes infinite-dimensional Lie (super)algebras and vertex operator algebras in this project: the classification of the integrable representations on affine Lie superalgebras and the weight modules over graded infinite-dimensional Lie algebras. Especially, we study the classification problems of the highest weight modules and modules of the intermediate series. In addition, we will give a unified method to describe the Whittaker modules for a wide range of graded Lie algebras. Futhermore, we will study the automorphism groups and fixed point subalgebras of several classes of important vertex operator algebras. Lastly, we will study rationality, regularity and C2-cofiniteness of vertex operator (super)algebras and their relationship.

仿射超代数、格顶点算子代数及相关代数的结构与表示理论是李理论的重要研究领域。这些代数都与无限维李代数，特别是与Virasoro代数有关。本课题主要研究无限维分次李（超）代数与顶点算子代数的几类重要表示：包括仿射型李超代数的可积表示的分类；分次无限维李代数的权模，特别是最高权模及中间序列模的分类问题；对尽可能广泛的分次李代数的Whittaker模构建一套统一刻画的方法；研究几类重要顶点算子代数的自同构群及不动点子代数的结构及性质；讨论一般顶点算子（超）代数的有理性、正则性和C2-余有限性及它们之间的关系。

仿射超代数、格顶点算子代数及相关代数的结构与表示理论是李理论的重要研究领域。这些代数都与无限维李代数，特别是与Virasoro 代数有关。本课题主要研究了无限维分次李（超）代数与顶点算子代数的几类重要表示：包括仿射型李超代数的可积表示的分类；分次无限维李代数(Schrödinger代数、W（2,2）代数、Schrödinger-Virasoro 代数、Schrödinger–Neveu–Schwarz代数、扭Heisenberg-Virasoro代数等)的权模，特别是最高权模及中间序列模的分类问题、Whittaker 模的构建；研究了抽象顶点算子代数表示的双模及其与Fusion rule计算的关系；给出了一个有效的方法来计算c1余有限顶点代数的Zhu代数A(V)。完成了一些二步幂零李代数的分类。

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