无限维△-分次李代数及相关量子顶点代数的研究

We are going to stuty the vertex operator representation, module category and the conjugacy problem of the Cartan type subalgebras for the infinite dimensional △-graded Lie algebras defined by Berman-Moody and Benkart-Zelmanov; to study the phi-imaginary Verma modules,Whittaker modules,Wakimoto modules and the boson-fermion representations for the lower rank gim algebras introduced by Slodowy; to explore the relations between the restricted module categories of the △-graded Lie algebras and the quasi-module categories of the vertex algebras by establishing quasi-module theory for the △-graded vertex algebras; to study the classification of irreducible modules of the simple toroidal vertex algebras; to study the relations among the Yangians, the centerless double Yangians and the quantum vertex algebras. Especially we need to find the quantum Serre relations for the higher rank simple Lie algebras; to establish the relations between the nontrivial central extensions of the double Yangians and the h-adic quantum vertex algebras, and the relations between the quantum affine algebras and the phi-coordinated modules of the quantum vertex algebras.

研究Berman-Moody及Benkart-Zelmanov无限维△-分次李代数的顶点算子表示、模范畴理论及Cartan型子代数的共轭性问题；研究低秩Slodowy交叉矩阵代数的结构及其phi-虚Verma模、Whittaker模、Wakimoto模及Boson-fermion表示等问题；通过△-分次顶点代数的quasi模理论建立起△-分次李代数的限制模范畴同顶点代数的quasi模范畴联系；研究单toroidal顶点代数的表示与不可约模的分类；研究杨氏代数和无中心双杨氏代数同量子顶点代数的自然联系，特别是建立高秩单李代数的量子Serre关系；建立非平凡中心扩张双杨氏代数同h-adic量子顶点代数的联系，及量子仿射代数同量子顶点代数及其phi-坐标模的联系。

本课题主要研究了以下问题：.1. 通过顶点代数的quasi 模理论建立起无穷维李代数（特别是各种△-分次李代数）的限制模范畴同仿射型顶点代数的quasi 模范畴之间的内在联系；.2. 研究无限维△-分次李代数及其相关李代数的表示理论，包括顶点算子表示构造、可积模分类、Harish-Chandra 模及Whittaker模分类等问题；.3. 研究单 toroidal 顶点代数的不可约模的分类，建立了扭Toroidal李代数的限制模与Toroidal顶点代数扭模的联系；.4.建立量子仿射代数同量子顶点代数及其phi-坐标模的联系；.5. 构造扭量子仿射代数的顶点算子表示。