顶点代数与无穷维李代数相关问题的研究

顶点（算子）代数和无穷维李代数是李理论的重要研究领域，顶点（算子）代数的起源和发展与无穷维李代数的表示紧密联系。仿射顶点算子代数是顶点算子代数中最重要的一类，它是顶点算子代数的基石。本项目主要研究仿射顶点算子代数的coset和orbifold理论及无穷维李代数与顶点代数的联系问题。具体地，我们将研究以下问题：1.有理仿射顶点算子代数的parafermion代数的orbifold理论；2.非有理仿射顶点算子代数的parafermion代数的结构和表示及与W-代数的联系问题；3.三角李代数与顶点代数的联系及模范畴等价问题。这些问题的解决对完善顶点代数的coset和orbifold理论，更好的理解物理logarithmic共形场论及无穷维李代数的表示理论有重要意义。

Both vertex (operator) algebras and infinite dimensional Lie algebras are important fields of the Lie theory. The origin and development of vertex (operator) algebras are closely related to the representation of infinite dimensional Lie algebras. Affine vertex operator algebras are important family of vertex operator algebras and they are the major building blocks in vertex operator algebra theory. This project will study the coset and orbifold theory of the affine vertex operator algebras and the connections between the infinite dimensional Lie algebras and vertex algebras. Specifically, we study the following problems: 1.The orbifold theory of the parafermion algebras of the rational affine vertex operator algebras. 2.The structures and representations of the parafermion algebras of the irrational affine vertex operator algebras and their connections to the W-algebras. 3.The connections between trigonometric Lie algebras and vertex algebras and problems of the equivalence between their module categories. Solving these problems will contribute the coset and orbifold theory of vertex algebras, and understand the logarithmic conformal field theory better and enrich the representation theory of infinite dimensional Lie algebras.