本课题研究顶点算子代数的扩张与整形式, 模顶点算子代数，以及模顶点算子代数与模李代数之间的关系。主要内容有四个方面：（1）研究顶点算子代数的扩张，拟证明大家熟知的猜想：任何有理顶点算子代数的单扩张都是有理的；（2）研究顶点算子代数的整形式，拟建立任何有理顶点算子代数都有整形式；（3）研究模顶点算子代数的结果与表示理论，其中包括最高权模理论，扭模及顶点算子代数的关系，如何利用整形式来构造模顶点算子代数；（4）研究模顶点算子代数与模李代数间的关系。本课题所研究的问题是顶点算子代数研究中的基本问题。其研究成果将极大促进顶点算子代数和模李代数的发展和交叉，为较为孤立的模李代数的研究提供新的想法和方法。

This is a proposal on the extensions of vertex operator algebras, the integral forms of vertex operator algebras, modular vertex operator algebras and its relations with modular Lie algebras. The proposal consists of four major parts: (1) Study the extensions of vertex operator algebras. Establish a well known conjecture in the field: Any simple extension of a rational vertex operator algebra is also rational. (2) Investigate the integral forms of vertex operator algebras. Prove that any rational vertex operator algebra has an integral form. (3) Establish the structure and representation theory for modular vertex operator algebra theory. The highest weight modules, the relation between twisted modules and modular vertex operator algebras and how to use the integral forms to construct modular vertex operator algebras will be studied. (4) Investigate the relation between modular vertex operator algebras and modular Lie algebras. These are the basic problems in the field. The results will greatly help the development of vertex operator algebra theory and its connections with modular Lie algebras, provide new ideas and methods for the relatively isolated research field of modular Lie algebras.