Expansion and application of representation theory of vertex operator algebras by means of the universal enveloping algebras.

引入一个概念，公理化了顶点算子代数的全称包络代数所满足的性质，并给出了该系统的有限条件。在此有限条件下，证明了系统上某类型模的范畴等价于有限维代数上的模的范畴。当系统是满足Zhu有限条件的顶点算子代数的全称包络代数时，我们的结果表明这样的顶点算子代数上的模的范畴等价于有限维代数上的模的范畴。然后考虑与顶点算子代数相关的当前李代数，得到了构造当前李代数的平面连接和李括号的定义在坐标变化下不变的新解释。在此基础上，利用Tsuchiya-Ueno-Yamada的方法考虑了与一系列模相关联的抽泡束，得到了抽泡束的相干性等预期性质。我们还考虑了朱氏代数的一般公式，从朱氏代数的第n个类似物中导出Verma型模的模块，以及构造非理性顶点算子代数示例的方法。主要结果是基于与土屋昭宏和永友清的联合研究。研究结果的部分灵感来自于与Abe Toshiyuki， Tomoyuki Arakara, Markus Rosellen， C.Y.Dong和John Duncan的讨论。

We introduced a concept which axiomatizes properties satisfied by the universal enveloping algebras of vertex operator algebras and formulated a certain finiteness condition for such a system. We then proved that the category of modules of certain type over such a system is equivalent to the category of modules over a finite-dimensional algebra under such a finiteness condition. In case the system is obtained as the universal enveloping algebra of a vertex operator algebra satisfying Zhu's finiteness condition, our result implies that the category of modules over such a vertex operator algebra is equivalent to the category of modules over a finite-dimensional algebra.We then considered the current Lie algebra associated with a vertex operator algebra and obtained a new interpretation of the fact that the flat connection used to construct the current Lie algebra and the definition of the Lie bracket is invariant under the change of coordinates. We then considered the sheaf of covacua associated with a series of modules attached to a family of punctured stable curves by using the method of Tsuchiya-Ueno-Yamada and established that some expected properties are satisfied, such as the coherency of the sheaf of covacua.We also considered a general formulation of Zhu's algebra, modules which induces the Verma type module from the n-th analogue of Zhu's algebra and a method of constructing examples of nonrational vertex operator algebras.The main results are based on joint research with Akihiro Tsuchiya, and Kiyokazu Nagatomo. The results are partially inspired by discussion with Toshiyuki Abe, Tomoyuki Arakara, Markus Rosellen, C.Y.Dong and John Duncan.