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

引入了顶点算子代数的万能包络代数所满足的性质的公理化概念，并给出了这类系统的一个有限条件。在这样的有限条件下，我们证明了这种系统上的某种类型的模范畴等价于有限维代数上的模范畴。当系统是满足朱的有限条件的顶点算子代数的泛包络代数时，我们的结果表明这样的顶点算子代数上的模范畴等价于有限维代数上的模范畴，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坦连接和李括号的定义在坐标变化下不变的新的解释.然后，利用Tsuhiya-Ueno-Yamada的方法，我们考虑了伴随在一族被穿孔的稳定曲线族上的一系列模所对应的余真空层，并证明了一些期望的性质，如余真空层的相干性。我们还考虑了朱氏代数的一般形式、由朱氏代数的第n次模拟导出Verma型模的模以及构造无理顶点算子代数实例的方法。主要结果是基于与Akihiro Tsuhiya和Kiyokazu Nagtom o的联合研究。这些结果的部分灵感来自于与安倍晋三、荒原智幸、马库斯·罗塞伦、董振英和约翰·邓肯的讨论。

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.