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

利用泛包络代数对顶点算子代数表示论进行扩展和应用。

基本信息

批准号：
17540012
负责人：
MATSUO Atsushi
金额：
$ 1.47万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540012/
关键词：
vertex operator algebra universal enveloping algebra conformal field theory finite-dimensional algebra Morita equivalence Lie algebra Riemann surface ポアソン代数トリプレット代数

项目摘要

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.

引入了顶点算子代数的泛包络代数所满足的性质公理化的概念，并给出了这类系统的有限性条件。证明了在这样的有限性条件下，系统上的某类模范畴等价于有限维代数上的模范畴。当系统是一个顶点算子代数的泛包络代数，且该顶点算子代数满足Zhu有限性条件时，我们的结果意味着这样的顶点算子代数上的模范畴等价于有限-然后，我们考虑了与顶点算子代数相关联的当前李代数，并得到了一个新的解释，即平坦连接使用构造了当前李代数，证明了李括号的定义在坐标变换下是不变的。利用Tsuchiya-Ueno-Yamada的方法，我们研究了一类收缩稳定曲线族上的模所对应的covacua层，证明了covacua层的凝聚性等性质，并给出了一个一般的朱氏代数公式，模的一种新方法，它从朱代数的n阶模导出Verma型模，并给出了非有理顶点算子代数的一种构造方法.主要结果是与土谷昭弘共同研究的结果，和长友清一这些结果部分受到了与Toshiyuki Abe、Tomoyuki Arakara、Markus Rosellen、C.Y.Dong和John邓肯讨论的启发。