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。

• 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

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.

我们介绍了一个概念，该概念通过顶点操作员代数的通用代数来满足的属性，并为该系统制定了一定的有限条件。然后，我们证明，在这样的系统上，某些类型的模块类别等于在这样的有限条件下的有限维代数上的模块类别。如果该系统被作为满足朱的有限条件的顶点操作员代数的普遍包围代数，那么我们的结果暗示，这种顶点操作员的模块类别等同于模块类别，而不是当前的lagebra。在用于构建当前谎言代数的平坦连接和躺椅的定义是在坐标的变化下不变的。 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朱的代数和构建非理性顶点操作员代数的示例的方法。主要结果基于与Akihiro Tsuchiya和Kiyokazu Nagatomo的联合研究。结果部分受到与Toshiyuki Abe，Tomoyuki Arakara，Markus Rosellen，C.Y.Dong和John Duncan的讨论的启发。

项目成果

A Z_2 orbifold model of the symplectic fermionic vertex perator superalgebra

辛费米子顶点算子超代数的Z_2轨道模型

• 发表时间: 2007
• 期刊: Matheroatische Zeitschrift 255(4)
• 作者: Komeda;J;Ohbuchi;A.;Toshiyuki Abe
• 通讯作者: Toshiyuki Abe

3-transposition groups of symplectic type and vertex operator algebras

• DOI: 10.2969/jmsj/1158241926
• 发表时间: 2003-11
• 期刊: Journal of The Mathematical Society of Japan
• 影响因子: 0.7
• 作者: A. Matsuo