| Project/Area Number |
17540012
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | The University of Tokyo |
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Principal Investigator |
MATSUO Atsushi The University of Tokyo, Department of Mathematical Sciences, Associate Professor, 大学院数理科学研究科, 助教授 (20238968)
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| Co-Investigator(Kenkyū-buntansha) |
NAGATOMO Kiyokazu Osaka University, Graduate School of Information Science and Technology, Associate Professor, 大学院情報科学研究科, 助教授 (90172543)
ABE Toshiyuki Ehime University, Graduate School of Science and Engineering, Lecturer, 大学院理工学研究科, 講師 (30380215)
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| Project Period (FY) |
2005 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | 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 |
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| Research Abstract |
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.
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