2007 Fiscal Year Final Research Report Summary
The study of integrable systems and Lie algebras associated with the period maps for primitive forms
| Project/Area Number |
16340016
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Kyoto University |
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Principal Investigator |
SAITO Kyoji Kyoto University, Mathematical SciResearch Institute for Mathematical Sciences, Professor (20012445)
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| Co-Investigator(Kenkyū-buntansha) |
MUKAI Shigeru Kyoto University, Research Institute for Mathematical Sciences, Professor (80115641)
SAITO Morihiko Kyoto University, Research Institute for Mathematical Sciences, Associate Professor (10186968)
TERAO Hiroaki Hokkaido University, Department of Mathematics, Professor (90119058)
SUWA Tatsuo Niigata University, Faculty of Engineering, Professor (40109418)
AOKI Hiroki Tokyo University of Science, Faculty of Science and Technology, Lecturer (10333189)
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| Project Period (FY) |
2004 – 2007
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| Keywords | primitive forms / Lie algebra / matrix factorization / derived category / reflection group / discontinuous group / partition function |
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| Research Abstract |
The subject of the present research program is the Lie theoretic study of primitive forms and their period maps. We obtain several results, which are briefly summarized as follows.
1. In a series of joint works with H. Kajiura and A. Takahashi, we developed a new method to approach to the construction of Lie algebras by a use of the category of matrix factorization of singularities associated with a regular system of weights. This has unified not only the construction of the semi-simple and elliptic Lie algebras, but also several concepts such as primitive forms, vanishing cycles, derived category of coherent sheaves, infinite Lie algebras, etc. In particular, we determine certain strongly exceptional collections for the category of ε=-1. This gives an essentially new (wild type) Lie algebras beyond Kac-Moody Lie algebras. The study of primitive forms associated with the Lie algebra should add new contents to mathematics. (3 published papers)
2. By a use of real algebraic geometry, we give a (part of) new proof of the classical fact that the complement of the discriminant loci of a finite reflection group is an Eilenberg-Mclane space with respect to an Artin group. As an application of the proof we obtain a solution to a combinatorial problem: the maximal number of linearization of an oriented tree. (2 published papers)
3. Even it was not stated explicitly in the program of the research, we obtain a remarkable progress, supported by the present fund, in the study of partition functions on non-commutative lattice. Namely, we develop a general theory of the high temperature development of the partition functions on non-commutative lattice. We introduce the compact set Ω (〓,G) of the limit functions, and obtain its comparison with limit set Ω (Pr,〓) associated with the growth function Pr.〓 and a trace formula. (Submitted)
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