The study of integrable systems and Lie algebras associated with the period maps for primitive forms

本课题的研究主题是对原始形式及其周期映射的测谎论研究。我们得到了几个结果，简要总结如下：1.在与H.Kajiura和A.Takahashi的一系列合作工作中，我们发展了一种新的方法，通过使用与正则权系相关的奇点的矩阵分解范畴来构造李代数。这不仅统一了半单李代数和椭圆李代数的构造，而且统一了本原形式、零圈、凝聚层导子范畴、无限李代数等概念，特别地，我们确定了ε=-1范畴的某些强例外集合。这给出了一个超越Kac-Moody李代数的本质上新的(野型)李代数。与李代数有关的本原形式的研究应该为数学增添新的内容。(3)2.利用实代数几何，给出了有限反射群的判别轨迹的补是关于Artin群的Eilenberg-McLane空间这一经典事实的(部分)新证明。作为证明的应用，我们得到了一个组合问题的解：有向树的最大线性化次数。(2篇已发表论文)3.即使在研究纲领中没有明确说明，在现有基金的支持下，我们在非对易格子上的配分函数的研究方面也取得了显著的进展。也就是说，我们发展了非对易晶格上配分函数高温发展的一般理论。引入了极限函数的紧集Ω(〓，G)，并将其与与增长函数有关的极限集Ω(Pr，〓)和一个迹公式进行了比较。(已提交)

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)