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

本研究计划的主题是原始形态及其时期图的李论研究。我们得到了几个结果，简要总结如下。在与H. Kajiura和a . Takahashi的一系列合作工作中，我们开发了一种利用正则权系统奇异点的矩阵分解范畴来构造李代数的新方法。这不仅统一了半简单李代数和椭圆李代数的构造，而且统一了原始形式、消失循环、相干束的派生范畴、无穷李代数等概念。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（野生型）李代数。与李代数有关的原始形式的研究应该给数学增添新的内容。（论文3篇）利用实代数几何给出了有限反射群的判别轨迹的补对Artin群是Eilenberg-Mclane空间这一经典事实的一个（部分）新证明。作为证明的一个应用，我们得到了一个组合问题的解：有向树线性化的最大次数。（发表论文2篇）尽管在研究计划中没有明确说明，但在现有基金的支持下，我们在非交换格上的配分函数研究方面取得了显著的进展。也就是说，我们提出了非交换格上配分函数高温展开的一般理论。引入了极限函数的紧集Ω（〓，G），并与生长函数Pr.〓相关联的极限集Ω （Pr，〓）进行了比较，并给出了迹式。(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)