Deformation Theory of Algebras and related topics

代数变形理论是Gerstenhaber在1963年发现的。现在它已经成为研究数学物理的重要理论和核心工具之一。本课题主要研究李三系的形变问题，主要成果如下：（1）奠定了李三系形变理论的基础。定义了变形的无穷小、等价、刚性、可积等基本概念。最重要的发现是发现Yamaguchi上同调理论非常适合于研究Lts的形变理论。(2)To已知理解扰动和变形之间区别是非常困难的。我的答复如下：在扰动任何计算是由原来的产品，但它是由变形的产品在变形的世界。完成这个项目已基本完成的讨论与Gerstenhaber教授。2003年9月，我去宾夕法尼亚大学拜访了他，2005年3月，他去广岛大学拜访了我。我们讨论的主要内容如下：1）Yamaguchi上同调，2）基于数学家Reimann，Kodaira-Spencer和Gerstenhaber的形变理论的历史，3）我对形变理论在反问题中的应用的想法，4）形变理论在量子力学中的应用。问题和展望在这个项目中，我刚刚展示了研究Lts形变理论的起点。我知道前进的道路上有许多障碍。即使是对于一个LTS的表示的定义，至少有三个建议，这似乎是不同的，我。山口是不是顺序之一，需要更多的工作。对于工程中的反问题，变形理论将起到非常重要的作用。

Algebraic deformation theory was found by Gerstenhaber in 1963. Now this has become one of the most important theory and the core tool to investigate mathematical physics. In this project I have mainly worked on the deformation of Lie triple systems (Lts).Achievements (1)I have laid the foundation of the deformation theory of Lts'. I defined the basic concepts such as infinitesimal, equivalence, rigid, integrable for a deformation of a Lts. The most important discovery is to find out that Yamaguchi cohomology theory is well suited for the investigation of the deformation theory of Lts'.(2)To understand a difference between perturbation and deformation has been known to be very difficult. My answer is as follow : In the perturbation any calculation is done by original product, but it is done by deformed product in the deformed world.Accomplishment This project has been basically done by the discussions with Professor Gerstenhaber. I visited him at University of Pennsylvania in September 2003 and he visited me at Hiroshima University in March 2005. Main subjects of our discussions are as follows : 1)Yamaguchi cohomology, 2)History of Deformation Theory based on the mathematicians Reimann, Kodaira-Spencer and Gerstenhaber, 3)My idea of the application of Deformation Theory to Inverse problems and 4)Deformation Theory in the quantum mechanics.Problems and forward In this project, I have just shown the starting point to research the deformation theory of Lts'. I know that there are many obstructions to go forward. Even for the definition of the representation of a Lts there are at least three proposals which seem to be different for me. That of Yamaguchi is not the ordinal one and requires more works. As for the inverse problem in engineering, the deformation theory will take the very important role, I believe.