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最重要的理论之一和核心工具。在这个项目中，我主要研究李氏三系（Lts）的变形。(1)奠定了Lts的变形理论基础。定义了Lts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上同理论非常适合于研究Lts的变形理论。众所周知，理解摄动和变形之间的区别是非常困难的。我的回答是：在摄动中，任何计算都是用原始积来完成的，但在变形的世界中，它是用变形积来完成的。通过与Gerstenhaber教授的讨论，这个项目已经基本完成。2003年9月，我在宾夕法尼亚大学拜访了他，2005年3月，他在广岛大学拜访了我。我们讨论的主要内容有：1)Yamaguchi上同；2)以Reimann、Kodaira-Spencer和Gerstenhaber为基础的变形理论的历史；3)我对变形理论在反问题中的应用的想法；4)变形理论在量子力学中的应用。在这个项目中，我只是展示了研究Lts变形理论的起点。我知道前进的道路上有许多障碍。甚至对于Lts表示的定义，至少有三种建议对我来说似乎是不同的。山口的那个不是普通的，需要更多的工作。对于工程中的逆问题，我相信变形理论将发挥非常重要的作用。

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.