Structure Theory and Deformation Theory of Algebras with two multiplications

具有Jacobson根平方为零的非交换泊松代数（ncPa）是类lie泊松代数与标准ncPa的半直积。我在ncPa中找到了一个好元素。我已经成功地用这个元和标准泊松子代数描述了一个ncPa的结构。这也是研究代数系统结构的一个新方向。(2)分类定理在这个定理中，我列出了有限维ncPa上的所有有限维简单泊松模。(3)将Gerstenhaber的代数变形理论应用于ncPa。然后我发现如果ncPa是无穷小刚性的那么它就是刚性的。我是通过与Prof. Gerstenhaber （university of PA）的讨论完成这个项目的。有了这笔资金，我在2001年3月邀请了他几天，并在2001年8月访问了美国。这里我将列出讨论的主要内容。(1)我们发现我已经理解了他的代数变形的本质(2)我自己发现的有限维ncPa理论在数学和数学科学中的作用(3)量子群、代数变形理论及其关系变形理论的传播这个项目的目的之一是在日本传播变形理论。我相信，这是通过在日本数学学会的一次会议上发表演讲和组织一次会议，让格斯滕哈伯教授在九州工业大学讲课来完成的。之后(1)构造了一个三重系统的变形理论。这是由与温斯坦教授（加州大学）的讨论引起的。(2)学习金宝的作品。我一直和格斯滕哈伯教授讨论这个问题。这门课当然是关于量子群和代数变形理论的。(3)由Yang-Baxter方程构造ncPa

New results(1) Structure theorem A noncommuative Poisson algebra ("ncPa") with Jacobson radicalsquered zero is a semidirect product of a Lie-like Poisson algebra and a standard ncPa.I found a nice element in a ncPa. I have succeeded in describing a structure of a ncPa in terms of this Element and standard Poisson subalgebras. This is also a new direction to investigate a structure on algebraic systems.(2) Classification Theorem In this theorem I list all the finite-dimensional simple Poisson modules over a Finite-dimensional ncPa.(3) I have applied the Gerstenhaber's algebraic deformation theory to ncPa's. Then I found that if a ncPa is infinitesimally rigid then it is rigid.How carry out the projectI accomplished the project based on discussion with Prof. Gerstenhaber (Univ of PA). With the grant I could invite him several days in March, 2001 and also I visited US inAugust, 2001. Here I shall list the main contents discussed.(1) We found that I had understood the essence of his algebraicdeformation(2) A role of the theory of finite-dimensional ncPa, which is found by myself, in math and math science(3) Quantum groups, algebraic deformation theory and their relationshipThe Spread of Deformation TheoryOne of the aims of the project is the spread of Deformation theory over Japan. It was done, I believe, by giving a talk at a meeting of Japan mathematical society and organizing a meeting to have a lecture of Prof. Gerstenhaber at Kyushu Institute of Technology.Afterward(1) Construction of a deformation theory of triple systems. This is motivated by a discussion with Prof. Weinstein (Univ of CA).(2) Study the works of Jimbo. I keep discussing about this subject with Prof. Gerstenhaber. A subject is of course About a quantum group and an algebraic deformation theory.(3) Construction of ncPa from a Yang-Baxter equation