Structure Theory and Deformation Theory of Algebras with two multiplications

新结果(1)结构定理具有Jacobson根平方的非交换Poisson代数(“NCPA”)是类Lie-Poisson代数和标准NCPA的半直积。我在NCPA中找到了一个很好的元素。我已经成功地用这个元和标准Poisson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上的所有有限维单Poisson模。(3)我把Gertenhaber的代数形变理论应用到NCPA上。然后我发现如果NCPA是无穷小刚性的，那么它就是刚性的。我是如何进行这个项目的？我是在与PA大学Gertenhaber教授讨论的基础上完成这个项目的。有了这笔钱，我可以在2001年3月邀请他几天，也可以在2001年8月访问美国。在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形变的本质(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、代数形变理论及其关系形变理论的传播该项目的目标之一是在日本推广形变理论。我相信，这是通过在日本数学学会的一次会议上发表演讲，并在九州理工学院组织一次格斯滕哈伯教授的讲座而完成的。之后(1)三重系形变理论的建立。这是由与温斯坦教授(加州大学)的一次讨论引起的。(2)研究金博的作品。我一直在和格斯滕哈伯教授讨论这个问题。当然，这门课是关于量子群和代数形变理论的。(3)由杨-巴克斯特方程构造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