Structure Theory and Deformation Theory of Algebras with two multiplications

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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