Structure Theory and Deformation Theory of Algebras with two multiplications
Project/Area Number 
12640028

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  KYUSHU INSTITUTE OF TECHNOLOGY 
Principal Investigator 
KUBO Fujio Kyushu Institute of Technology, Professor, Faculty of Engineering, Dept of Mathematical Science, 工学部, 教授 (80112168)

Project Period (FY) 
2000 – 2001

Project Status 
Completed (Fiscal Year 2001)

Budget Amount *help 
¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2001: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2000: ¥700,000 (Direct Cost: ¥700,000)

Keywords  noncommutative Poisson algebra / algebraic deformation tgheory / Gerstenhaber / inverse scattering method / quantum group / cohomology / Lie triple system / YanBaxter equation / ポアソン代数 / ポアソン加群 / 代数的変形 / 変形理論 / ゲルンステンハーバー 
Research Abstract 
New results (1) Structure theorem A noncommuative Poisson algebra ("ncPa") with Jacobson radicalsquered zero is a semidirect product of a Lielike 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 finitedimensional simple Poisson modules over a Finitedimensional 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 project I 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 finitedimensional ncPa, which is found by myself, in math and math science (3) Quantum groups, algebraic deformation theory and their relationship The Spread of Deformation Theory One 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 YangBaxter equation

Report
(3 results)
Research Products
(6 results)