2001 Fiscal Year Final Research Report Summary
Study of Algebraic groups and Lie Algebras and Applications
| Project/Area Number |
11640008
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | University of Tsukuba |
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Principal Investigator |
MORITA Jun University of Tsukuba, Institute of Mathematics, Professr, 数学系, 教授 (20166416)
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| Co-Investigator(Kenkyū-buntansha) |
MIYASHITA Yoichi Kagoshima University, Department of education, Professor, 教育学部, 教授 (00000795)
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| Project Period (FY) |
1999 – 2001
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| Keywords | Kac-Moody groups / Kac-Moody algebra / Gauss decomposition |
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| Research Abstract |
The existence of strong Gauss decompositions for general Kac-Moody groups has been proved. In the case of finite dimensional semisimple algebraic groups such a result was given before by V. Chernousov etc. In the infinite dimensional case, several ,new properties as well,as strong Gauss decompositions have been established. More explicitely, we let
G = a Kac-Moody group,
Z(G) = the center of G,
T = the standard maximal torus,
U = the standard maximal upper triangular unipotent subgroup,
V = the standard maximal lower triangular unipotent subgroup.
Then the following has been shown to be-true for every h[0x81b8(Shift-JIS)]T :
G=Z(G)[0x81be(Shift-JIS)][0x81be(Shift-JIS)]__<g[0x81b8(Shift-JIS)]G>g(VhU)g^<-1>.
Furthermore, using this, it has been proved that every noncentral element is able to be expressed as a product of two unipotent elements, which is a very strong result to study the group structure of a Kac-Moody group. As related topics, positive cones and semigroups have been discussed, and Matsumoto type presentations have been given for certain K-seniigroups. Also, some quasi-periodic structures have been studied as applications of algebraic group theory and algebraic number theory.
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