Study of Algebraic groups and Lie Algebras and Applications

点击翻译按钮获取中文摘要

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 letG = 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.