Study of Algebraic groups and Lie Algebras and Applications

代数群和李代数的研究及其应用

• 批准号: 11640008
• 负责人: MORITA Jun
• 金额: $ 2.05万
• 依托单位: University of Tsukuba
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640008/
• 关键词: Kac-Moody groups; Kac-Moody algebra; Gauss decomposition; カッツ・ムーディ群; ガウス分解; 単純群; 準結晶

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.

一般Kac-Moody群的强高斯分解的存在性已被证明。在有限维半简单代数群的情况下，V. Chernousov 等人之前给出了这样的结果。在无限维情况下，还建立了一些新的性质，如强高斯分解。更明确地，我们令 G = Kac-Moody 群，Z(G) = G 的中心，T = 标准最大环面，U = 标准最大上三角单能子群，V = 标准最大下三角单能子群。然后对于每个 h[0x81b8(Shift-JIS)]T，以下内容已被证明是正确的:G=Z(G)[0x81be(Shift-JIS)][0x81be(Shift-JIS)]__<g[0x81b8(Shift-JIS)]G>g(VhU)g^<-1>。此外，使用由此证明，每个非中心元素都可以表示为两个单能元素的乘积，这对于研究 Kac-Moody 群的群结构来说是一个非常有力的结果。作为相关主题，我们讨论了正锥和半群，并且针对某些 K-seniigroups 给出了 Matsumoto 类型的演示。此外，还研究了一些准周期结构作为代数群论和代数数论的应用。

项目成果

Robert Moody, Jun Morita: "Positivity for K_1 and K_2"Journal of Algebra. 229. 1-24 (2000)

罗伯特·穆迪 (Robert Moody)、森田淳 (Jun Morita)：“K_1 和 K_2 的正性”代数杂志。

Tatsuya Kimijima, Jun Morita: "A certain algebraic construction of quasicrystals and their isomorphism classes"Journal of Physics A : Math.Gen. 33. 8483-8487 (2000)

Tatsuya Kimijima、Jun Morita：“准晶体及其同构类的某种代数构造”物理学杂志 A ：Math.Gen。

Jun Morita, Kuniko Sakamoto: "Shell structure of dodecagonal quasicrystals associated with root system F_4 and cyclotomic field Q(ζ_ )"Communications in Algebra. 28. 256-263 (2000)

Jun Morita、Kuniko Sakamoto：“与根系 F_4 和分圆场 Q(ζ_<12> ) 相关的十二角准晶体的壳结构”通讯代数 28. 256-263 (2000)

Robert Moody, Jun Morita: "Positivity for K_1 and K_2"Journal of Algebra. 229. 1-24 (2000)

罗伯特·穆迪 (Robert Moody)、森田淳 (Jun Morita)：“K_1 和 K_2 的正性”代数杂志。

Jun Morita, Eugene Plotkin: "Prescribed Gauss decompositions for Kac-Moody groups over fields"Rendiconfi del Seminario Matematico de lla Universifa di Padova. 106. 153-163 (2001)

Jun Morita，Eugene Plotkin：“域上 Kac-Moody 群的规定高斯分解”Rendiconfi del Seminario Matematico de lla Universifa di Padova。