A research of algebraic groups and Lie algebras

代数群和李代数的研究

• 批准号: 08454002
• 负责人: MORITA Jun
• 金额: $ 4.22万
• 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 (B)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 无数据
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454002/
• 关键词: Kac-Moody group; algebraic group; alternating group; Coxeter group; Lie algebra; Lie algebra; algebraic K-theory; alternating group; algebraic group; Lie algebtra

We studied the group structure of Kac-Moody groups and associated K-groups. Especially we are interested in the Kac-Moody groups over Z.We selected several types of Kac-Moody groups and studied their relations. In particular, we determined the group structure of their abelian quotients. Also, we studied the group structure of K-group K_2 (2, Z [1/p]). One of our results is as follows. For a prime number p<greater than or equal>5, we found that K_2 (2, Z [1/p]) is not central, and furthermore K_2 (2, Z [1/p]) *Z_* x Z_<p-1>. To obtain these results, we use several braid relations and meta-abelianizations. The main tool is the so-called Dennis-Stein symbol. If p=6k+1, k=2^lm, (2, m) =1, then we choose the Dennis-Stein symbol named d (-2^<l+1>,3m). If p=6k-1, k=2^lm, (2, m), then we choose the Dennis-Stein symbol named d (2^<l+1>,3m). In the meta-abelian quotient of St (2, Z [1/p]), these Dennis-Stein symbols work well, which gave the non-centrality of K_2 (2, Z [1/p]) and so on. And, we studied the relation between alternating groups and hyperbolic Coxeter groups. Then, we constructed a compact Riemann Surface of genus 1+ {(p-1) ! (p-6) /24} for every prime number p<greater than or equal>11. As well as these results, we studied several related topics. For example, a research of quantum groups, a research of algebraic geometry, a research of finite groups, a research of Lie groups, a research of number theory, a research of representation theory of rings, etc. And we announced these results in several international conferences, and published these results in several international academic journals. Also we are prepairing to publish some new results in certain international academic journals.

研究了Kac-Moody群及其相关K-群的群结构。特别是Z上的Kac-Moody群，我们选取了几类Kac-Moody群并研究了它们之间的关系。特别地，我们确定了它们的交换子的群结构。研究了K-群K_2（2，Z [1/p]）的群结构。我们的结果之一如下。对于素数p5<greater than or equal>，我们发现K_2（2，Z [1/p]）不是中心的，而且K_2（2，Z [1/p]）*Z_* x Z_<p-1>.为了得到这些结果，我们使用了几个辫子关系和元阿贝尔化。主要工具是所谓的丹尼斯-斯坦符号。如果p= 6 k +1，k= 2 ^lm，（2，m）=1，那么我们选择命名为d（-2 ^&lt;l+1&gt;，3 m）的Dennis-Stein符号。如果p= 6 k-1，k= 2 ^lm，（2，m），那么我们选择命名为d（2^&lt;l+1&gt;，3 m）的Dennis-Stein符号。在St（2，Z [1/p]）的亚交换商中，这些Dennis-Stein符号起了很好的作用，给出了K_2（2，Z [1/p]）的非中心性等，并研究了交错群与双曲Coxeter群的关系。然后，我们构造了亏格为1+ {（p-1）！(p-6)/24}，对于每一个素数p <greater than or equal>11。除了这些结果，我们还研究了几个相关的主题。例如，量子群的研究，代数几何的研究，有限群的研究，李群的研究，数论的研究，环的表示论的研究等，我们在几个国际会议上宣布了这些结果，并在几个国际学术期刊上发表了这些结果。我们也准备在国际学术期刊上发表一些新的成果。

项目成果

Jun Morita: "Meta-abelianizations of SL (2, Z[1/]) and Dennis-Stein symbols" Tsukuba J.Math. 20-1. 71-76 (1996)

Jun Morita：“SL (2, Z[1/]) 和 Dennis-Stein 符号的元阿贝尔化” Tsukuba J.Math。

Mitsuhiro Takeuchi: "Cocycle deformations of coordinate rings of quantum matrices" J.Algebra. (to appear).

Mitsuhiro Takeuchi：“量子矩阵坐标环的余循环变形”J.代数。

Mitsuo Hoshino: "Selfinjectivity of rings relative to Lambek torsion theory" Tsukuba J.Math.(to appear).

Mitsuo Hoshino：“环的自射性相对于兰贝克挠场理论”Tsukuba J.Math.（即将出现）。

Jun Morita: "Meta-abelianizations of SL (2,Z[1/p]) and Dennis-Stein symbols" Tsukuba J. Math.20(1). 71-76 (1996)

Jun Morita：“SL (2,Z[1/p]) 和 Dennis-Stein 符号的元阿贝尔化”Tsukuba J. Math.20(1)。

A.Skowronski,Kunio Yamagata: "Socle deformations of self-injective algebras" Proc.London Math.Soc.72(3). 545-566 (1996)

A.Skowronski，Kunio Yamagata：“自注入代数的 Socle 变形”Proc.London Math.Soc.72(3)。