A research of algebraic groups and Lie algebras

代数群和李代数的研究

基本信息

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

项目摘要

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.

我们研究了 Kac-Moody 群和相关 K 群的群结构。我们特别对Z上的Kac-Moody群感兴趣。我们选择了几种类型的Kac-Moody群并研究了它们的关系。特别是，我们确定了它们的阿贝尔商的群结构。此外，我们还研究了 K-群 K_2 (2, Z [1/p]) 的群结构。我们的结果之一如下。对于质数p<大于等于>5，我们发现K_2(2,Z[1/p])不是中心的，而且K_2(2,Z[1/p])*Z_*xZ_< p-1>。为了获得这些结果，我们使用了几种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=6k+1, k=2^lm, (2, m) =1，则我们选择名为 d (-2^<l+1>,3m) 的 Dennis-Stein 符号。如果 p=6k-1, k=2^lm, (2, m)，那么我们选择名为 d (2^<l+1>,3m) 的 Dennis-Stein 符号。在 St (2, Z [1/p]) 的元阿贝尔商中，这些 Dennis-Stein 符号效果很好，这给出了 K_2 (2, Z [1/p]) 的非中心性等等。并且，我们研究了交替群和双曲考克斯特群之间的关系。然后，我们构造了一个属 1+ {(p-1) ! (p-6) /24} 对于每个素数 p<大于或等于>11。除了这些结果之外，我们还研究了几个相关主题。比如量子群的研究，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等。我们在几个方面公布了这些成果。国际会议，并在多个国际学术期刊上发表这些成果。我们还准备在某些国际学术期刊上发表一些新成果。