A research of algebraic groups and Lie algebras

我们研究了Kac-Moody基团和相关k -基团的基团结构。我们对z上的Kac-Moody群特别感兴趣，选取了几种类型的Kac-Moody群，研究了它们之间的关系。特别地，我们确定了它们的阿贝尔商的群结构。同时研究了k -基团K_2 （2, Z [1/p]）的基团结构。我们的结果之一如下。对于p< >或等于>的素数，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，我们使用了几个辫状关系和元离化。主要的工具是所谓的丹尼斯-斯坦符号。如果p=6k+1, k=2^lm, (2, m) =1，那么我们选择命名为d （-2^<l+1>,3m）的Dennis-Stein符号。如果p=6k-1, k=2^lm, (2, m)，那么我们选择命名为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符号很好地证明了K_2 （2, Z [1/p]）等的非中心性。并研究了交替群与双曲Coxeter群之间的关系。然后构造了1+ {（p-1）的紧致黎曼曲面！（p-6） /24}对于每一个素数p<大于等于bbb11。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。此外，我们正准备在某些国际学术期刊上发表一些新的研究成果。

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.