A research of algebraic groups and Lie algebras

研究了Kac-Moody群及其伴随K-群的群结构。特别是，我们对Z上的Kac-Moody群感兴趣。我们选择了几种类型的Kac-Moody群，并研究了它们之间的关系。特别地，我们确定了它们的交换商的群结构。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。对于大于或等于&gt；5的素数p&lt；，我们发现K_2(2，Z[1/p])不是中心的，并且K_2(2，Z[1/p])*Z_*xZ_&lt；p-1&gt；为了得到这些结果，我们使用了几个辫子关系和亚简化法。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，则我们选择丹尼斯-斯坦符号d(-2^&lt；L+1&gt；，3m)。如果p=6k-1，k=2^lm，(2，m)，则我们选择丹尼斯-斯坦符号d(2^&lt；L+1&gt；，3m)。在ST(2，Z[1/p])的亚阿贝尔商中，这些Dennis-Stein符号工作得很好，从而给出了K_2(2，Z[1/p])的非中心性等。并研究了交错群与双曲Coxeter群之间的关系。然后，我们构造了亏格为1+{(p-1)！(p-6)/24}对于每个大于或等于&gt；11的素数p&lt；，我们研究了几个相关的问题。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，环的表示理论的研究等，我们在几次国际会议上公布了这些结果，并在几个国际学术期刊上发表了这些结果。此外，我们还准备在某些国际学术期刊上发表一些新的成果。

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.