| Co-Investigator(Kenkyū-buntansha) |
KOGISO Takeyoshi Univ.of Tsukuba, Inst.of Mathematics, Research Associate, 数学系, 助手 (20282296)
YAMAGATA Kunio Univ.of Tsukuba, Inst.of Mathematics, Associate Professor, 数学系, 助教授 (60015849)
MIYAMOTO Masahiko Univ.of Tsukuba, Inst.of Mathematics, Professor, 数学系, 教授 (30125356)
KIMURA Tatsuo Univ.of Tsukuba, Inst.of Mathematics, Professor, 数学系, 教授 (30022726)
TAKEUCHI Mitsuhiro Univ.of Tsukuba, Inst.of Mathematics, Professor, 数学系, 教授 (00015950)
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| Research Abstract |
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.
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