CHARACTERIZING SEMISIMPLE LIE GROUPS BY CERTAIN FINITE SUBGROUPS

[摘要] This thesis is concerned with a certain class of finite subgroups of Lie groups. Let G be a compact connected simply connected simple Lie group, T be a maximal torus in G and N be the normalizer of T in G. Then we have a topological group extension i p (1) O (--->) T (--->) N (--->) W (--->) 1, with W = N/T being the Weyl group. The Weyl group is a finite group which, while it gives information about G does not characterize G. (We can have isomorphic Weyl groups for nonisomorphic groups.) We are concerned with a finite group extension sitting inside (1); namely i p (2) O (--->) F (--->) (;;)J(G) (--->) W (--->) 1. Here F is the group of all fourth roots of unity in T and (;;)J(G) is a finite subgroup of N (which is an extension of a subgroup J(G) of N defined by J. Tits). It is known that if the (;;)J(G) groups are isomorphic for two Lie groups then the Lie groups are isomorphic. We seek a group-theoretic characterization of those finite groups which arise as (;;)J(G) for some Lie group G (compact, connected, simply-connected and simple). A really good characterization would allow one to prove the Cartan-Killing classification as a theorem in finite group theory. We know presentations for all (;;)J(G) and from these we calculate their abelianizations. Given a group extension (2), the set of all other extensions with the same F and W and the same action (phi) of W on F form a cohomology group H(,(phi))(;;2)(W,F). We study how the presentation of (;;)J(G) may be modified to give other extensions in this cohomology class, and obtain some fairly definitive results along these lines. Some preliminary results are computed for (;;)J(G) where G is an exceptional Lie group.

[发布机构] Rice University