On G-Modules as Lie Algebra Representations

Abstract

We first observe that for a finite group G, the integral group ring Z[G] has a natural bracket. Then, we choose a particularly nice Lie subalgebra, Z[G]−. Extending to a characteristic zero field F, we obtain a semisimple Lie algebra F[G]−. We show that irreducible representations of G extend to irreducible Lie algebra representations of F[G]−. For G-modules M, we restrict the action of Z[G] on M to Z[G]− to obtain a Z[G]− representation. We compare the group cohomology of M as a G˜-module, where G˜ is the commutator subgroup of G, to the Lie algebra cohomology of M as a Z[G]−-representation. We compare these cohomology groups, noting some striking similarities. Finally, we construct what we call Lie-Tate cohomology groups for arbitrary G-modules, and we prove that they are similar to the ordinary Tate cohomology groups in that they send short exact sequences to doubly-infinite exact sequences.