Tannaka Duality for Discrete Groups

Abstract

Introduction. Let A be a commutative ring, G a group, and let RePA(G) be the category of representations pv:G AutA(V) of G on finitely generated modules over A. We shall deal with the question to what extent G can be recovered from RepA(G). Note that Pontryagin's classical duality theorem asserts, essentially, that every compact abelian group can be reconstructed from its irreducible unitary representations with the operation of tensor product between them (namely, the group of the characters). Following Pontryagin, several duality theories (i.e., theorems about the possibility and a way to reconstruct a group from its representations) were established. The first one was the Duality of for compact Lie groups [24] (see also [71). The work of Tannaka was generalized in several directions (see for example [1, 6, 10, 171). The principal work for Lie groups was done by Hochschild and Mostow [11]. They used the algebra of representative functions RF(G) (the F-linear span of the coefficients of all the finite dimensional representations of G over a field F) and the group MF(G) of proper automorphisms of RF(G) (i.e., automorphisms which fix the constant functions and commute with the G-right translations of RF(G)). From their results in [11] one may deduce the following Theorem which is cited here in order to illustrate what we mean by Tannaka duality: