Invariant Algebras and Completely Reducible Representations

We give a general construction of affine noetherian algebras with the property that every finite dimensional representation is completely reducible. Starting from enveloping algebras of semi simple Lie algebras in characteristic zero we obtain explicit examples and describe some of their properties. In the following an algebra A will always mean an associative algebra over a field k with a unit element. Mostly, we will assume that A is affine and (left and right) noetherian. An A-module M is a left A-module if not otherwise stated, and we will not distinguish between the module M and the corresponding linear representation of A on the vector space M . 1. The condition FCR It is well known that the enveloping algebra U(g) of a semisimple Lie algebra g in characteristic zero has the following two important properties: • Every finite dimensional representation of U(g) is completely reducible. • U(g) has enough finite dimensional representations, i.e., the intersection of the kernels of all finite dimensional representations is zero. The first is due to H. Weyl, the second to Harish-Chandra (see [Dix] 2.5.7, p. 84). Algebras satisfying the second condition will be called residually finite-dimensional. In the following we consider algebras A which satisfy these two properties. For this purpose we introduce the following condition FCR (= “Finite dimensional representations are Completely Reducible”): (FCR) A is residually finite-dimensional and every finite dimensional representation of A is completely reducible. Received February 10, 1994.