The Scheme of Finite Dimensional Representations of an Algebra

For a finitely generated /̂ -algebra A and a finite dimensional ^-vector space M the representations of A on M form an affine ^-scheme Mod^(M). Of particular interest for this scheme are the connected components, the irreducible components, and the open and closed orbits under the natural action of the general linear group AutA(M), since the orbits are the equivalence classes of representations. The connected components are known for a finite dimensional algebra A. In this paper we characterize the connected components when A is commutative or an enveloping algebra of a Lie algebra in characteristic zero. For the algebra k[x> y]/(x, y) we describe the open orbits and the irreducible components. Finally, we examine the connection with the theory of deformations of algebra representations.