For a finitely generated
k-algebra A and a finite dimensional k-vector space M the representations of A on M
form an affine k-scheme ModA(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 Autk(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)2 we describe the open orbits and the irreducible components.
Finally, we examine the connection with the theory of deformations of algebra
representations.
