The scheme of finite-dimensional representations of an algebra

Abstract

For a finitely generated /^-algebra A and a finite dimension- al ^-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 equiva- lence classes of representatio ns. The connected components are known for a finite dimensional algebra A. In this paper we characterize the connected components when A is com- mutative or an enveloping algebra of a Lie algebra in chara- cteristic zero. For the algebra k(x> y)/(x, y)2 we describe the open orbits and the irreducible components. Finally, we ex- amine the connection with the theory of deformations of al- gebra representations. Introduction* For almost any ^-algebra A it is an impossible task to classify all finite dimensional A-modules. There are some standard exceptions such as the finite dimensional semisimple algebras, A = k(x)/(xm), A = k(x, y)/(x, yf, and A = k(x). Even for the polynomial algebra in two indeterminates, k(x, y), Gelfand and Ponomarev have shown that the classification problem is as difficult as classifying the modules over the free algebra k{x19 •••, xm}. (See (7) for details.) If we restrict our attention to modules of a given dimension, say n, then to classify the ^-dimensional A-modules is to classify the orbits of Aut}χkn) = GL(n) acting on the space of ^-dimensional A-module structures. If we take A to be a finitely generated k- algebra then this space of module structures is an affine fc-scheme, which we call Mod^(M) where M = kn. Although we cannot hope to determine the orbits in most cases, we may be able to describe coarser features of ModA(M) such as the irreducible components and the connected components. It is surprising that even for the con- nected components there is not a complete answer for an arbitrary finitely generated ^-algebra, while the irreducible components are not at all understood. Only for one interesting algebra, namely A = k(x, y)/(x, y)2, have the irreducible components been determined, and for this it is essential to know the classification of the finite dimensional A-modules. (In § 5 we recall the description of the indecomposable modules from (9), determine all open orbits, and from them describe the irreducible components as found by Flanigan