Vol. 91, No. 1, 1980

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The scheme of finite-dimensional representations of an algebra

Kent Morrison

Vol. 91 (1980), No. 1, 199–218

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.

Mathematical Subject Classification 2000
Primary: 16A64, 16A64
Secondary: 14L30
Received: 1 November 1978
Published: 1 November 1980
Kent Morrison