Irreducible components of module varieties: an example
Let \(Q\) be a quiver consisting of a single vertex along with two loops \(\alpha\), \(\beta.\) For \(k\) an algebraically closed field, let \(kQ\) be the quiver algebra for \(Q\). For a given dimension \(d\) one has the quiver variety rep\((Q,d) =k^{d^{2}}\times k^{d^{2}}\) (where each term on the right corresponds to a matrix representation of a linear operator on \(\alpha\) and \(\beta\) respectively) upon which \(\text{GL}(d)\) acts: the orbits of this action correspond to isomorphism classes of representations. For nonzero \(q\in k\) let \(I_{q}\) be the ideal generated by \(\{\alpha^{2},\beta^{2},\beta\alpha+q\alpha\beta\}\): one then can define the closed subset rep\((Q,I_{q},d)\) of quiver representations which are trivial on \(I_{q}.\) The work being reviewed is a study of the irreducible components of rep\((Q,I_{q},d).\) Initially, these components are described in terms of orbit closures of modules related to the Kronecker quiver (two vertices, \(1,2\) and two arrows \(\alpha,\beta:1\rightarrow2\)), but this formulation is shown to be equivalent to a more direct description involving representations satisfying numerical criteria depending on the parity of \(d\). Along with this description, results are obtained concerning intersections of irreducible representations; in particular it is shown that a nontrivial intersection of irreducible components of rep\((Q,I_{q},d)\) is irreducible. The class of examples presented here are noteworthy for two reasons. First, it provides an description of irreducible components in terms of equations for modules over an algebra which is neither representation finite nor hereditary. Second, applying the results to the case \(q=-1\) and \(d=4\) provides for a description of a famous example by Carlson -- see [\textit{C. Riedtmann}, Ann.\ Sci. Èc. Norm. Supér. (4) 19, No. 2, 275--301 (1986; Zbl 0603.16025)].
scientific article; zbMATH DE number 5949285