Framed moduli and Grassmannians of submodules (Q2847033)

To a finite dimensional algebra we can associate certain quiver, which is a set of vertices with arrows between them. \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)] constructed a moduli space of representations of quivers on vector spaces which gives a solution to the problem of classification of representations of finite dimensional algebras. In the construction of Nakajima varieties, there is an extension of the parameter space of representations by considering framed representations, i.e. representations together with a homomorphism from the vector space of each vertex (in a representation of the quiver) to a fixed vector space. \textit{M. Reineke} [J. Algebra 320, No. 1, 94--115 (2008; Zbl 1153.14033)] realized a frame moduli space of representations of acyclic quivers on vector spaces, as a Grassmannian of subrepresentations of an injective representation depending only on the dimensions of the vector spaces of the framed representation.NEWLINENEWLINE The paper under review generalizes Reineke's construction to quivers with relations, for finite dimensional algebras, and with possibly oriented cycles, proving that the quotient space (using some version of Geometric Invariant Theory) of the space of representations of a finite dimensional algebra \(A\) with fixed dimension vectors, is isomorphic to the Grassmannian of submodules of an injective \(A\)-module.NEWLINENEWLINEThe generalization to quivers with oriented cycles studies the fibers of the moduli space over the categorical quotient, by studying the quiver \(A_{n-1}^{(1)}\) with cyclic orientation. Explicit presentations of the fibers, by equations in projective space, are given, when the ground field is \(\mathbb{C}\) or \(\mathbb{R}\). Finally, the same technique is applied for a quiver where all oriented cycles pairwise commute.