In this paper we determine the derived representation type of quadratic string algebras and prove that every tame algebra whose quiver has cycles is equivalent to some skewed-gentle algebra.

We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the ‘split-quotient quiver’ introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the qui...

We construct reflection functors on categories of modules over deformed wreath products of the preprojective algebra of a quiver. These functors give equivalences of categories associated to generic parameters which are in the same orbit under the Weyl group action. We give applications to the representation theory of symplectic reflection algebras of wreath product groups.

We review the definition of a Calabi-Yau triangulated category and survey examples coming from the representation theory of quivers and finite-dimensional algebras. Our main motivation comes from the links between quiver representations and Fomin-Zelevinsky’s cluster algebras. Mathematics Subject Classification (2000). Primary 18E30; Secondary 16D90, 18G10.

We establish a connection between knot theory and cluster algebras via representation theory. To every diagram (or link diagram), we associate algebra by constructing quiver with potential. The rank of the is 2n, where n number crossing points in diagram. then construct 2n indecomposable modules T(i) over Jacobian For each T(i), show that submodule lattice isomorphic to corresponding Kauffman s...

This note provides a specific quiver which does not admit a maximal green sequence, and which is mutation-equivalent to a quiver which does admit a maximal green sequence. This provides a counterexample to the conjecture that the existence of maximal green sequences is invariant under quiver mutation. The proof uses the ‘scattering diagrams’ of Gross-Hacking-Keel-Kontsevich to show that a maxim...

The paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs. To include gradings of all basic finite-dimensional algebras over an algebraically closed field, a theory of coverings of graphs with relations is introduced. The object of this paper is to relate group gradings on algebras to coverings of a gr...

Given an ultragraph in the sense of Tomforde, we construct a topological quiver in the sense of Muhly and Tomforde in such a way that the universal C∗-algebras associated to the two objects coincide. We apply results of Muhly and Tomforde for topological quiver algebras and of Katsura for topological graph C∗-algebras to study the K-theory and gauge-invariant ideal structure of ultragraph C∗-al...

We completely describe the tree classes of the components of the stable Auslander-Reiten quiver of a quantum complete intersection. In particular, we show that the tree class is always A∞ whenever the algebra is of wild representation type. Moreover, in the tame case, there is one component of tree class Ã12, whereas all the others are of tree class A∞.

We develop the theory of special biserial and string coalgebras and other concepts from the representation theory of quivers. These theoretical tools are then used to describe the finite dimensional comodules and Auslander-Reiten quiver for the coordinate Hopf algebra of quantum SL(2) at a root of unity. We also compute quantum dimensions and describe the stable Green ring. Let C = k ζ [SL(2)] ...