In this article we prove explicit formulae for the number of non-isomorphic cluster-tilted algebras of type Ãn in the derived equivalence classes. In particular, we obtain the number of elements in the mutation classes of quivers of type Ãn. As a by-product, this provides an alternative proof for the number of quivers mutation equivalent to a quiver of Dynkin type Dn which was first determined ...

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...

Quiver Grassmannians and quiver flags are natural generalisations of usual Grassmannians and flags. They arise in the study of quiver representations and Hall algebras. In general, they are projective varieties which are neither smooth nor irreducible. We use a scheme theoretic approach to calculate their tangent space and to obtain a dimension estimate similar to the one of Reineke in [Rei02]....

Brane tilings are introduced by string theorists to study four-dimensional N = 1 superconformal field theories. See e.g. a review by Kennaway [5] and references therein for a physical background. A brane tiling is a bipartite graph on a real two-torus which encodes the information of a quiver with relations. A typical example of such a quiver is the McKay quiver determined by a finite abelian s...

We introduce a framework of translation quiver varieties which includes Nakajima as well their graded and cyclic versions. An important feature is that the sets fixed points under toric actions can be again realized varieties. This allows one to simplify in several steps. prove are smooth, pure have Tate motivic classes. also describe an algorithm compute those

One of the main tools for the study of the category of finite dimensional modules over a basic algebra, over an algebraically closed field k is its presentation as quiver and relations. This theory is mainly due to P. Gabriel (see for example [GRo]). More precisely, it has been proved that for all finite dimensional and basic algebras over an algebraically closed field k, there exists a unique ...