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$\tau$-tilting theory can be thought of as a generalization of the classical tilting theory which allows mutations at any indecomposable summand of a support $\tau$-tilting pair. Indeed, for any algebra $\Lambda$ its tilting modules…

In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras.…

We associate a quiver to a quasi-triangulation of a non-orientable marked surface and define a notion of quiver mutation that is compatible with quasi-cluster algebra mutation defined by Dupont and Palesi. Moreover, we use our quiver to…

In this paper we study multiserial and special multiserial algebras. These algebras are a natural generalization of biserial and special biserial algebras to algebras of wild representation type. We define a module to be multiserial if its…

For a finite dimensional hereditary algebra, we consider: exceptional sequences in the category of finite dimensional modules, silting objects in the bounded derived category, and m-cluster tilting objects in the m-cluster category. There…

This is a survey article for "Handbook of Linear Algebra", 2nd ed., Chapman & Hall/CRC, 2014. An informal introduction to representations of quivers and finite dimensional algebras from a linear algebraist's point of view is given. The…

Cluster-tilted algebras are trivial extensions of tilted algebras. This correspondence induces a surjective map from tilted algebras to cluster-tilted algebras. If B is a cluster-tilted algebra, we use the fibre of B under this map to study…

In this paper, we present an explicit and purely combinatorial characterization of the $m$-coloured quivers that appear within the $m$-coloured mutation class of a quiver of type $\mathbb{D}_n$. The $m$-coloured mutation, as defined by Buan…

We provide a complete classification of all tame and wild symmetric special multiserial algebras in terms of the underlying Brauer configuration. Our classification contains the symmetric special multiserial algebras of finite…

We initiate the study of applications of machine learning to Seiberg duality, focusing on the case of quiver gauge theories, a problem also of interest in mathematics in the context of cluster algebras. Within the general theme of Seiberg…

Kauer moves are local moves of an edge in a Brauer graph that yield derived equivalences between Brauer graph algebras [Kau98]. These derived equivalences may be interpreted in terms of silting mutations. In this paper, we generalize the…

We define an operation which associates to a pair (B,M) where B is a cluster-tilted algebra and M is a B-module which lies in a local slice of B, a new cluster-tilted algebra B'. In terms of the quivers, this operation corresponds to adding…

From the viewpoint of mutation, we will give a brief survey of tilting theory and cluster-tilting theory together with a motivation from cluster algebras. Then we will give an introdution to \tau-tilting theory which was recently developed…

Let $A$ be a finite dimensional algebra over an algebraically closed field $k$, $\mathcal {D}^b(A)$ be the bounded derived category of $A$-mod and $A^{(m)}$ be the $m$-replicated algebra of $A$. In this paper, we investigate the structure…

Comparing the module categories of an algebra and of the endomorphism algebra of a given support $\tau$-tilting module, we give a generalization of the Brenner-Butler's tilting theorem in the framework of $\tau$-tilting theory. Afterwards…

We introduce a new class of symmetric algebras, which we call hybrid algebras. This class contains on one extreme Brauer graph algebras, and on the other extreme general weighted surface algebras. We show that hybrid algebras are precisely…

Let $\Lambda$ be a hereditary algebra, $B_0=End_\Lambda(T_0)$ be a tilted algebra. We will construct tilting $B_0$-modules from tilting $\Lambda$-modules and use this result to show how tilting quivers of BB-tilted algebras can be obtained…

The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type $A_n$, by counting the mutation class of any quiver with underlying graph $A_n$. It will also follow that if $T$ and…

Let $Q$ be a finite quiver without oriented cycles and $k$ an algebraically closed field.In this paper we establish a connection between cluster algebras and the representation theory of the path algebra $kQ$, in terms of the spectral…

Quivers (directed graphs) and species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their…