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We study automorphisms and representations of quasi polynomial algebras (QPAs) and quasi Laurent polynomial algebras (QLPAs). For any QLPA defined by an arbitrary skew symmetric integral matrix, we explicitly describe its automorphism…

In \cite{FT19}, Finkelberg and Tsymbaliuk introduced the notion of shifted quantum affine algebras and described their role in the study of quantized Coulomb branches associated to certain 3D $N = 4$ quiver gauge theories. We describe a new…

We begin the study of Khovanov-Lauda-Rouquier type algebras associated to moduli stacks of coherent sheaves on smooth projective curves. We consider the case of $\mathbb{P}^1$ and define, for any pair $(r,d)$ of a rank and a degree, the KLR…

We provide a topological characterization of quivers whose path algebra satisfies a polynomial identity. This class includes the oriented cycle and acyclic quivers and, in the latter case, we describe the associated T-ideal. We introduce a…

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which…

Given a triangulation of a polygon P with n vertices, we associate an ice quiver with potential such that the associated Jacobian algebra has the structure of a Gorenstein tiled K[x]-order L. Then we show that the stable category of the…

We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove…

We consider the dual space of linear groups over Dynkinian and Euclidean algebras, i.e. finite dimensional algebras derived equivalent to the path algebra of Dynkin or Euclidean quiver. We prove that this space contains an open dense subset…

We develop a method to compute certain mutations of quivers with potentials and use this to construct an explicit family of non-degenerate potentials on the exceptional quiver $X_7$. We confirm a conjecture of Geiss-Labardini-Schroer by…

To every minimal model of a complete local isolated cDV singularity Donovan--Wemyss associate a finite dimensional symmetric algebra known as the contraction algebra. We construct the first known standard derived equivalences between these…

In this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles. We do this in two settings, first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal…

We describe certain quiver Hopf algebras by parameters. This leads to the classification of multiple Taft algebras as well as pointed Yetter-Drinfeld modules and their corresponding Nichols algebras. In particular, when the ground-field $k$…

Let $\mathbb{X}$ be a weighted projective line of tubular type and $\operatorname{coh}\mathbb{X}$ the category of coherent sheaves on $\mathbb{X}$. The main purpose of this note is to show that the subgraph of the tilting graph consisting…

This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin…

This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer…

We define an integral form of shifted quantum affine algebras of type $A$ and construct Poincar\'e-Birkhoff-Witt-Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these…

Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. These results are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object $T$ in a…

It is shown that certain transformations on quiver-dimension vector pairs induce isomorphisms on the corresponding moduli spaces of quiver representations and map a stable dimension vector to a stable dimension vector. This result combined…

We define algebras of quasi-quaternion type, which are symmetric algebras of tame representation type whose stable module category has certain structure similar to that of the algebras of quaternion type introduced by Erdmann. We observe…

In this survey paper we give an overview of a generalization, introduced by R. Bautista and the author, of the theory of mutation of quivers with potential developed in 2007 by Derksen-Weyman-Zelevinsky. This new construction allows us to…