Related papers: Quivers with potentials for Grassmannian cluster a…
Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the associated preprojective algebra. To each terminal representation M of Q (these are certain preinjective representations), we attach a natural subcategory $C_M$ of…
We generalize the Caldero-Chapoton formula for cluster algebras of finite type to the skew-symmetrizable case. This is done by replacing representation categories of Dynkin quivers by categories of locally free modules over certain…
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 consider two non-degenerate potentials for the quiver arising from the once-punctured torus, which are a natural choice to study and compare: the first is the Labardini-potential, yielding a finite-dimensional Jacobian algebra, whereas…
We apply our previous work on cluster characters for Hom-infinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky's Cluster algebras IV for…
We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of `extended quivers' which are oriented hypergraphs. We describe mutations of such…
We show that many cluster-theoretic properties of the Markov quiver hold also for adjacency quivers of triangulations of once-punctured closed surfaces of arbitrary genus. Along the way we consider the class P of quivers introduced by…
Given a quiver with potential $(Q,W)$, Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of $(Q,W)$. As shown by Davison-Meinhardt, this algebra comes with a filtration whose associated graded…
We introduce a family of cluster algebras of infinite rank associated with root systems of type $A$, $D$, $E$. We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories…
Finite-dimensional Jacobian algebras are studied from the perspective of representation types. We establish that (like other representation types) the notions of $E$-finiteness and $E$-tameness are invariant under mutations of quivers with…
Motivated by the mutation theory of quivers with potentials developed by Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster algebras it provides, we propose a mutation theory of species with potentials for…
We provide a transformation formula of non-commutative Donaldson-Thomas invariants under a composition of mutations. Consequently, we get a description of a composition of cluster transformations in terms of quiver Grassmannians. As an…
Let $V$ be a finite-dimensional unitary representation of a compact quantum group $\mathrm{G}$ and denote by $\mathrm{G}_W$ the isotropy subgroup of a linear subspace $W\le V$ regarded as a point in the Grassmannian $\mathbb{G}(V)$. We show…
We show that the endomorphism ring of each cluster tilting object in a tubular cluster category is a finite dimensional Jacobian algebra which is tame of polynomial growth. Moreover, these Jacobian algebras are given by a quiver with a…
Let $\mathbb{X}$ be a weighted projective line and $\mathcal{C}_\mathbb{X}$ the associated cluster category. It is known that $\mathcal{C}_\mathbb{X}$ can be realized as a generalized cluster category of quiver with potential. In this note,…
We construct a cluster algebra structure within the quantum cohomology ring of a quiver variety associated with an $A$-type quiver. Specifically, let $Fl:=Fl(N_1,\ldots,N_{n+1})$ denote a partial flag variety of length $n$, and…
We associate a dimer algebra A to a Postnikov diagram D (in a disk) corresponding to a cluster of minors in the cluster structure of the Grassmannian Gr(k,n). We show that A is isomorphic to the endomorphism algebra of a corresponding…
Given a quiver associated to a cluster algebra and a sequence of vertices, iterative mutation leads to $F$-Polynomials which appear in numerous places in the cluster algebraic literature. The coefficients of the monomials in these…
We introduce a new cluster character with coefficients for a cluster category $\mathcal{C}$ and rather than using a Frobenius $2$-Calabi-Yau realization to incorporate coefficients into the representation-theoretic model for a cluster…
Let a finite group G act on a differential graded algebra A. This article presents necessary conditions and sufficient conditions for the skew group algebra A*G to be Calabi-Yau. In particular, when A is the Ginzburg dg algebra of a quiver…