Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A_n. We associate to each cluster C of U an abelian category Cat_C such that the indecomposable objects of Cat_C are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic realization and a geometric realization of Cat_C. Then, we generalize the ``denominator Theorem'' of Fomin and Zelevinsky to any cluster.
Let $\mathfrak g$ be a complex simple Lie algebra and let $Ψ$ be an extremal set of positive roots. One associates with $Ψ$ an infinite dimensional Koszul algebra $\bold S_Ψ^{\lie g}$ which is a graded subalgebra of the locally finite part of $((\bold V)^{op}\tensor S(\lie g))^{\lie g}$, where $\bold V$ is the direct sum of all simple finite dimensional $\lie g$-modules. We describe the structure of the algebra $\bold S_Ψ^{\lie g}$ explicitly in terms of an infinite quiver with relations for $\lie g$ of types $A$ and $C$. We also describe several infinite families of quivers and finite dimensi…
It has been proved in \cite{LS} that cluster variables in cluster algebras of every skew-symmetric cluster algebra are positive. We prove that any regular generalized cluster variable of an affine quiver is positive. As a corollary, we obtain that generalized cluster variables of affine quivers are positive and construct various positive bases. This generalizes the results in \cite{Dupont} and \cite{DXX}.
Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding quiver representation category. This also provides some explicit formulas for cluster variables.
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of…
Let $B(Λ_0)$ be the level 1 highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\mathbb{B}(Λ_0)$ of $B(Λ_0)$ via quiver varieties and the path realization ${\mathcal P}^{\rm ad}(Λ_0)$ of $B(Λ_0)$ arising from the adjoint crystal $\adjoint$.
Acyclic cluster algebras have an interpretation in terms of tilting objects in a Calabi-Yau category defined by some hereditary algebra. For a given quiver $Q$ it is thus desirable to decide if the cluster algebra defined by $Q$ is acyclic. We call $Q$ cluster-acyclic in this case, otherwise cluster-cyclic. In this note we classify the cluster-cyclic quivers with three vertices using a Diophantine equation studied by Markov.
We define an analogue of the Caldero-Chapoton map (\cite{CC}) for the cluster category of finite dimensional nilpotent representations over a cyclic quiver. We prove that it is a cluster character (in the sense of \cite{Palu}) and satisfies some inductive formulas for the multiplication between the generalized cluster variables (the images of objects of the cluster category under the map). Moreover, we construct a $\mathbb{Z}$-basis for the algebras generated by all generalized cluster variables.
In this paper we show that there is a link between the combinatorics of the canonical basis of a quantized enveloping algebra and the monomial bases of the second author arising from representations of quivers. We prove that some reparametrization functions of the canonical basis, arising from the link between Lusztig's approach to the canonical basis and the string parametrization of the canonical basis, are given on a large region of linearity by linear functions arising from these monomial basis for a quantized enveloping algebra.
Sherman-Zelevinsky and Cerulli constructed canonically positive bases in cluster algebras associated to affine quivers having at most three vertices. Both constructions involve cluster monomials and normalized Chebyshev polynomials of the first kind evaluated at a certain "imaginary" element in the cluster algebra. Using this combinatorial description, it is possible to define for any affine quiver $Q$ a set $\mathcal B(Q)$ which is conjectured to be the canonically positive basis of the acyclic cluster algebra $\mathcal A(Q)$. In this article, we provide a geometric realization of the element…
In \cite{CK2005} and \cite{SZ}, the authors constructed the bases of cluster algebras of finite types and of type $\widetilde{A}_{1,1}$, respectively. In this paper, we will deduce $\mathbb{Z}$-bases for cluster algebras of affine types.
Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretical interpretation of cluster mutation in case of acyclic cluster algebras of finite type.
The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in \cite{CK2005} and \cite{SZ} respectively. In this paper, we will deduce $\mathbb{Z}$-bases for cluster algebras for affine types $\widetilde{A}_{n,n},\widetilde{D}$ and $\widetilde{E}$. Moreover, we give an inductive formula for computing the multiplication between two generalized cluster variables associated to objects in a tube.
We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the "Cluster algebras IV" paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F-polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these g-vectors and F-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpr…
The theory of Caldero-Chapoton algebras of Cerulli-Irelli, Labardini-Fragoso and Schroer leads to a refinement of the notions of cluster variables and clusters, via so called component clusters. In this paper we compare component clusters to classical clusters for the cluster algebra of an acyclic quiver. We propose a definition of mutation between component clusters and determine the mutation relations of component clusters for affine quivers. In the case of a wild quiver, we provide bounds for the size of component clusters.
The main objective of this paper is to provide a theory for computing the Hochschild cohomology of algebras arising from a linear category with finitely many objects and zero compositions. For this purpose, we consider such a category using an ad hoc quiver $Q$, with an algebra associated to each vertex and a bimodule to each arrow. The computation relies on cohomological functors that we introduce, and on the combinatorics of the quiver. One point extensions are occurrences of this situation, and Happel's long exact sequence is a particular case of the long exact sequence of cohomology that w…
We introduce a multivariate generalization of normalized Chebyshev polynomials of the second kind. We prove that these polynomials arise in the context of cluster characters associated to Dynkin quivers of type $\mathbb A$ and representation-infinite quivers. This allows to obtain a simple combinatorial description of cluster algebras of type $\mathbb A$. We also provide explicit multiplication formulas for cluster characters associated to regular modules over the path algebra of any representation-infinite quiver.
We consider (iced) quiver with potential $(\bar{Q}(D), F(D), \bar{W}(D))$ associated to a Postnilov Diagram $D$ and prove the mutation of the quiver with potential $(\bare{Q}(D), F(D), \bar{W}(D))$ is compatible with the geometric exchange of the Postnikov diagram $D$. This ensures we may define a quiver with potential for a Grassmannian cluster algebra. We show such quiver with potential is always rigid (thus non-degenerate) and Jacobian-finite. And in fact, it is the unique non-degenerate (thus unique rigid) quiver with potential associated to the Grassmannian cluster algebra up to right-equ…
We consider $m$-cluster tilted algebras arising from quivers of Euclidean type and we give necessary and sufficient conditions for those algebras to be representation finite. For the case $\widetilde{A}$, using the geometric realization, we get a description of representation finite type in terms of $(m+2)$-angulations. We establish which $m$-cluster tilted algebras arise at the same time from quivers of type $A$ and $\widetilde{A}$. Finally, we characterize representation infinite $m$-cluster tilted algebras arising from a quiver of type $\widetilde{A}$ as $m$-relations extensions of some ite…
In this paper we explain how and why the list of Happel-Vossieck of tame concealed algebras is closely related to the list of A. Seven of minimal infinite cluster quivers.
We study quiver Grassmannians associated with indecomposable representations of the Kronecker quiver. We find a cellular decomposition of them and we compute their Betti numbers. As an application, we give a geometric realization of the "canonical basis" of cluster algebras of Kronecker type (found by Sherman and Zelevinsky) and of type $A_2^{(1)}$.
Given a brane tiling, that is a bipartite graph on a torus, we can associate with it a quiver potential and a quiver potential algebra. Under certain consistency conditions on a brane tiling, we prove a formula for the Donaldson-Thomas type invariants of the moduli space of framed cyclic modules over the corresponding quiver potential algebra. We relate this formula with the counting of perfect matchings of the periodic plane tiling corresponding to the brane tiling. We prove that the same consistency conditions imply that the quiver potential algebra is a 3-Calabi-Yau algebra. We also formula…
Let $(Q,W)$ be a quiver with a non degenerate potential. We give a new description of the \textbf{c}-vectors of $Q$. We use it to show that, if $Q$ is mutation equivalent to a Dynkin quiver, then the set of positive $\mathbf{c}$-vectors of the cluster algebra associated to $Q^{\text{op}}$ coincides with the set of dimension vectors of the indecomposable modules over the Jacobian algebra of $(Q,W)$.
We show that the mutation class of a finite quiver without oriented cycles is finite if and only is the quiver is either Dynkin, extended Dynkin or has at most two vertices.
Let $Q$ be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra $\mathcal A(Q)$. We prove that the set $\mathcal G(Q)$ of generic variables contains naturally the set $\mathcal M(Q)$ of cluster monomials in $\mathcal A(Q)$ and that these two sets coincide if and only if $Q$ is a Dynkin quiver. We establish multiplicative properties of these generic variables analogous to multiplicative properties of Lusztig's dual semicanonical basis. This allows to compute explicitly the generic variables when $Q$ is a quiver of affine type. When $Q$…
Given a tagged triangulation of a once-punctured polygon $P^*$ with $n$ vertices, we associate an ice quiver with potential such that the frozen part of the associated frozen Jacobian algebra has the structure of a Gorenstein $K[X]$-order $Λ$. Then we show that the stable category of the category of Cohen-Macaulay $Λ$-modules is equivalent to the cluster category $\mathcal{C}$ of type $D_n$. It gives a natural interpretation of the usual indexation of cluster tilting objects of $\mathcal{C}$ by tagged triangulations of $P^*$. Moreover, it extends naturally the triangulated categorification by …
We consider the general notion of coloured quiver mutation and show that the mutation class of a coloured quiver $Q$, arising from an $m$-cluster tilting object associated with $H$, is finite if and only if $H$ is of finite or tame representation type, or it has at most 2 simples. This generalizes a result known for 1-cluster categories.
We prove a multiplication theorem of a quantum Caldero-Chapoton map associated to valued quivers which extends the results in \cite{DX}\cite{D}. As an application, when $Q$ is a valued quiver of finite type or rank 2, we obtain that the algebra $\mathcal{AH}_{|k|}(Q)$ generated by all cluster characters (see Definition \ref{def}) is exactly the quantum cluster algebra $\mathcal{EH}_{|k|}(Q)$ and various bases of the quantum cluster algebras of rank 2 can naturally be deduced.
We define mutation on coloured quivers associated to tilting objects in higher cluster categories. We show that this operation is compatible with the mutation operation on the tilting objects. This gives a combinatorial approach to tilting in higher cluster categories and especially an algorithm to determine the Gabriel quivers of tilting objects in such categories.
We show that the cone of finite stability conditions of a quiver Q without oriented cycles has a fan covering given by (the dual of) the cluster fan of Q. Along the way, we give new proofs of Schofield's results on perpendicular categories. We also study domains of semi-invariants of quivers via quiver exceptional sequences. In particular, we recover Igusa-Orr-Todorov-Weyman's theorem on cluster complexes and domains of semi-invariants for Dynkin quivers.
In association with a finite dimensional algebra A of global dimension two, we consider the endomorphism algebra of A, viewed as an object in the triangulated hull of the orbit category of the bounded derived category, in the sense of Amiot. We characterize the algebras A of global dimension two such that its endomorphism algebra is isomorphic to a cluster-tilted algebra with a cyclically oriented quiver.Furthermore, in the case that the cluster tilted algebra with a cyclically oriented quiver is of Dynkin or extended Dynkin type then A is derived equivalent to a hereditary algebra of the same…
We prove that the frieze sequences of cluster variables associated with the vertices of an affine quiver satisfy linear recurrence relations. In particular, we obtain a proof of a recent conjecture by Assem-Reutenauer-Smith.
The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang-Zelevinsky in finite type. In type $A_n^{\!(1)}$ and fin…
Inspirited by the importance of the spectral theory of graphs, we introduce the spectral theory of valued cluster quiver of a cluster algebra. Our aim is to characterize a cluster algebra via its spectrum so as to use the spectral theory as a tool. First, we give the relations between exchange spectrum of a valued cluster quiver and adjacency spectrum of its underlying valued graph, and between exchange spectra of a valued cluster quiver and its full valued subquivers. The key point is to find some invariants from the spectrum theory under mutations of cluster algebras, which is the second par…
Additive functions on translation quivers have played an important role in the representation theory of finite dimensional algebras, the most prominent ones are the hammock functions introduced by S. Brenner. When dealing with cluster categories (and cluster-tilted algebras), one should look at a corresponding class of functions defined on stable translation quivers, namely the cluster-additive ones. We conjecture that the cluster-additive functions on a stable translation quiver of Dynkin type A_n, D_n, E_6, E_7, E_8 are non-negative linear combinations of cluster-hammock functions (with inde…
Let $Q$ be a finite quiver without oriented cycles and $\mathcal A(Q)$ be the coefficient-free cluster algebra with initial seed $(Q,\textbf u)$. Using the Caldero-Chapoton map, we introduce and investigate a family of generic variables in $\Z[\textbf u^{\pm 1}]$ containing the cluster monomials of $\mathcal A(Q)$. The aim of these generic variables is to give an explicit new method for constructing $\Z$-bases in the cluster algebra $\mathcal A(Q)$. If $Q$ is an affine quiver with minimal imaginary root $δ$, we investigate differences between cluster characters associated to indecomposable rep…
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 Kontsevich and Soibelman, characterize the mutation-finite quivers that belong to that class and draw some conclusions regarding non-degenerate potentials on them.
We study Laurent expansions of cluster variables in a cluster algebra of rank 2 associated to a generalized Kronecker quiver. In the case of the ordinary Kronecker quiver, we obtain explicit expressions for Laurent expansions of the elements of the canonical basis for the corresponding cluster algebra.
We construct bar-invariant $\mathbb{Z}[q^{\pm 1/2}]-$bases of the quantum cluster algebra of the Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the cluster algebra of the Kronecker quiver in the sense of \cite{sherzel},\cite{calzel} and \cite{gls} respectively. As a byproduct, we prove the positivity of the elements in these bases.
Let $Q$ be an euclidean quiver. Using friezes in the sense of Assem-Reutenauer-Smith, we provide an algorithm for computing the (canonical) cluster character associated to any object in the cluster category of $Q$. In particular, this algorithm allows to compute all the cluster variables in the cluster algebra associated to $Q$. It also allows to compute the sum of the Euler characteristics of the quiver grassmannians of any module $M$ over the path algebra of $Q$.
For any valued quiver, by using BGP-reflection functors, an injection from the set of preprojective objects in the cluster category to the set of cluster variables of the corresponding cluster algebra is given, the images are called preprojective cluster variables. It is proved that all preprojective cluster variables have denominators $u^{\underline{dim}M}$ in their irreducible fractions of integral polynomials, where $M$ is the corresponding preprojective module or preinjective module. If the quiver is of Dynkin type, we generalize the denominator theorem in [FZ2] to any seed, and also gener…
We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. By using $d-$cluster categories which are defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a $d-$compatibility degree $(-||-)$ on any pair of ``colored'' almost positive real Schur roots which generalizes previous definitions on the non-colored case, and call two such roots compatible provided the $d-$compatibility degree of them is zero. Associated to the root system $Φ$ corresponding to the va…
In this paper, we study the distribution of the genuses of cluster quivers of finite mutation type. First, we prove that in the $11$ exceptional cases, the distribution of genuses is $0$ or $1$. Next, we consider the relationship between the genus of an oriented surface and that of cluster quivers from this surface. It is verified that the genus of an oriented surface is an upper bound for the genuses of cluster quivers from this surface. Furthermore, for any non-negative integer $n$ and a closed oriented surface of genus $n$, we show that there always exist a set of punctures and a triangulat…
We construct a special embedding of the translation quiver $\mathbb{Z}Q$ in the three-dimensional affine space $\mathbb{R}^{3}$ where $Q$ is a finite connected acyclic quiver and $\mathbb{Z}Q$ contains a local slice whose quiver is isomorphic to the opposite quiver $Q^{op}$ of $Q.$ Via this embedding, we show that there exists an involutive anti-automorphism of the translation quiver $\mathbb{Z}Q.$ As an immediate consequence, we characterize explicitly the group of cluster automorphisms of the cluster algebras of seed $(X,Q)$, where $Q$ and $Q^{op}$ are mutation equivalent.
Let ${\rm SI}_β(Q)$ be the semi-invariant ring of $β$-dimensional representations of a quiver $Q$. Suppose that $(Q,β)$ projects to another quiver with dimension vector $(Q',β')$ through an exceptional representation $E$. We show that if ${\rm SI}_β(Q)$ is the upper cluster algebra associated to an ice quiver $Δ$, then ${\rm SI}_{β'}(Q')$ is the upper cluster algebra associated to $Δ'$, where $Δ'$ is obtained from $Δ$ through simple operations depending on $E$. We also study the relation of their basis using the quiver with potential model.
Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\leq 2$. We construct a triangulated category $\Cc_A$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$. When $\Cc_A$ is $\Hom$-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schr{ö}er and by Buan-Iyama-Reiten-Scott. Our results also apply to quivers with potential. Namely, we introduce a cluste…
We characterize the marked bordered unpunctured oriented surfaces with the property that all the Jacobian algebras of the quivers with potentials arising from their triangulations are derived equivalent. These are either surfaces of genus g with b boundary components and one marked point on each component, or the disc with 4 or 5 points on its boundary. We show that for each such marked surface, all the quivers in the mutation class have the same number of arrows, and the corresponding Jacobian algebras constitute a complete derived equivalence class of finite-dimensional algebras whose member…
We show that a reddening sequence exists for any quiver which is Banff. Our proof is combinatorial and relies on the triangular extension construction for quivers. The other facts needed are that the existence of a reddening sequence is mutation invariant and passes to induced subquivers. Banff quivers define locally acyclic cluster algebras which are known to coincide with their upper cluster algebras.The existence of reddening sequences for these quivers is consistent with a conjectural relationship between the existence of a reddening sequence and a cluster algebra's equality with its upper…
We consider frieze sequences corresponding to sequences of cluster mutations for affine D and E type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient relations found by Keller and Scherotzke. Viewing the frieze sequence as a discrete dynamical system, we reduce it to a symplectic map on a lower dimensional space and prove Liouville integrability of the latter.
In a recent article we introduced a mechanism for producing a presentation of the descent algebra of the symmetric group as a quiver with relations, the mechanism arising from a new construction of the descent algebra as a homomorphic image of an algebra of binary forests. Here we extend the method to construct a similar presentation of the descent algebra of the hyperoctahedral group, providing a simple proof of the known formula for the quiver of this algebra and a straightforward method for calculating the relations.
Motivated by work of Barot, Geiss and Zelevinsky, we study a collection of Z-bases (which we call companion bases) of the integral root lattice of a root system of simply-laced Dynkin type. Each companion basis is associated with the quiver of a cluster-tilted algebra of the corresponding type. In type A, we establish that the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra may be obtained, up to sign, by expanding the positive roots in terms of any companion basis for the quiver of that algebra. This generalises part of Gabriel's Theorem. Also,…
We survey and compare various generalizations of braid groups for quivers with superpotential and focus on the cluster braid groups, which are introduced in a joint work with A.~King. Our motivations come from the study of cluster algebras, Calabi-Yau categories and Bridgeland stability conditions.
Let $\tilde{Q}$ (resp. $Q$) be an extended exchange (resp. exchange) cluster quiver of finite mutation type. We introduce the distribution set of the number of arrows for $Mut[\tilde{Q}]$ (resp. $Mut[Q]$), give the maximum and minimum numbers of the distribution set and establish the existence of an extended complete walk (resp. a complete walk). As a consequence, we prove that the distribution set for $Mut[\tilde{Q}]$ (resp. $Mut[Q]$) is continuous except the exceptional cases. In case of cluster quivers $Q_{inf}$ of infinite mutation type, the number of arrows does not present a continuous d…
Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg) of a symmetric Kac-Moody Lie algebra g, studied earlier by the author via perverse sheaves on graded quiver varieties [arXiv:math/9912158]. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, …
