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Register a new userCluster categories for algebras of global dimension 2 and quivers with potential
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Authors
Claire Amiot
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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{o}er and by Buan-Iyama-Reiten-Scott. Our results also apply to quivers with potential. Namely, we introduce a cluster category $Cc_{(Q,W)}$ associated to a quiver with potential $(Q,W)$. When it is Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra $Jj(Q,W)$.
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Building on work by Geiss-Leclerc-Schroer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories associated with acyclic quivers and certain Frobenius subcategories of module categories over preprojective algebras. Our motivation comes from the conjectures formulated by Fomin and Zelevinsky in `Cluster algebras IV: Coefficients. We provide new evidence for Conjectures 5.4, 6.10, 7.2, 7.10 and 7.12 and show by an example that the statement of Conjecture 7.17 does not always hold.
This is an introduction to some aspects of Fomin-Zelevinskys 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 schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.
As a generalization of acyclic 2-Calabi-Yau categories, we consider 2-Calabi-Yau categories with a directed cluster-tilting subcategory; we study their cluster-tilting subcategories and the cluster combinatorics that they encode. We show that such categories have a cluster structure. Triangulated 2-Calabi-Yau categories with a directed cluster-tilting subcategory are closely related to representations of certain semi-hereditary categories, more specifically to representations of thread quivers. Thread quivers are a tool to classify and study certain semi-hereditary categories using both quivers and linearly ordered sets (threads). We study the case where the thread quiver consists of a single thread (so that representations of this thread quiver correspond to representations of some linearly ordered set), and show that, similar to the case of a Dynkin quiver of type $A$, the cluster-tilting subcategories can be understood via triangulations of an associated cyclically ordered set. In this way, we gain insight into the structure of the cluster-tilting subcategories of 2-Calabi-Yau categories with a directed cluster-tilting subcategory. As an application, we show that every 2-Calabi-Yau category which admits a directed cluster-tilting subcategory with countably many isomorphism classes of indecomposable objects has a cluster-tilting subcategory $mathcal{V}$ with the following property: any rigid object in the cluster category can be reached from $mathcal{V}$ by finitely many mutations. This implies that there is a cluster map which is defined on all rigid objects, and thus that there is a cluster algebra whose cluster variables are exactly given by the rigid indecomposable objects.
In this paper, we use basic formal variable techniques to study certain categories of modules for the toroidal Lie algebra $tau$. More specifically, we define and study two categories $mathcal{E}_{tau}$ and $mathcal{C}_{tau}$ of $tau$-modules using generating functions, where $mathcal{E}_{tau}$ is proved to contain the evaluation modules while $mathcal{C}_{tau}$ contains certain restricted $tau$-modules, the evaluation modules, and their tensor product modules. Furthermore, we classify the irreducible integrable modules in categories $mathcal{E}_{tau}$ and $mathcal{C}_{tau}$.
In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t^2). Using these results together with results of Geiss, Leclerc and Schroer we give, when k is algebraically closed, a classification of pairs (Q,R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally, when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra F_q[t]/(t^r). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.
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