This paper is a representation-theoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by Fomin-Shapiro-Thurston, the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky, and string modules associated to arcs on unpunctured surfaces by Assem-Brustle-Charbonneau-Plamondon. Modifying the latter construction, to each arc and each ideal triangulation of a bordered marked surface we associate in an explicit way a representation of the quiver with potential constructed in Part I, so that whenever two ideal triangulations are related by a flip, the associated representations are related by the corresponding mutation.
We attempt to relate two recent developments: cluster algebras associated to triangulations of surfaces by Fomin-Shapiro-Thurston, and quivers with potentials and their mutations introduced by Derksen-Weyman-Zelevinsky. To each ideal triangulation of a bordered surface with marked points we associate a quiver with potential, in such a way that whenever two ideal triangulations are related by a flip of an arc, the respective quivers with potentials are related by a mutation with respect to the flipped arc. We prove that if the surface has non-empty boundary, then the quivers with potentials ass…
In this paper, we introduce generalized quiver varieties which include as special cases classical and cyclic quiver varieties. The geometry of generalized quiver varieties is governed by a finitely generated algebra P: the algebra P is self-injective if the quiver Q is of Dynkin type, and coincides with the preprojective algebra in the case of classical quiver varieties. We show that in the Dynkin case the strata of generalized quiver varieties are in bijection with the isomorphism classes of objects in proj P, and that their degeneration order coincides with the Jensen-Su-Zimmermann's degener…
The aim of this note is to answer several open problems arising from the geometric description of the $m$-cluster categories of type $A_n$ and their realization in terms of the $m$-th power of a translation quiver. In particular, we give a geometric interpretation of the triangulated structure of $m$-cluster categories. Furthermore, we characterize all the connected components arising from a cluster category when taking the $m$-th power of its Auslander-Reiten quiver.
We show that if a connected, Hom-finite, Krull-Schmidt triangulated category has an Auslander-Reiten quiver component with Dynkin tree class then the category has Auslander-Reiten triangles and that component is the entire quiver. This is an analogue for triangulated categories of a theorem of Auslander, and extends a previous result of Scherotzke. We also show that if there is a quiver component with extended Dynkin tree class, then other components must also have extended Dynkin class or one of a small set of infinite trees, provided there is a non-zero homomorphism between the components. T…
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 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.
To each tagged triangulation of a surface with marked points and non-empty boundary we associate a quiver with potential, in such a way that whenever we apply a flip to a tagged triangulation, the Jacobian algebra of the QP associated to the resulting tagged triangulation is isomorphic to the Jacobian algebra of the QP obtained by mutating the QP of the original one. Furthermore, we show that any two tagged triangulations are related by a sequence of flips compatible with QP-mutation. We also prove that for each of the QPs constructed, the ideal of the non-completed path algebra generated by t…
Let T be a locally finite triangulated category with an autoequivalence F such that the orbit category T/F is triangulated. We show that if X is an m-cluster tilting subcategory, then the image of X in T/F is an m-cluster tilting subcategory if and only if X is F-perodic. We show that for path-algebras of Dynking quivers δone may study the periodic properties of n-cluster tilting objects in the n-cluster category Cn(kδ) to obtain information on periodicity of the preimage as n-cluster tilting subcategories of Db(kδ). Finally we classify the periodic properties of all 2-cluster tilting objects …
This is a companion paper of arXiv:1901.09461, where different notions of dimension for triangulated categories are discussed. Here we compute dimensions for some examples of triangulated categories and thus illustrate and motivate material from loc. cit. Our examples include path algebras of finite ordered quivers, orbifold projective lines, some tensor powers of path algebras in Dynkin quivers of type A and categories, generated by an exceptional pair.
We study a triangulated category of graded matrix factorizations for a polynomial of type ADE. We show that it is equivalent to the derived category of finitely generated modules over the path algebra of the corresponding Dynkin quiver. Also, we discuss a special stability condition for the triangulated category in the sense of T. Bridgeland, which is naturally defined by the grading.
We derive an algorithm for mutating quivers of 2-CY tilted algebras that have loops and 2-cycles, under certain specific conditions. Further, we give the classification of the 2-CY tilted algebras coming from standard algebraic 2-CY triangulated categories with a finite number of indecomposables. These form a class of algebras that satisfy the setup for our mutation algorithm.
We show that an algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category if it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As a first application, we show that the stable category of maximal Cohen-Macaulay modules over a certain isolated singularity of dimension three is a cluster category. As a second application, we prove the non-acyclicity of the quivers of endomorphism algebras of cluster-tilting objects in the stable categories of representation-i…
Tropical friezes are the tropical analogues of Coxeter-Conway frieze patterns. In this note, we study them using triangulated categories. A tropical frieze on a 2-Calabi-Yau triangulated category $\mathcal{C}$ is a function satisfying a certain addition formula. We show that when $\mathcal{C}$ is the cluster category of a Dynkin quiver, the tropical friezes on ${\mathcal{C}}$ are in bijection with the $n$-tuples in ${\mathbb{Z}}^n$, any tropical frieze $f$ on $\mathcal{C}$ is of a special form, and there exists a cluster-tilting object such that $f$ simultaneously takes non-negative values or …
The aim of this paper is to give analogs of the cluster expansion formula of Musiker and Schiffler for cluster algebras of type A with coefficients arising from boundary arcs of the corresponding triangulated polygon. Indeed, we give three cluster expansion formulas by perfect matchings of angles in triangulated polygon, by discrete subsets of arrows of the corresponding ice quiver and by minimal cuts of the corresponding quiver with potential.
In this short paper we introduce a new triangulated category for rational surface singularities which in the non-Gorenstein case acts as a substitute for the stable category of matrix factorizations. The category is formed as a Frobenius quotient of the category of special CM modules, and we classify the relatively projective-injective objects and thus describe the AR quiver of the quotient. Connections to the corresponding reconstruction algebras are also discussed.
In this survey article we give a brief account of constructions and results concerning the quivers with potentials associated to triangulations of surfaces with marked points. Besides the fact that the mutations of these quivers with potentials are compatible with the flips of triangulations, we mention some recent results on the representation type of Jacobian algebras and the uniqueness of non-degenerate potentials. We also mention how the the quivers with potentials associated to triangulations give rise to CY2 and CY3 triangulated categories that have turned out to be useful in the subject…
We compute the Grothendieck group of certain 2-Calabi--Yau triangulated categories appearing naturally in the study of the link between quiver representations and Fomin--Zelevinsky's cluster algebras. In this setup, we also prove a generalization of Fomin--Zelevinsky's mutation rule.
We embed triangulated categories defined by quivers with potential arising from ideal triangulations of marked bordered surfaces into Fukaya categories of quasi-projective 3-folds associated to meromorphic quadratic differentials. Together with previous results, this yields non-trivial computations of spaces of stability conditions on Fukaya categories of symplectic 6-manifolds.
We show that Derksen-Weyman-Zelevinsky's mutations of quivers with potential yield equivalences of suitable 3-Calabi-Yau triangulated categories. Our approach is related to that of Iyama-Reiten and Koszul dual to that of Kontsevich-Soibelman. It improves on previous work by Vitoria. In the appendix, the first-named author studies pseudo-compact derived categories of certain pseudo-compact dg algebras.
We describe presentations of braid groups of type ADE and show how these presentations are compatible with mutation of quivers, building on work of Barot and Marsh for Coxeter groups. In types A and D these presentations can be understood geometrically using triangulated surfaces. We then give a categorical interpretation of the presentations, with the new generators acting as spherical twists at simple modules on derived categories of Ginzburg dg-algebras of quivers with potential.
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…
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 category of Cohen-Macaulay L-modules is equivalent to the cluster category C of Dynkin type A(n-3). It gives a natural interpretation of the usual indexation of cluster tilting objects of C by triangulations of P. Moreover, it extends naturally the triangulated categorification by C of the cluster algebra of type A(n-3) to an exact categorification by adding co…
This paper was inspired by four articles: surface cluster algebras studied by Fomin-Shapiro-Thurston \cite{fst}, the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky \cite{dwz}, string modules associated to arcs on unpunctured surfaces by Assem-Br$\ddot{u}$stle-Charbonneau-Plamondon \cite{acbp} and Quivers with potentials associated to triangulated surfaces, part II: Arc representations by Labardini-Fragoso. \cite{lf2}. For a surface with marked points ($Σ,M$) Labardini-Fragoso associated a quiver with potential $(Q(τ),S(τ))$ then for an ideal triangulation of …
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 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…
We associate a coloured quiver to a rigid object in a Hom-finite 2-Calabi--Yau triangulated category and to a partial triangulation on a marked (unpunctured) Riemann surface. We show that, in the case where the category is the generalised cluster category associated to a surface, the coloured quivers coincide. We also show that compatible notions of mutation can be defined and give an explicit description in the case of a disk. A partial description is given in the general 2-Calabi-Yau case. We show further that Iyama-Yoshino reduction can be interpreted as cutting along an arc in the surface.
The properties of the preprojective algebra are very di fferent whether the associated quiver is of Dynkin type or not. However in both cases, one can construct from it a triangulated category of Calabi-Yau dimension 2. In this note we explain the generalizations of this fact in the context of higher preprojective algebra, and we give some homological properties that characterize preprojective algebras.
The representations of a quiver Q over a field k have been studied for a long time. It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A. Here we draw the attention to the case when A = k[ε] is the algebra of dual numbers, thus to the Λ-modules with Λ= kQ[ε]. The algebra Λ is a 1-Gorenstein algebra, therefore the torsionless Λ-modules are known to be of special interest (as the Gorenstein-projective or maximal Cohen-Macaulay modules). They form a Frobenius category \Cal L, thus the corresponding stable category is a triangulated categor…
We consider tilting mutations of a weakly symmetric algebra at a subset of simple modules, as recently introduced by T. Aihara. These mutations are defined as the endomorphism rings of certain tilting complexes of length 1. Starting from a weakly symmetric algebra A, presented by a quiver with relations, we give a detailed description of the quiver and relations of the algebra obtained by mutating at a single loopless vertex of the quiver of A. In this form the mutation procedure appears similar to, although significantly more complicated than, the mutation procedure of Derksen, Weyman and Zel…
We prove that moduli spaces of meromorphic quadratic differentials with simple zeroes on compact Riemann surfaces can be identified with spaces of stability conditions on a class of CY3 triangulated categories defined using quivers with potential associated to triangulated surfaces. We relate the finite-length trajectories of such quadratic differentials to the stable objects of the corresponding stability condition.
In this article we study the triangulated category of singularities associated with a non-commutative resolution of singularities. In particular, we give a complete description of this category in the case of a curve with nodal singularities, classifying its indecomposable objects and computing its Auslander-Reiten quiver and K-group.
We prove that mutation of cluster-tilting objects in triangulated 2-Calabi-Yau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2-CY-tilted algebras and Jacobian algebras associated with quivers with potentials. We show that cluster-tilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3-CY algebras. The nearly Morita equivalence for 2-CY-tilted algebras is shown to hold for the finite length modules over Jacobian algebras.
We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster tilting objects in 2-Calabi-Yau triangulated categories, hence all their non-projective indecomposable modules are $Ω$-periodic of period dividing 4. Our construction is based on the combinatorial notion of triangulation quivers, which arise naturally from triangulations of oriented surfaces with marked points. This class of algebras contains the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other…
We consider stable representations of non-Dynkin quivers with respect to a central charge. On one condition the existence of a stable representation with self-extensions implies the existence of infinitely many stables without self-extensions. In this case the phases of the stable representations approach one or two limit points. In particular, the phases are not dense in two arcs.
Given a certain triangulation of a punctured surface with boundary, we construct a new triangulated surface without punctures which covers it. This new surface is naturally equipped with an action of a group of order two, and its quotient by this action recovers the original surface. We show that the group acts on the quivers with potentials associated to the surfaces, and that their Ginzburg dg algebras are skew group algebras of each other, up to Morita equivalence. We then use these results to construct functors between the generalized cluster categories associated to the triangulations. Th…
In the first part of the paper we defined and studied a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. For type $A$ we conjecture, following Geiss and Schröer, that this operation is compatible with taking the socle of parabolic induction of representations of general linear groups over a local non-archimedean field, at least when one of the irreducible components is rigid. We verify this conjecture in special cases.
Perverse schobers, stability conditions and quadratic differentials I
Mathematisches Institut, Universität Bonn（波恩大学数学研究所）；Centre for Quantum Mathematics, Department of Mathematics and Computer Science, University of Southern Denmark（南丹麦大学数学与计算机科学系量子数学中心）；Yau Mathematical Sciences Center and Department of Mathematical Sciences, Tsinghua University（清华大学丘成桐数学科学中心及数学科学系）；Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）
