Equivalences between cluster categories

Cluster categories were introduced in the paper “Tilting theory and cluster combinatorics” [8] in order to better understand the combinatorics of cluster algebras, by giving new, module theoretic and categorical meanings to the combinatorics of the well known cluster algebras [20]. Subsequently, correspondences between the many notions in these two areas were given, e.g. [5, 6, 11, 9] and others.

We give an example of a cluster-tilted algebra $$\Lambda $$ with quiver Q, such that the associated cluster algebra $$\mathcal {A}(Q)$$ has a denominator vector which is not the dimension vector of any indecomposable $$\Lambda $$ -module. This answers a question posed by T. Nakanishi. The relevant example is a cluster-tilted algebra associated with a tame hereditary algebra. We show that for such a cluster-tilted algebra $$\Lambda $$ , we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid $$\Lambda $$ -modules. In order to do this it is necessary, and of independent interest, to first classify the indecomposable rigid $$\Lambda $$ -modules in this case.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Here, employing a notion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="asterisk"> <mml:semantics> <mml:mo> ∗ </mml:mo> <mml:annotation encoding="application/x-tex">\ast</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.

We study the stable category of the graded Cohen–Macaulay modules of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\boldsymbol{w})$ of this category associated with each reduced expression $\boldsymbol{w}$ of $w$ and give a sufficient condition on $\boldsymbol{w}$ such that $M(\boldsymbol{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\boldsymbol{w})$. Moreover, we compare it with a triangle equivalence given by Amiot–Reiten–Todorov for a cluster category.

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 these results, we answer several conjectures on the connection between cluster algebras and quiver representations.

Cluster algebras are commutative algebras that were introduced by Fomin and Zelevinsky in order to model the dual canonical basis of a quantum group and total positivity in algebraic groups. Cluster categories were introduced as a representation-theoretic model for cluster algebras. In this article we use this representation-theoretic approach to prove a conjecture of Fomin and Zelevinsky, that for cluster algebras with no coefficients associated to quivers with no oriented cycles, a seed is determined by its cluster. We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.

In Berenstein and Rupel (2015), the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category \(\mathcal {A}\) to an appropriate q-polynomial algebra. In the case that \(\mathcal {A}\) is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in (Ding and Xu, Sci. China Math. 55(10) 2045–2066, 2012). Moreover, if the underlying graph of Q associated with \(\mathcal {A}\) is bipartite and the matrix B associated to the quiver Q is of full rank, we show that the image of the algebra homomorphism is in the corresponding quantum cluster algebra.

An approach to non-simply laced cluster algebras

These notes are an expanded version of a mini-course given in the CIMPA School ‘Homological Methods, Representation Theory and Cluster Algebras’, held from the 7th to the 18th of March 2016 in Mar del Plata (Argentina). The aim of the course was to introduce the participants to cluster tilted algebras and their applications in the representation theory of algebras.

Hereditary Abelian Categories and Almost Split Sequences

Exceptional sequences and clusters

A strong connection between cluster algebras and representation theory was established by the cluster category. Cluster characters, like the original Caldero-Chapoton map, are maps from certain triangulated categories to cluster algebras and they have generated much interest. Holm and Jørgensen constructed a modified Caldero-Chapoton map from a sufficiently nice triangulated category to a commutative ring, which is a generalised frieze under some conditions. In their construction, a quotient K0sp(T)/M of a Grothendieck group of a cluster tilting subcategory T is used. In this article, we show that this quotient is the Grothendieck group of a certain extriangulated category, thereby exposing the significance of it and the relevance of extriangulated structures. We use this to define another modified Caldero-Chapoton map that recovers the one of Holm–Jørgensen.We prove our results in a higher homological context. Suppose S is a (d+2)-angulated category with subcategories X⊆T⊆S, where X is functorially finite and T is 2d-cluster tilting, satisfying some mild conditions. We show there is an isomorphism between the Grothendieck group K0(S,EX,sX) of the category S, equipped with the d-exangulated structure induced by X, and the quotient K0sp(T)/N, where N is the higher analogue of M above. When X=T the isomorphism is induced by the higher index with respect to T introduced recently by Jørgensen. Thus, in the general case, we can understand the map taking an object in S to its K0-class in K0(S,EX,sX) as a higher index with respect to the rigid subcategory X.

Tilting modules arising from knot invariants

We introduce a relative tilting theory in abelian categories and show that this work offers a unified framework of different previous notions of tilting, ranging from Auslander–Solberg relative tilting modules on Artin algebras to infinitely generated tilting modules on arbitrary rings. Furthermore, we see that it presents a tool for developing new tilting theories in categories that can be embedded nicely in an abelian category. In particular, we will show how the tilting theory in exact categories built this way, coincides with tilting objects in extriangulated categories introduced recently. We will review Bazzoni’s tilting characterization, the relative homological dimensions on the induced tilting classes and parametrise certain cotorsion-like pairs by using n-X-tilting classes. As an application, we show how to construct relative tilting classes and cotorsion pairs in Rep(Q,C) (the category of representations of a quiver Q in an abelian category C) from tilting classes in C, where Q is finite-cone-shape.