Rigid and Schurian modules over cluster-tilted algebras of tame type

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.     

Cluster Tilted Algebras with a Cyclically Oriented Quiver

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 type.     

Cluster-tilted algebras and slices

We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the Auslander–Reiten quiver of the cluster-tilted algebra, which we call local slice.     

Cluster-concealed algebras

The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects T in cluster categories; we call such an algebra cluster-concealed in case T is obtained from a preprojective tilting module. For example, all representation-finite cluster-tilted algebras are cluster-concealed. If C is a representation-finite cluster-tilted algebra, then the indecomposable C-modules are shown to be determined by their dimension vectors. For a general cluster-tilted algebra C, we are going to describe the dimension vectors of the indecomposable C-modules in terms of the root system of a quadratic form. The roots may have both positive and negative coordinates and we have to take absolute values.     

Generic variables in acyclic cluster algebras

Let Q be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra A(Q). We prove that the set G(Q) of generic variables contains naturally the set M(Q) of cluster monomials in 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 is the Kronecker quiver, the set G(Q) is a Z-basis of A(Q) and this basis is compared to Sherman-Zelevinsky and Caldero-Zelevinsky bases.     

Pairwise compatibility for 2-simple minded collections

In τ-tilting theory, it is often difficult to determine when a set of bricks forms a 2-simple minded collection. The aim of this paper is to determine when a set of bricks is contained in a 2-simple minded collection for a τ-tilting finite algebra. We begin by extending the definition of mutation from 2-simple minded collections to more general sets of bricks (which we call semibrick pairs). This gives us an algorithm to check if a semibrick pair is contained in a 2-simple minded collection. We then use this algorithm to show that the 2-simple minded collections of a τ-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) are given by pairwise compatibility conditions if and only if every vertex in the corresponding quiver has degree at most 2. As an application, we show that the classifying space of the τ-cluster morphism category of a τ-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) is an Eilenberg-MacLane space if every vertex in the corresponding quiver has degree at most 2.     

On the Galois coverings of a cluster-tilted algebra

We study the module category of a certain Galois covering of a cluster-tilted algebra which we call the cluster repetitive algebra. Our main result compares the module categories of the cluster repetitive algebra of a tilted algebra C and the repetitive algebra of C, in the sense of Hughes and Waschbüsch.     

On the quiver with relations of a quasitilted algebra and applications

In this paper we discuss, in terms of quiver with relations, su?cient and necessary conditions for an algebra to be a quasitilted algebra. We start with an algebra with global dimension at most two and we give a su?cient condition to be a quasitilted algebra. We show that this condition is not necessary. In the case of a strongly simply connected schurian algebra, we discuss necessary conditions, and combining both types of conditions, we are able to analyze if some given algebra is quasitilted. As an application we obtain the quiver with relations of all the tilted and cluster tilted algebras of Dynkin type E_p.     

Constructing tilted algebras from cluster-tilted algebras

Any cluster-tilted algebra is the relation extension of a tilted algebra. Given the distribution of a cluster-tilting object in the Auslander–Reiten quiver of the cluster category, we present a method to construct all tilted algebras whose relation extension is the endomorphism ring of this cluster-tilting object.     

Higher level affine Schur and Hecke algebras

We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur algebra surjects to the Dipper-James-Mathas cyclotomic q-Schur algebra. Moreover, we give nice diagrammatic presentations for all the algebras introduced in this paper.     

Endomorphism Algebras of Maximal Rigid Objects in Cluster Tubes

Given a maximal rigid object T of the cluster tube, we determine the objects finitely presented by T. We then use the method of Keller and Reiten to show that the endomorphism algebra of T is Gorenstein and of finite representation type, as first shown by Vatne. This algebra turns out to be the Jacobian algebra of a certain quiver with potential, when the characteristic of the base field is not 3. We study how this quiver with potential changes when T is mutated. We also provide a derived equivalence classification for the endomorphism algebras of maximal rigid objects.     

Subcategories of fixed points of mutations in root categories with type _(n-1)

For any n 3, let R(n) denote the root category of finite-dimensional nilpotent representations of cyclic quiver with n vertices. In the present paper, we prove that R(n-1) is triangle-equivalent to the subcategory of fixed points of certain left (or right) mutation in R(n). As an application, it is shown that the affine Kac-Moody algebra of type n-2 is isomorphic to a Lie subalgebra of the Kac-Moody algebra of type n-1.     

Quiver varieties and Frenkel–Kac construction

An affine Lie algebra acts on cohomology groups of quiver varieties of affine type. A Heisenberg algebra acts on cohomology groups of Hilbert schemes of points on a minimal resolution of a Kleinian singularity. We show that in the case of type A the former is obtained by Frenkel–Kac construction from the latter.     

AR quivers, exceptional sequences and algorithms in derived Hall algebras

Consider the canonical isomorphism between the positive part U ⁺ of the quantum group U _q (g) and the Hall algebra H(Λ), where the semisimple Lie algebra g and the finite-dimensional hereditary algebra Λ share a Dynkin diagram. Chen and Xiao have given two algorithms to decompose the root vectors into linear combinations of monomials of Chevalley generators of U ⁺, respectively induced by the braid group action on the exceptional sequences of Λ-modules and the structure of the Auslander-Reiten quiver of Λ. In this paper, we obtain the corresponding algorithms for the derived Hall algebra DH(Λ), which was introduced by Toën. We show that both algorithms are applicable to the lattice algebra and Heisenberg double in the sense of Kapranov. All the new recursive formulae have the same flavor with the quantum Serre relations.     

Maximal Green Sequences for Preprojective Algebras

Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller. We take the categorical notion and introduce maximal green sequences for hearts of bounded t-structures of triangulated categories that can be tilted indefinitely. We study the case where the heart is the category of modules over the preprojective algebra of a quiver without loops. The combinatorical counterpart of maximal green sequences for Dynkin quivers are maximal chains in the Hasse quiver of basic support τ-tilting modules. We show that a quiver has a maximal green sequence if and only if it is of Dynkin type. More generally, we study module categories for finite-dimensional algebras with finitely many bricks.     

Cell Decompositions of Quiver Flag Varieties for Nilpotent Representations of the Cyclic Quiver

Generalizing Schubert cells in type A and a cell decomposition of Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of a cyclic quiver admit an affine cell decomposition parametrized by multi-tableaux. We show that they carry a torus operation with finitely many fixpoints. As an application of the cell decomposition we obtain a vector space basis of certain modules (for quiver Hecke algebras of nilpotent representations of this quiver), similar modules have been studied by Kato as analogues of standard modules.     

A quiver presentation for Solomon's descent algebra

The descent algebra Σ(W) is a subalgebra of the group algebra QW of a finite Coxeter group W, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of W. Thus Σ(W) is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct Σ(W) as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of S, the set of simple reflections in W. From this construction we obtain some general information about the quiver of Σ(W) and an algorithm for the construction of a quiver presentation for the descent algebra Σ(W) of any given finite Coxeter group W.     

On the noncommutative Donaldson-Thomas invariants arising from brane tilings

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 formulate a rationality conjecture for the generating functions of the Donaldson-Thomas type invariants.