Open Frobenius Cluster-Tilted Algebras

In this paper, we compute the Frobenius dimension of any cluster-tilted algebra of finite type. Moreover, we give conditions on the bound quiver of a cluster-tilted algebra [Formula: see text] such that [Formula: see text] has non-trivial open Frobenius structures.

τ-Tilting finite cluster-tilted algebras

We prove if B is a cluster-tilted algebra, then B is τB-tilting finite if and only if B is representation-finite.

Hochschild cohomology of m-cluster tilted algebras of type 𝔸̃

In this paper, we compute the dimension of the Hochschild cohomology groups of any [Formula: see text]-cluster tilted algebra of type [Formula: see text]. Moreover, we give conditions on the bounded quiver of an [Formula: see text]-cluster tilted algebra [Formula: see text] of type [Formula: see text] such that the Gerstenhaber algebra [Formula: see text] has nontrivial multiplicative structures. We also show that the derived class of gentle [Formula: see text]-cluster tilted algebras is not always completely determined by the dimension of the Hochschild cohomology.

Split t-structures and torsion pairs in hereditary categories

We construct a bijection between split torsion pairs in the module category of a tilted algebra having a complete slice in the preinjective component with corresponding [Formula: see text]-structures. We also classify split [Formula: see text]-structures in the derived category of a hereditary algebra.

EXPLICIT CONSTRUCTION OF COMPANION BASES

A companion basis for a quiver Γ mutation equivalent to a simply-laced Dynkin quiver is a subset of the associated root system which is a$\mathbb{Z}$-basis for the integral root lattice with the property that the non-zero inner products of pairs of its elements correspond to the edges in the underlying graph of Γ. It is known in typeA(and conjectured for all simply-laced Dynkin cases) that any companion basis can be used to compute the dimension vectors of the finitely generated indecomposable modules over the associated cluster-tilted algebra. Here, we present a procedure for explicitly constructing a companion basis for any quiver of mutation typeAorD.

TILTED ALGEBRAS AND CROSSED PRODUCTS

We consider an artin algebra A and its crossed product algebra Aα#σG, where G is a finite group with its order invertible in A. Then, we prove that A is a tilted algebra if and only if so is Aα#σG.

The Auslander–Reiten components of $\mathcal{K}^b({\rm proj}\, \Lambda)$ for a cluster-tilted algebra of type Ã

We classify the Auslander–Reiten components of [Formula: see text], where Λ is a cluster-tilted algebra of type Ã. The main tool is the combinatoric description of the indecomposable complexes in [Formula: see text] via homotopy strings and homotopy bands.

THE FIRST HOCHSCHILD COHOMOLOGY GROUP OF A CLUSTER TILTED ALGEBRA REVISITED

Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH 1(B) with coefficients in the B–B-bimodule B. If C is a tilted algebra such that B is the relation extension of C by [Formula: see text], then we prove that HH 1(B) is isomorphic, as a vector space, to the direct sum of HH 1(C) with HH 1(B,E). This yields homological interpretations for results of the first and the fourth authors with M. J. Redondo.