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.

Universal Deformation Rings of Modules over Self-injective Cluster-Tilted Algebras Are Trivial

Let [Formula: see text] be a fixed algebraically closed field of arbitrary characteristic, let [Formula: see text] be a finite dimensional self-injective [Formula: see text]-algebra, and let [Formula: see text] be an indecomposable non-projective left [Formula: see text]-module with finite dimension over [Formula: see text]. We prove that if [Formula: see text] is the Auslander–Reiten translation of [Formula: see text], then the versal deformation rings [Formula: see text] and [Formula: see text] (in the sense of F.M. Bleher and the second author) are isomorphic. We use this to prove that if [Formula: see text] is further a cluster-tilted [Formula: see text]-algebra, then [Formula: see text] is universal and isomorphic to [Formula: see text].

On representation dimension of tame cluster tilted algebras

The aim of this work is to study the representation dimension of cluster tilted algebras. We prove that the weak representation dimension of tame cluster tilted algebras is equal to three. We construct a generator module that reaches the weak representation dimension, unfortunately this module is not always a cogenerator. We show for which algebras this module gives the representation dimension.

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.

Extending silted algebras to cluster-tilted algebras

It is well known that the relation extensions of tilted algebras are cluster-tilted algebras. In this paper, we extend the result to silted algebras and prove that some extension of silted algebras are cluster-tilted algebras.

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.

Gentle m-Calabi-Yau tilted algebras

We prove that all gentle 2-Calabi-Yau tilted algebras are Jacobian, moreover their bound quiver can be obtained via block decomposition. For two related families, the m-cluster-tilted algebras of type A and A~, we prove that a module M is stable Cohen-Macaulay if and only if Ωm+1τM≃M.