Cluster-tilted algebras as trivial extensions

Abstract

Given a finite dimensional algebra C (over an algebraically closed field) of global dimension at most two, we define its relation-extension algebra to be the trivial extension C ⋉ Ext 2 C (DC, C) of C by the C-C-bimodule Ext 2 C (DC, C). We give a construction for the quiver of the relationextension algebra in case the quiver of C has no oriented cycles. Our main result says that an algebraC is cluster-tilted if and only if there exists a tilted algebra C such thatC is isomorphic to the relation-extension of C.

Induced and coinduced modules over cluster-tilted algebras

“…Relation extensions where introduced in [3]. In the special case where C is a tilted algebra, we have the following result.…”

“…The corresponding cluster-tilted algebra B is of typeà (3,2) and it is given by the quiver with relations below. , the projective B-module at vertex 2, as in Theorem 4.2.1.…”

“…Tilted algebras are the endomorphism algebras of tilting modules over hereditary algebras, whereas clustertilted algebras are the endomorphism algebras of cluster-tilting objects over cluster categories of hereditary algebras. This similarity in the two definitions lead to the following precise relation between tilted and cluster-tilted algebras, which was established in [3].…”

The strong global dimension of piecewise hereditary algebras

“…On the other hand, these algebras are used in many parts of representation theory. For instance, in order to develop the representation theory of other classes of algebras such as the selfinjective algebras (see [28]) or the cluster tilted algebras (see [4]), in order to investigate singularity theory (see [20]), or in order to categorify cluster algebras (see [8]). …”

Mutating seeds: types \mathbb{A} and \widetilde{\mathbb{A}}.

“…Their tree-like shape comes from the classification of the tilted algebras of the linearly oriented quivers of type A n , see [29]. The additional arrows from right to left children arise because of [2,12]. Note that since the original tree τ is planar, we may use for T the terminology of planar binary trees: subtree at a vertex (which is a precontinuant tree), left and right child, parent.…”