On tilting modules over cluster-tilted algebras

TL;DR

This paper investigates how tilting modules over cluster-tilted algebras relate to tilting objects in associated cluster categories, providing new insights into exchange relations and endomorphism algebras.

Contribution

It demonstrates that tilting modules lift to tilting objects in cluster categories and explores their exchange relations and endomorphism algebras.

Findings

Tilting modules over cluster-tilted algebras lift to cluster categories.

Describes exchange relations for tilting modules.

Identifies tilting modules with cluster-tilted endomorphism algebras.

Abstract

In this paper, we show that the tilting modules over a cluster-tilted algebra $A$ lift to tilting objects in the associated cluster category $\mathcal{C}_H$. As a first application, we describe the induced exchange relation for tilting $A$-modules arising from the exchange relation for tilting object in $\mathcal{C}_H$. As a second application, we exhibit tilting $A$-modules having cluster-tilted endomorphism algebras.