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 $C\ltimes \Ext_C^2(DC,C)$ of $C$ by the $C$-$C$-bimodule $\Ext_C^2(DC,C)$. We give a construction for the quiver of the relation-extension algebra in case the quiver of $C$ has no oriented cycles. Our main result says that an algebra $\tilde{C}$ is cluster-tilted if and only if there exists a tilted algebra $C$ such that $\tilde{C}$ is isomorphic to the relation-extension of $C$.

14 pages

Cited by in corpus (1)

• On tilting modules over cluster-tilted algebras