Representation theory of partial relation extensions
Let $C$ be a finite dimensional algebra of global dimension at most two. A partial relation extension is any trivial extension of $C$ by a direct summand of its relation $C$-$C$-bimodule. When $C$ is a tilted algebra, this construction provides an intermediate class of algebras between tilted and cluster tilted algebras. The text investigates the representation theory of partial relation extensions. When $C$ is tilted, any complete slice in the Auslander–Reiten quiver of $C$ embeds as a local slice in the Auslander–Reiten quiver of the partial relation extension. Moreover, a systematic way of producing partial relation extensions is introduced by considering direct sum decompositions of the potential arising from a minimal system of relations of $C$.