Representation theory of partial relation extensions

TL;DR

This paper explores the representation theory of partial relation extensions of finite-dimensional algebras, revealing their structural properties and introducing methods to systematically construct such extensions.

Contribution

It introduces the concept of partial relation extensions, analyzes their properties when C is tilted, and provides a systematic construction method.

Findings

Complete slices in C embed as local slices in extensions

Partial relation extensions form an intermediate class between tilted and cluster tilted algebras

A systematic construction method for partial relation extensions is developed

Abstract

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.