Mutation of Auslander generators

TL;DR

This paper explores how to mutate Auslander generators in artin algebras with representation dimension three, resulting in new generators with derived equivalent endomorphism rings, and constructs infinite sets of such generators for specific algebras.

Contribution

It introduces a mutation process for Auslander generators that preserves derived equivalence of their endomorphism rings, expanding the understanding of their structure.

Findings

Mutation can produce new Auslander generators with derived equivalent endomorphism rings.

Applied to selfinjective algebras with radical cube zero, an infinite set of Auslander generators is constructed.

The method broadens the class of known Auslander generators in certain algebra families.

Abstract

Let $\Lambda$ be an artin algebra with representation dimension equal to three and $M$ an Auslander generator of $\Lambda$. We show how, under certain assumptions, we can mutate $M$ to get a new Auslander generator whose endomorphism ring is derived equivalent to the endomorphism ring of $M$. We apply our results to selfinjective algebras with radical cube zero of infinite representation type, where we construct an infinite set of Auslander generators.