Derived autoequivalences from periodic algebras

TL;DR

This paper introduces a new method for constructing autoequivalences of derived categories of symmetric algebras using projective modules with periodic endomorphism algebras, expanding on previous work and connecting to known autoequivalence frameworks.

Contribution

It generalizes existing autoequivalence constructions for derived categories of symmetric algebras through a novel approach involving periodic endomorphism algebras.

Findings

Constructs autoequivalences from periodic endomorphism algebras.

Shows compositions and inverses are controlled by algebra resolutions.

Demonstrates autoequivalences can be built from non-Morita equivalent algebra derived equivalences.

Abstract

We present a construction of autoequivalences of derived categories of symmetric algebras based on projective modules with periodic endomorphism algebras. This construction generalises autoequivalences previously constructed by Rouquier-Zimmermann and is related to the autoequivalences of Seidel-Thomas and Huybrechts-Thomas. We show that compositions and inverses of these equivalences are controlled by the resolutions of our endomorphism algebra and that each autoequivalence can be obtained by certain compositions of derived equivalences between algebras which are in general not Morita equivalent.