The Tilting Theory of Contraction Algebras

TL;DR

This paper develops a comprehensive understanding of the derived equivalences of contraction algebras associated with cDV singularities, introducing new functors and describing their autoequivalence groups.

Contribution

It constructs the first known standard derived equivalences for contraction algebras and characterizes their autoequivalence groups using hyperplane arrangements.

Findings

Established a faithful group action on the derived category

Identified all standard equivalences as compositions of constructed functors

Provided a complete description of the autoequivalence group

Abstract

To every minimal model of a complete local isolated cDV singularity Donovan--Wemyss associate a finite dimensional symmetric algebra known as the contraction algebra. We construct the first known standard derived equivalences between these algebras and then use the structure of an associated hyperplane arrangement to control the compositions, obtaining a faithful group action on the bounded derived category. Further, we determine precisely those standard equivalences which are induced by two-term tilting complexes and show that any standard equivalence between contraction algebras (up to algebra isomorphism) can be viewed as the composition of our constructed functors. Thus, for a contraction algebra, we obtain a complete picture of its derived equivalence class and, in particular, of its derived autoequivalence group.