Minimal ${A}_{\infty}$-algebras of endomorphisms: The case of $d\mathbb{Z}$-cluster tilting objects

TL;DR

This paper explores the role of minimal A-infinity algebra structures in classifying derived endomorphism algebras of dZ-cluster tilting objects within algebraic triangulated categories, building on recent classification results.

Contribution

It emphasizes the importance of minimal A-infinity structures and enhanced obstruction theory in the proof of the Derived Auslander--Iyama Correspondence.

Findings

Minimal A-infinity structures are crucial in classifying derived endomorphism algebras.

Enhanced A-infinity obstruction theory aids in understanding algebraic classifications.

The work extends the understanding of dZ-cluster tilting objects in triangulated categories.

Abstract

The Derived Auslander--Iyama Corresponence, a recent result of the authors, provides a classification up to quasi-isomorphism of the derived endomorphism algebras of basic $d\mathbb{Z}$-cluster tilting objects in $\operatorname{Hom}$-finite algebraic triangulated categories in terms of a small amount of algebraic data. In this note we highlight the role of minimal $A_\infty$-algebra structures in the proof of this result, as well as the crucial role of the enhanced $A_\infty$-obstruction theory developed by the second-named author.