Uniqueness of $A_\infty$-structures and Hochschild cohomology

TL;DR

This paper explores conditions under which a differential graded algebra's $A_ abla$-structure is unique up to quasi-isomorphism, utilizing Hochschild cohomology, and extends these results to derived $A_ abla$-algebras.

Contribution

It provides a sufficient condition for the uniqueness of $A_ abla$-structures using Hochschild cohomology and extends this to derived $A_ abla$-algebras.

Findings

Provides a criterion for $A_ abla$-structure uniqueness

Extends Hochschild cohomology to derived $A_ abla$-algebras

Establishes conditions for quasi-isomorphism uniqueness

Abstract

This paper investigates if a differential graded algebra can have more than one $A_\infty$-structure extending the given differential graded algebra structure. We give a sufficient condition for uniqueness of such an $A_\infty$-structure up to quasi-isomorphism using Hochschild cohomology. We then extend this condition to Sagave's notion of derived $A_\infty$-algebras after introducing a notion of Hochschild cohomology that applies to this.