The Derived Deligne Conjecture

TL;DR

This paper introduces a new framework using brace algebras on operads to study derived $A_ inf$-algebras, enabling generalizations of the Deligne conjecture and improving their practical handling.

Contribution

It develops a novel brace algebra framework on operads that facilitates the study of derived $A_ inf$-algebras and generalizes the Deligne conjecture.

Findings

New conceptual framework for derived $A_ inf$-algebras

Generalized Lie algebra structure on Hochschild complex

Rigorous versions of the Deligne conjecture

Abstract

Derived $A_\infty$-algebras have a wealth of theoretical advantages over regular $A_\infty$-algebras. However, due to their bigraded nature, in practice they are often unwieldy to work with. We develop a framework involving brace algebras on operads which allows us to study derived $A_\infty$ algebras in a new conceptual context. One particular advantage is that this construction allows us to generalize the Lie algebra structure on the Hochschild complex of an $A_\infty$-algebra, obtaining new and rigorous versions of the Deligne conjecture.