The Derived Deligne Conjecture

Derived $A_{\mathrm{\infty }}$-algebras have a wealth of theoretical advantages over regular $A_{\mathrm{\infty }}$-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 $A_{\mathrm{\infty }}$ 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 $A_{\mathrm{\infty }}$-algebra, obtaining new and rigorous versions of the Deligne conjecture.