A note on the differential calculus of Hochschild theory for $ A_{\infty} $-algebras

Abstract

We show by constructing explicit homotopy operators that the Hochschild (co)homology of an $ A_{\infty} $-algebra of Stasheff admits a differential calculus structure. As an application, we reproduce a result of Tradler which says that the Hochschild cohomology of a cyclic $ A_\infty $-algebra admits a Batalin-Vilkovisky algebra structure.

Keywords

• differential calculus
• $ a_{\infty} $-algebra
• cyclic $ a_{\infty} $-algebra
• hochschild (co)homology
• batalin-vilkovisky algebra