Compatibility with Cap-Products in Tsygan's Formality and Homological Duflo Isomorphism

Abstract

In this paper we prove, with details and in full generality, that the isomorphism induced on tangent homology by the Shoikhet-Tsygan formality L∞-quasi-isomorphism for Hochschild chains is compatible with cap-products. This is a homological analogue of the compatibility with cup-products of the isomorphism induced on tangent cohomology by Kontsevich formality L∞-quasi-isomorphism for Hochschild cochains. As in the cohomological situation our proof relies on a homotopy argument involving a variant of Kontsevich eye. In particular, we clarify the rôle played by the I-cube introduced in Calaque and Rossi (SIGMA 4, paper 060, 17 2008). Since we treat here the case of a most possibly general Maurer-Cartan element, not forced to be a bidifferential operator, we take this opportunity to recall the natural algebraic structures on the pair of Hochschild cochain and chain complexes of an A∞-algebra. In particular we prove that they naturally inherit the structure of an A∞-algebra with an A∞-(bi)module. © 2010 Springer.