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

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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.

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Authors and Affiliations

  1. Institut Camille Jordan, Université de Lyon, Université Lyon 1, CNRS, UMR5208, 43 blvd du 11 novembre 1918, 69622, Villeurbanne-Cedex, France

    Damien Calaque

  2. Department of Mathematics, ETH Zurich, 8092, Zurich, Switzerland

    Carlo A. Rossi

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  1. Damien Calaque
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  2. Carlo A. Rossi
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Correspondence to Damien Calaque.

Additional information

The results of this paper were mainly obtained when D.C. was working in ETH (on leave of absence from Université Lyon 1). His research was fully supported by the European Union thanks to a Marie Curie Intra-European Fellowship (contract number MEIF-CT-2007-042212).

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Calaque, D., Rossi, C.A. Compatibility with Cap-Products in Tsygan’s Formality and Homological Duflo Isomorphism. Lett Math Phys 95, 135–209 (2011). https://doi.org/10.1007/s11005-010-0451-z

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  • DOI: https://doi.org/10.1007/s11005-010-0451-z

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