Two-colored noncommutative Gerstenhaber formality and infinity Duflo isomorphism

Abstract

Using new configuration spaces, we give an explicit construction that extends Kontsevich's Lie-infinity quasi-isomorphism from polyvector fields to Hochschild cochains to a quasi-isomorphism of A-infinity algebras equipped with actions by homotopy derivations of the Lie algebra of polyvector fields. One may term this formality a formality of two-colored noncommutative Gerstenhaber homotopy algebras. In our result the action of polyvector fields by homotopy derivations of the wedge product on polyvector fields is not the adjoint action by the Schouten bracket, but a homotopy nontrivial and, in a sense, unique deformation of that action. As an application we give an explicit Duflo-type construction for Lie-infinity algebras that generalizes the Duflo-Kontsevich isomorphism between the Chevalley-Eilenberg cohomology of the symmetric algebra on a Lie algebra and the Chevalley-Eilenberg cohomology of the universal enveloping algebra of the Lie algebra.

16 pages

References in corpus (8)

• Operads and Motives in Deformation Quantization
• Operads, homotopy algebra and iterated integrals for double loop spaces
• Deformation theory of representations of prop(erad)s
• Bimodules and branes in deformation quantization
• Isomorphisme de Duflo et la cohomologie tangentielle
• Geometric realizations of the multiplihedron and its complexification
• OCHA and the swiss-cheese operad
• On the $A_{\infty }$-Formality conjecture

Cited by in corpus (5)

• The Homotopy Braces Formality Morphism
• OCHA and Leibniz Pairs, towards a Koszul duality
• Operads, configuration spaces and quantization
• On the compatibility between cup products, the Alekseev--Torossian connection and the Kashiwara--Vergne conjecture
• Grothendieck-Teichmueller group and Poisson cohomologies