An $L_{\infty }$ algebra structure on polyvector fields

Abstract

It is well-known that the Kontsevich formality [K97] for Hochschild cochains of the polynomial algebra $A=S(V^{\ast })$ fails if the vector space $V$ is infinite-dimensional. In the present paper, we study the corresponding obstructions. We construct an $L_{\infty }$ structure on polyvector fields on $V$ having the even degree Taylor components, with the degree 2 component given by the Schouten-Nijenhuis bracket, but having as well higher non-vanishing Taylor components. We prove that this $L_{\infty }$ algebra is quasi-isomorphic to the corresponding Hochschild cochain complex. We prove that our $L_{\infty }$ algebra is $L_{\infty }$ quasi-isomorphic to the Lie algebra of polyvector fields on $V$ with the Schouten-Nijenhuis bracket, if $V$ is finite-dimensional.

40 pages, v7. The paper is essentially edited. The exposition in Section 1 is improved. Appendices A and B are added

References in corpus (4)

• M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra
• Kontsevich formality and PBW algebras
• On quantizable odd Lie bialgebras
• An explicit two step quantization of Poisson structures and Lie bialgebras

Cited by in corpus (10)

• Exotic automorphisms of the Schouten algebra of polyvector fields
• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• Deformation quantization and the Gerstenhaber structure on the homology of knot spaces
• Grothendieck-Teichmueller group, operads and graph complexes: a survey
• On quantizable odd Lie bialgebras
• Operads, configuration spaces and quantization
• Formality of a higher-codimensional Swiss-Cheese operad
• An explicit two step quantization of Poisson structures and Lie bialgebras
• A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids
• Boardman-Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules