Selecta Mathematica

, Volume 24, Issue 2, pp 1691–1728 | Cite as

An \(L_\infty \) algebra structure on polyvector fields

  • Boris Shoikhet


It is well-known that the Kontsevich formality (Kontsevich in Deformation quantization of Poisson manifolds, 2003) for Hochschild cochains of the polynomial algebra \(A=S(V^*)\) 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. The degree 2 component is given by the Schouten–Nijenhuis bracket, but all its higher even degree components are non-zero. 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.

Mathematics Subject Classification

53D55 46L65 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Wiskunde en Informatica, Campus MiddelheimUniversiteit AntwerpenAntwerpBelgium

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