An \(L_\infty \) algebra structure on polyvector fields
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Abstract
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 46L65Preview
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References
- 1.Goldman, W., Millson, J.: The deformation theory of representations of fundamental groups of compact Kahler manifolds. Publ. Math. lI.H.E.S. 67, 43–96 (1988)CrossRefzbMATHGoogle Scholar
- 2.Khoroshkin, A., Merkulov, S., Willwacher, T.: On quantizable odd Lie bialgebras. (2015). archive preprint arXiv:1512.04710
- 3.Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2003)Google Scholar
- 4.Merkulov, S., Willwacher, T.: An explicit two step quantization of Poisson structures and Lie bialgebras. (2016). archive preprint arXiv:1612.00368
- 5.Merkulov, S., Wilwacher, T.: Deformation theory of Lie bialgebra properads. (2016). archive preprint arXiv:1605.01282
- 6.Tamarkin, D.: Another proof of M. Kontsevich formality theorem. (1998). Preprint math arXiv:math/9803025
- 7.Willwacher, T.: M. Kontsevich’s graph complex and the Grothendieck–Teichmller Lie algebra. Invent. Math. 200(3), 671–760 (2015)MathSciNetCrossRefzbMATHGoogle Scholar