Koszul duality in deformation quantization and Tamarkin's approach to Kontsevich formality

Details Koszul duality in deformation quantization and Tamarkin's approach to Kontsevich formality Koszul duality in deformation quantization and Tamarkin's approach to Kontsevich formality

2008-05-02

arxiv.org/abs/0805.0174v2

Advances in Math. 224(2010) 731-771

Mathematics

Quantum Algebra

49 pages, 2 figures

Scientific paper

Let $\alpha$ be a quadratic Poisson bivector on a vector space $V$. Then one can also consider $\alpha$ as a quadratic Poisson bivector on the vector space $V^*[1]$. Fixed a universal deformation quantization (prediction some weights to all Kontsevich graphs [K97]), we have deformation quantization of the both algebras $S(V^*)$ and $\Lambda(V)$. These are graded quadratic algebras, and therefore Koszul algebras. We prove that for some universal deformation quantization, independent on $\alpha$, these two algebras are Koszul dual. We characterize some deformation quantizations for which this theorem is true in the framework of the Tamarkin's theory [T1].

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