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

TL;DR

This paper explores the Koszul duality between deformed symmetric and exterior algebras arising from quadratic Poisson structures, using Tamarkin's approach to establish universal deformation quantizations that preserve this duality.

Contribution

It demonstrates that for certain universal deformation quantizations, the deformed symmetric and exterior algebras are Koszul dual, extending Tamarkin's framework to quadratic Poisson structures.

Findings

Deformed symmetric and exterior algebras are Koszul dual under certain quantizations.

Characterization of deformation quantizations preserving Koszul duality.

Application of Tamarkin's theory to quadratic Poisson bivectors.

Abstract

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].