Koszul duality in deformation quantization, I

TL;DR

This paper constructs a PBW algebra from polynomial Poisson structures on vector spaces, generalizing classical theorems and relating to quantum R-matrices, with a conjecture linking it to Kontsevich's star-algebra.

Contribution

It introduces a new algebraic construction with PBW property from polynomial Poisson structures, extending classical results and connecting to quantum algebra concepts.

Findings

Constructed algebra obeys PBW property for polynomial Poisson structures.

Generalizes PBW theorem to quadratic Poisson structures.

Provides a free resolution of the deformed algebra.

Abstract

Let $\alpha$ be a polynomial Poisson bivector on a finite-dimensional vector space $V$ over $\mathbb{C}$. Then Kontsevich [K97] gives a formula for a quantization $f\star g$ of the algebra $S(V)^*$. We give a construction of an algebra with the PBW property defined from $\alpha$ by generators and relations. Namely, we define an algebra as the quotient of the free tensor algebra $T(V^*)$ by relations $x_i\otimes x_j-x_j\otimes x_i=R_{ij}(\hbar)$ where $R_{ij}(\hbar)\in T(V^*)\otimes\hbar \mathbb{C}[[\hbar]]$, $R_{ij}=\hbar \Sym(\alpha_{ij})+\mathcal{O}(\hbar^2)$, with one relation for each pair of $i,j=1...\dim V$. We prove that the constructed algebra obeys the PBW property, and this is a generalization of the Poincar\'{e}-Birkhoff-Witt theorem. In the case of a linear Poisson structure we get the PBW theorem itself, and for a quadratic Poisson structure we get an object closely related to a quantum $R$-matrix on $V$. At the same time we get a free resolution of the deformed algebra (for an arbitrary $\alpha$). The construction of this PBW algebra is rather simple, as well as the proof of the PBW property. The major efforts should be undertaken to prove the conjecture that in this way we get an algebra isomorphic to the Kontsevich star-algebra.