  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.
