An explicit formula for PBW quantization

Abstract

Let $k$ be a field of characteristic zero, $\g$ a $k$-Lie algebra, $e:S\g@>>>U\g$ the symmetrization map. The PBW quantization is the one parameter family of associative products: $$x{\star }_{t}y=\sum _{p=0}^{\infty }{B}_p(x,y)t^p(t\in k)$$ where $B_p$ is the homogeneous component of degree $-p$ of the map $B:S\g\otimes_kS\g@>>>S\g$, $B(x,y)=e^{-1}(exey)$. In this paper we give an explicit formula for $B$. As an application, we prove that for each $p\ge 0$, $B_p$ is a bidifferential operator of order $\le p$.

7 pages, AmsTex; more typos corrected, an ambiguous definition made precise