An explicit formula for PBW quantization

Details An explicit formula for PBW quantization An explicit formula for PBW quantization

2000-01-24

arxiv.org/abs/math/0001127v3

Comm. Algebra 30, (2002) 1705-1713

Mathematics

Quantum Algebra

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

Scientific paper

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: $$ x\star_t y=\sum_{p=0}^\infty B_p(x,y)t^p\qquad (t\in k) $$ 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 $\le p$.

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