Short star-products on filtered quantizations

Pavel Etingof

Let $A$ be a filtered Poisson algebra with Poisson bracket $\lbrace{,\rbrace}$ of degree $-2$. A {\it star product} on $A$ is an associative product $*: A\otimes A\to A$ given by $$a*b=ab+\sum_{i\ge 1}C_i(a,b),$$ where $C_i$ has degree $-2i$ and $C_1(a,b)-C_1(b,a)=\lbrace{a,b\rbrace}$. We call the product * is {\it even} if $C_i(a,b)=(-1)^iC_i(b,a)$ for all $i$, and call it {\it short} if $C_i(a,b)=0$ whenever $i>{\rm min}({\rm deg}(a), {\rm deg}(b))$.

http://scgp.stonybrook.edu/video_portal/video.php?id=3912

Simons- Workshop: Vertex Algebras and Gauge Theory