On the local structure of quantizations in characteristic p

TL;DR

This paper demonstrates that the local structure of central quantizations of affine Poisson varieties in characteristic p resembles a tensor product of a Weyl algebra and a local Poisson algebra, extending characteristic 0 results.

Contribution

It establishes a positive characteristic analogue of slice algebra structures in quantizations, showing local isomorphisms to tensor products involving Weyl algebras.

Findings

Completion of the quantization at a closed point is isomorphic to a tensor product of a Weyl algebra and a local Poisson algebra.

The result extends known characteristic 0 structures to characteristic p.

Provides a new understanding of local quantization structures in positive characteristic.

Abstract

Let $A$ be a central quantization of an affine Poisson variety $X$ over a field of characteristic $p>0.$ We show that the completion of $A$ with respect to a closed point $y\in X$ is isomorphic to the tensor product of the Weyl algebra with a local Poisson algebra. This result can be thought of as a positive characteristic analogue of results of Losev and Kaledin about slice algebras of quantizations in characteristic 0.