Lifting of Polynomial Symplectomorphisms and Deformation Quantization

TL;DR

This paper investigates how polynomial symplectomorphisms can be lifted to Weyl algebra automorphisms using approximation techniques, addressing a key problem in deformation quantization and algebraic geometry.

Contribution

It adapts Anick's approximation result to symplectomorphisms and formulates the lifting problem within the context of deformation quantization.

Findings

Reproved Anick's approximation theorem for polynomial automorphisms.

Formulated the lifting problem for symplectomorphisms in deformation quantization.

Connected the lifting problem to major open problems in algebraic geometry.

Abstract

We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. We utilize -- and reprove -- D. Anick's fundamental result on approximation of polynomial automorphisms, adapt it to the case of symplectomorphisms, and formulate the lifting problem. The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory.