Endomorphisms of quantized Weyl algebras

TL;DR

This paper explores the connection between automorphisms of Weyl algebras and polynomial symplectomorphisms using quantized Weyl algebras at roots of unity, aiming to address a conjecture by Belov-Kanel and Kontsevich.

Contribution

It proposes a novel approach to the conjecture by analyzing endomorphisms of quantized Weyl algebras at roots of unity.

Findings

Provides a framework linking Weyl algebra automorphisms to symplectomorphisms

Suggests methods for approaching the conjecture using quantized algebra structures

Offers insights into the structure of automorphism groups at roots of unity

Abstract

Belov-Kanel and Kontsevich conjectured that the group of automorphisms of the n'th Weyl algebra and the group of polynomial symplectomorphisms of C^2 are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity.