A conjecture of Bavula on homomorphisms of the Weyl algebra

TL;DR

This paper investigates Bavula's conjecture on endomorphisms of Weyl algebras, proving it for the case n=1, disproving it for higher dimensions, and extending similar ideas to symplectic Poisson algebras.

Contribution

The paper proves Bavula's conjecture for the Weyl algebra in one dimension, shows it does not hold in higher dimensions, and establishes an analogous result for symplectic Poisson algebras.

Findings

Bavula's conjecture holds for A_1

The conjecture is false for A_n when n > 1

An analogue of the conjecture is proven for symplectic Poisson algebras

Abstract

In the paper {\em The inversion formulae for automorphisms of polynomial algebras and differential operators in prime characteristic}, J. Pure Appl. Algebra 212 (2008), no. 10, 2320-2337, see also arXiv:math/0604477, Vladimir Bavula states the following Conjecture: (BC) Any endomorphism of a Weyl algebra (in a finite characteristic case) is a monomorphism. The purpose of this preprint is to prove BC for $A_1$, show that BC is wrong for $A_n$ when $n > 1$, and prove an analogue of $BC$ for symplectic Poisson algebras.