# The graph of a Weyl algebra endomorphism

## TL;DR

This paper investigates the structure of endomorphisms of Weyl algebras using bimodules, proving the simplicity of their graphs and linking this to the Dixmier conjecture, while also developing algorithms for invertibility detection.

## Contribution

It introduces the graph of an endomorphism as a simple bimodule and connects its properties to the Dixmier conjecture, also providing a non-commutative Groebner basis algorithm.

## Key findings

- The graph of an endomorphism of a Weyl algebra is a simple bimodule.
- The tensor product of the dual graph and the graph relates to the Dixmier conjecture.
- A non-commutative Groebner basis algorithm is developed for invertibility detection.

## Abstract

Endomorphisms of Weyl algebras are studied using bimodules. Initially, for a Weyl algebra over a field of characteristic zero, Bernstein's inequality implies that holonomic bimodules finitely generated from the right or left form a monoidal category. The most important bimodule in this paper is the graph of an endomorphism. We prove that the graph of an endomorphism of a Weyl algebra over a field of characteristic zero is a simple bimodule. The simplicity of the tensor product of the dual graph and the graph is equivalent to the Dixmier conjecture. It is also shown how the graph construction leads to a non-commutative Groebner basis algorithm for detecting invertibility of an endomorphism for Weyl algebras and computing the inverse over arbitrary fields.