# On the hom-associative Weyl algebras

## TL;DR

This paper demonstrates that the classical Weyl algebra can be nontrivially deformed into a hom-associative algebra, preserving some properties while altering others, and establishes a hom-associative analogue of the Dixmier conjecture.

## Contribution

It introduces a novel deformation of the Weyl algebra into a hom-associative algebra and proves an analogue of the Dixmier conjecture within this framework.

## Key findings

- Weyl algebra admits nontrivial hom-associative deformations.
- Hom-associative Weyl algebras have only isomorphisms as homomorphisms.
- Hom-associative deformation induces a hom-Lie algebra structure.

## Abstract

The first (associative) Weyl algebra is formally rigid in the classical sense. In this paper, we show that it can however be formally deformed in a nontrivial way when considered as a so-called hom-associative algebra, and that this deformation preserves properties such as the commuter, while deforming others, such as the center, power associativity, the set of derivations, and some commutation relations. We then show that this deformation induces a formal deformation of the corresponding Lie algebra into what is known as a hom-Lie algebra, when using the commutator as bracket. We also prove that all homomorphisms between any two purely hom-associative Weyl algebras are in fact isomorphisms. In particular, all endomorphisms are automorphisms in this case, hence proving a hom-associative analogue of the Dixmier conjecture to hold true.