The higher-order hom-associative Weyl algebras

TL;DR

This paper demonstrates that higher-order Weyl algebras can be nontrivially deformed into hom-associative algebras, revealing new structural properties and linking to major conjectures in algebra.

Contribution

It introduces nontrivial deformations of higher-order Weyl algebras as hom-associative algebras and classifies these structures, connecting to the Dixmier and Jacobian conjectures.

Findings

Higher-order Weyl algebras can be deformed into hom-associative algebras.

Hom-associative Weyl algebras are simple and contain no zero divisors.

Classification of all hom-associative Weyl algebras up to isomorphism.

Abstract

We show that the higher-order Weyl algebras over a field of characteristic zero, which are formally rigid as associative algebras, can be formally deformed in a nontrivial way as hom-associative algebras. We also show that these hom-associative Weyl algebras arise naturally as hom-associative iterated differential polynomial rings, that they contain no zero divisors, are power-associative only when associative, and that they are simple. We then determine their commuters, nuclei, centers, and derivations. Last, we classify all hom-associative Weyl algebras up to isomorphism and conjecture that all nonzero homomorphisms between any two isomorphic hom-associative Weyl algebras are isomorphisms. The latter conjecture turns out to be stably equivalent to the Dixmier Conjecture, and hence also to the Jacobian Conjecture.