Hom-associative Ore extensions and weak unitalizations

TL;DR

This paper introduces hom-associative Ore extensions, explores their properties, constructs various hom-associative algebraic structures, and demonstrates how to embed these algebras into weakly unital versions, broadening algebraic frameworks.

Contribution

It defines hom-associative Ore extensions, constructs hom-associative quantum planes and Weyl algebras, and introduces the concept of weak unitalization for hom-associative algebras.

Findings

Hom-associative Ore extensions characterized by necessary and sufficient conditions.

Constructed hom-associative quantum planes, enveloping algebras, and Weyl algebras.

Proved the simplicity of certain hom-associative Weyl algebras.

Abstract

We introduce hom-associative Ore extensions as non-unital, non-associative Ore extensions with a hom-associative multiplication, and give some necessary and sufficient conditions when such exist. Within this framework, we construct families of hom-associative quantum planes, universal enveloping algebras of a Lie algebra, and Weyl algebras, all being hom-associative generalizations of their classical counterparts, as well as prove that the latter are simple. We also provide a way of embedding any multiplicative hom-associative algebra into a multiplicative, weakly unital hom-associative algebra, which we call a weak unitalization.