On unitality conditions for Hom-associative algebras

TL;DR

This paper explores different unitality conditions in hom-associative algebras, analyzing their properties, embedding possibilities, and connections to associative algebras, with implications for algebraic structure and classification.

Contribution

It introduces and compares classical and weak unitality conditions, investigates embedding issues, and characterizes weakly unital hom-associative algebras with bijective twisting maps.

Findings

Classical unitality is stronger than weak unitality.

Associativity relates to the image of the twisting map.

Weakly unital hom-associative algebras with bijective twistings are twisted associative algebras.

Abstract

In hom-associative structures, the associativity condition $(xy)z=x(yz)$ is twisted to $\alpha(x)(yz) = (xy)\alpha(z)$, with $\alpha$ a map in the appropriate category. In the present paper, we consider two different unitality conditions for hom-associative algebras. The first one, existence of a unit in the classical sense, is stronger than the second one, which we call weak unitality. We show associativity conditions connected to the size of the image of the twisting map for unital hom-associative algebras. Also the problem of embedding arbitrary hom-associative algebras into unital or weakly unital ones is investigated. Finally, we show that weakly unital hom-associative algebras with bijective twisting map are twisted versions of associative algebras.