On Hom type algebras

TL;DR

This paper systematically explores all possible Hom-type algebra structures in Lie and associative categories, analyzing their relations and conditions for associativity, especially focusing on unital cases.

Contribution

It provides a comprehensive enumeration of Hom-algebra variants and establishes conditions linking Hom-associative and associative algebras, including classification of unital types.

Findings

Enumerated all Hom-Lie and Hom-associative algebra structures

Identified conditions under which Hom-associative algebras are actually associative

Classified implications between different unital Hom-associative algebra types

Abstract

Hom-algebras are generalizations of algebras obtained using a twisting by a linear map. But there is a priori a freedom on where to twist. We enumerate here all the possible choices in the Lie and associative categories and study the relations between the obtained algebras. The associative case is richer since it admits the notion of unit element. We use this fact to find sufficient conditions for hom-associative algebras to be associative and classify the implications between the hom-associative types of unital algebras.