Paradigm of Nonassociative Hom-algebras and Hom-superalgebras

TL;DR

This paper surveys the development of nonassociative Hom-algebra and Hom-superalgebra structures, highlighting their defining twisted identities and recent research progress in the field.

Contribution

It provides a comprehensive overview of various Hom-algebra structures and their developments, emphasizing the role of homomorphism-twisted identities.

Findings

Overview of Hom-associative, Hom-Lie, and Hom-Poisson algebras

Discussion of $ ext{Z}_2$-graded Hom-superalgebras

Summary of recent research trends in Hom-algebra structures

Abstract

The aim of this paper is to give a survey of nonassociative Hom-algebra and Hom-superalgebra structures. The main feature of these algebras is that the identities defining the structures are twisted by homomorphisms. We discuss Hom-associative algebras, Hom-Flexible algebras, Hom-Lie algebras, $G$-hom-associative algebras, Hom-Poisson algebras, Hom-alternative algebras and Hom-Jordan algebras and $\mathbb{Z}_2$-graded versions. We give an overview of the development of Hom-algebras structures which have been intensively investigated recently.