Nearly associative and nearly Hom-associative algebras and bialgebras

TL;DR

This paper explores nearly associative and nearly Hom-associative algebras, establishing their properties, classifications, and connections with bialgebras, and introduces related bimodules, matched pairs, and identities.

Contribution

It provides the first comprehensive study of nearly associative and nearly Hom-associative algebras, including classifications and their relation to bialgebras.

Findings

Nearly associative algebras are Lie-admissible.

Two-dimensional nearly associative algebras are classified.

Bimodules, matched pairs, and Manin triples are characterized and linked to nearly associative bialgebras.

Abstract

Basic definitions and properties of nearly associative algebras are described. Nearly associative algebras are proved to be Lie-admissible algebras. Two-dimensional nearly associative algebras are classified, and its main classes are derived. The bimodules, matched pairs and Manin triple of a nearly associative algebras are derived and their equivalence with nearly associative bialgebras is proved. Basic definitions and properties of nearly Hom-associative algebras are described. Related bimodules and matched pairs are given, and associated identities are established.