BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras

TL;DR

This paper introduces and explores new algebraic structures called BiHom-associative, BiHom-Lie, and BiHom-bialgebras, extending classical algebra concepts with two commuting linear maps, and discusses their properties and constructions.

Contribution

It presents the definitions and foundational properties of BiHom-Lie algebras and BiHom-bialgebras, expanding the framework of Hom-type algebras with categorical approaches.

Findings

Defined BiHom-Lie algebras and BiHom-bialgebras.

Established basic properties and constructions.

Explored representations and tensor product structures.

Abstract

A BiHom-associative algebra is a (nonassociative) algebra $A$ endowed with two commuting multiplicative linear maps $\alpha,\beta\colon A\rightarrow A$ such that $\alpha (a)(bc)=(ab)\beta (c)$, for all $a, b, c\in A$. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).