BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras

A BiHom-associative algebra is a (nonassociative) algebra A endowed with two commuting multiplicative linear maps α,β : A → A such that α(a)(bc) = (ab)β(c), for all a,b,c ∈ 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).