  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).
