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)

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. Let us consider a Hopf algebra $A$ such that its Jacobson radical $J$ is a nilpotent Hopf ideal and $H:=A/J$ is a semisimple algebra. We prove that the canonical projection of $A$ on $H$ has a section which is an $H$ –colinear algebra map. Furthermore, if $H$ is cosemisimple too, then we can choose this section to be an $(H,H)$ –bicolinear algebra morphism. This fact allows us to describe $A$ as a ‘generalized bosonization’ of a certain algebra $R$ in the category of Yetter–Drinfeld modules over $H$ . As an application we give a categorical proof of Radford’s result about Hopf algebras with projections. We also consider the dual situation. Let $A$ be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of $H$ into $A$ which is an $H$ –linear coalgebra morphism. Furthermore, if $H$ is semisimple too, then we can choose this retraction to be an $(H,H)$ –bilinear coalgebra morphism. Then, also in this case, we can describe $A$ as a ‘generalized bosonization’ of a certain coalgebra $R$ in the category of Yetter–Drinfeld modules over $H$ .

The purpose of this paper is to study Rota–Baxter operators for BiHom-associative algebras. Moreover, we introduce and discuss the properties of the notions of BiHom-(tri)dendriform algebra, BiHom-Zinbiel algebra and BiHom-quadri-algebra. We construct the