Some articles on bialgebroid:

Hopf Algebras - Related Concepts
... A left Hopf algebroid (H,R) is a left bialgebroid together with an antipode the bialgebroid (H,R) consists of a total algebra H and a base algebra R and two mappings, an algebra homomorphism s R → H ... In particular a left bialgebroid (H,R) has an R-R-bimodule structure on H which prefers the left side as follows for all h in H ... Additionally the bialgebroid (H,R) must satisfy Δ(ab) = Δ(a)Δ(b) for all a,b in H, and a condition to make sure this last condition makes sense every image point Δ(a ...