Hopf algebroids with bijective antipodes: axioms, integrals, and duals

Motivated by the study of depth 2 Frobenius extensions, we introduce a new notion of Hopf algebroid. It is a 2-sided bialgebroid with a bijective antipode which connects the two, left and right handed, structures. While all the interesting examples of the Hopf algebroid of J.H. Lu turn out to be Hopf algebroids in the sense of this paper, there exist simple examples showing that our definition is not a special case of Luʹs. Our Hopf algebroids, however, belong to the class of ×L-Hopf algebras proposed by P. Schauenburg. After discussing the axioms and some examples, we study the theory of non-degenerate integrals in order to obtain duals of Hopf algebroids.