Algebraic Versions of a Finite‐Dimensional Quantum Groupoid
We establish the equivalence of three approaches to the theoiy of finite dimensional quantum groupoids. These are the generalized Kac algebras of T. Yamanouchi, the weak Kac algebras, i.e., the weak C x‐Hopf algebras introduced by G. Böhm, F. Nill, and K. Szlachányi which hase an involutive antipode, and the Kac bimodules. The latter are an algebraic version of the Hopf bimodules of J.‐M. Vallin. We also study the structure and construct examples of finite dimensional quantum groupoids.