Nikshych, Dmitri; Vainerman, Leonid Algebraic versions of a finite-dimensional quantum groupoid. (English) Zbl 1032.46537 Caenepeel, Stefaan (ed.) et al., Hopf algebras and quantum groups. Proceedings of the Brussels conference, Brussels, Belgium. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 209, 189-220 (2000). Summary: We establish the equivalence of three approaches to the theory of finite-dimensional quantum groupoids. These are the generalized Kac algebras of T. Yamanouchi [J. Algebra 163, 9-50 (1994; Zbl 0830.46047)], the weak Kac algebras, i.e., the weak \(C^*\)-Hopf algebras introduced by G. Böhm, F. Nill and K. Szlachányi [J. Algebra 221, 385-438 (1999; Zbl 0949.16037)] which have an involutive antipode, and the Kac bimodules. The latter are an algebraic version of the Hopf bimodules of J.-M. Vallin [J. Oper. Theory 35, 39-65 (1996; Zbl 0849.22002)]. We also study the structure and construct examples of finite-dimensional quantum groupoids.For the entire collection see [Zbl 0958.00024]. Cited in 20 Documents MSC: 46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.) 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 46L05 General theory of \(C^*\)-algebras 46L65 Quantizations, deformations for selfadjoint operator algebras Keywords:generalized Kac algebras; weak Kac algebras; weak \(C^*\)-Hopf algebras; Kac bimodules Citations:Zbl 0830.46047; Zbl 0949.16037; Zbl 0849.22002 PDFBibTeX XMLCite \textit{D. Nikshych} and \textit{L. Vainerman}, Lect. Notes Pure Appl. Math. 209, 189--220 (2000; Zbl 1032.46537) Full Text: arXiv