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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].

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
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