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Multiplier Hopf algebras and duality. (English) Zbl 0872.17008

Budzyński, Robert (ed.) et al., Quantum groups and quantum spaces. Lectures delivered during the minisemester, Warsaw, Poland, December 1, 1995. Warszawa: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 40, 51-58 (1997).
Summary: We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups).
The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf *-algebra with positive invariant functionals, then also the dual is a multiplier Hopf *-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of \(C^*\)-algebras.
For the entire collection see [Zbl 0865.00041].

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
22D35 Duality theorems for locally compact groups
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