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Traces on multiplier Hopf algebras. (English) Zbl 1216.16020
Quasitriangular ribbon Hopf algebras have a typical element with special properties in connection with the topology of knots and links. In the “finite-dimensional” case, the notion of a ribbon Hopf algebra is formulated in terms of grouplike elements. Moreover, in the “unimodular” finite-dimensional case, the ribbon structure determines all traces which are invariant for the antipode.
In the present paper, the author generalizes these algebraic structures to infinite-dimensional (multiplier) Hopf algebras. The problem is solved within the framework of multiplier Hopf algebras with integrals, the so-called “algebraic quantum groups”. This article lays the algebraic foundations to establish the existence of trace functions on infinite-dimensional (multiplier) Hopf algebras. By applying this theory to group-cograded multiplier Hopf algebras, the existence of group-traces on group-cograded multiplier Hopf algebras with possibly infinite-dimensional components is proved. The results as obtained by A. Virelizier are generalized in the case of finite-type Hopf group-coalgebras.
Reviewer: Li Fang (Hangzhou)

MSC:
16T05 Hopf algebras and their applications
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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