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A note on the antipode for algebraic quantum groups. (English) Zbl 1252.16028
D. E. Radford [Am. J. Math. 98, 333-355 (1976; Zbl 0332.16007)] proved a formula for the fourth power of the antipode of a finite dimensional Hopf algebra \(H\) in terms of the inner actions determined by the distinguished grouplike elements of \(H\) and its dual \(H^*\) on \(H\). M. Beattie, D. Bulacu and B. Torrecillas [J. Algebra 307, No. 1, 330-342 (2007; Zbl 1115.16016)] extended this formula to the case where \(H\) is a Hopf algebra with non-zero integrals. – In the paper under review, the formula is extended even more, to the case of regular multiplier Hopf algebras with integrals.

MSC:
16T05 Hopf algebras and their applications
16T20 Ring-theoretic aspects of quantum groups
17B37 Quantum groups (quantized enveloping algebras) and related deformations
46L65 Quantizations, deformations for selfadjoint operator algebras
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