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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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