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# A Note on the Antipode for Algebraic Quantum Groups

Published:2011-04-25
Printed: Jun 2012
• L. Delvaux,
Department of Mathematics, University of Hasselt, Agoralaan, B-3590 Diepenbeek, Belgium
• A. Van Daele,
Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium
• Shuanhong Wang,
Department of Mathematics, Southeast University, Nanjing 210096, China
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## Abstract

Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra. In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals. We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.
 Keywords: multiplier Hopf algebras, algebraic quantum groups, the antipode
 MSC Classifications: 16W30 - Coalgebras, bialgebras, Hopf algebras (See also 16S40, 57T05); rings, modules, etc. on which these act46L65 - Quantizations, deformations

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