A Note on the Antipode for Algebraic Quantum Groups
Printed: Jun 2012
A. Van Daele,
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.
multiplier Hopf algebras, algebraic quantum groups, the antipode
16W30 - Coalgebras, bialgebras, Hopf algebras (See also 16S40, 57T05); rings, modules, etc. on which these act
46L65 - Quantizations, deformations