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Radford’s \(S^4\) formula for co-Frobenius Hopf algebras. (English) Zbl 1115.16016
Let \(H\) be a co-Frobenius Hopf algebra, i.e., a Hopf algebra with non-zero left (or right) integrals, over a field. Let \(S\) be the antipode of \(H\), and let \(\alpha\), \(g\) be the distinguished grouplike elements of \(H^*\), \(H\), respectively. The authors prove that \(S^4(h)=g(\alpha\rightharpoonup h\leftharpoonup\alpha ^{-1})g^{-1}\), where \(\rightharpoonup\) and \(\leftharpoonup\) denote the usual left and right actions of \(H^*\) on \(H\). This extends the result of D. E. Radford [Am. J. Math. 98, 333-355 (1976; Zbl 0332.16007)] for finite dimensional \(H\).
Several proofs have been given to Radford’s result by other authors, some of them in situations more general than finite dimensional Hopf algebras over fields. The proof in the co-Frobenius case given in the paper under review provides itself a new proof in the finite dimensional case. The authors also find equivalent conditions for a co-Frobenius Hopf algebra to be cosemisimple and involutory in terms of integrals.

MSC:
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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