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On the antipode of a co-Frobenius (co)quasitriangular Hopf algebra. (English) Zbl 1186.16022

If \(H\) is a quasitriangular Hopf algebra with antipode \(S\), then \(S^2\) is the inner automorphism induced by the Drinfeld element \(u\), and \(S^4\) is the inner automorphism induced by the grouplike element \(g=uS(u)^{-1}\). If \(H\) is finite-dimensional, then it is proved by V. G. Drinfel’d [Leningr. Math. J. 1, No. 2, 321-342 (1990); translation from Algebra Anal. 1, No. 2, 30-46 (1989; Zbl 0718.16035)] and D. E. Radford [J. Algebra 151, No. 1, 1-11 (1992; Zbl 0767.16016)] that \(g\) can be expressed in terms of the modular elements of \(H\) and \(H^*\).
The paper under review presents a generalization of this result in the case when there exist non-zero integrals on \(H\), i.e. \(H\) is a co-Frobenius Hopf algebra. The method of proof is new even in the finite-dimensional case. Similar results are proved for the infinite-dimensional coquasitriangular case.

MSC:

16T05 Hopf algebras and their applications
16W20 Automorphisms and endomorphisms
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References:

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