Ribbon transformations, integrals, and triangular decompositions.
This paper discusses distinct facts related to integrals in Yetter-Drinfel'd Hopf algebras, i.e., Hopf algebras in the braided category of Yetter-Drinfel'd modules over a Hopf algebra. It introduces the twisted Nakayama automorphism and proves an analogue, in the Yetter-Drinfel'd context, of a formula due to \textit{D. E. Radford} for the fourth power of the antipode [Am. J. Math. 98, 333-355 (1976; Zbl 0332.16007)]. This formula is compared in an Appendix with the formula obtained by \textit{Yu. Bespalov, T. Kerler, V. Lyubashenko} and \textit{V. Turaev} [in J. Pure Appl. Algebra 148, No. 2, 113-164 (2000; Zbl 0961.16023)]. Integrals for two kinds of Hopf algebras with triangular decomposition that arise from Yetter-Drinfel'd Hopf algebras are computed, and this is applied to the case of the Frobenius-Lusztig kernel of \(sl(2)\).
scientific article; zbMATH DE number 2153372
scientific article