Sovereign and ribbon weak Hopf algebras.
The authors extend some results on Hopf algebras to weak Hopf algebras. A ribbon structure in a rigid braided category \(\mathcal C\) is a self-dual twist which is a natural isomorphism from the identity functor to itself compatible with the duality and the braiding. A sovereign structure in an autonomous category is a monoidal natural isomorphism from the left duality functor to the right duality functor. Deligne showed that there is a twist in \(\mathcal C\) if and only if \(\mathcal C\) admits a sovereign structure. Thus a ribbon structure in \(\mathcal C\) is a sovereign structure satisfying some axioms. From a Hopf algebra point of view of Deligne's theorem, \textit{L. H. Kauffman} and \textit{D. E. Radford} gave a bijective map between the ribbon elements and the sovereign elements (not using this language) in a quasitriangular Hopf algebra satisfying axioms related to the Drinfeld element [J. Algebra 159, No. 1, 98-114 (1993; Zbl 0802.16035)]. \textit{J. Bichon} introduced cosovereign Hopf algebras and described their relation with coquasitriandular Hopf algebras [J. Pure Appl. Algebra 157, No. 2-3, 121-133 (2001; Zbl 0976.16027)]. \textit{A. Bruguières} and \textit{A. Virelizier} provided a Hopf monad version of Deligne's theorem [Adv. Math. 215, No. 2, 679-733 (2007; Zbl 1168.18002)]. The paper under review discusses how Deligne's theorem appears in a weak Hopf algebra version. After reviewing (co)ribbon weak Hopf algebras, the authors define (co)sovereign weak Hopf algebras. Their main result is a necessary and sufficient condition based on Deligne's theorem for a (co)quasitriangular finite-dimensional weak Hopf algebra with bijective antipode to be (co)ribbon. The condition involves a sovereign character. Some examples (positive and negative) are given.
scientific article; zbMATH DE number 6481144