Remarks on bicovariant differential calculi and exterior Hopf algebras
This paper studies algebras of differential forms associated to quantum groups (Hopf algebras). The main point is to compare two natural notions of compatibility of the algebra of differential forms with the Hopf algebra structure. First there is the notion of bicovariance of the bimodule of one-forms and second there is the condition that there is an induced Hopf algebra structure on the algebra of all differential forms. The main result of the paper is that given a bicovariant bimodule of one- forms there is an induced Hopf algebra structure on the exterior algebra over this bimodule and, conversely, if one has a Hopf algebra structure on an algebra of differential forms over a quantum group then the bimodule of one forms is a bicovariant differential calculus. Moreover, the author studies the following question: Given an exterior Hopf algebra of differential forms over a quasitriangular quantum group, when is the \({\mathcal R}\)-matrix of the quantum group also a quasitriangular structure for the Hopf algebra of differential forms. Finally, examples of bicovariant differential calculi on simple quantum groups and on the quantum plane are given.
exterior Hopf algebra of differential forms
scientific article; zbMATH DE number 202412
scientific article