Remarks on bicovariant differential calculi and exterior Hopf algebras
type: Item
instance of: scholarly article
title: Remarks on bicovariant differential calculi and exterior Hopf algebras
zbMATH Open document ID: 0782.17009
DOI: 10.1007/BF00777376
author: Tomasz Brzeziński
published in: Letters in Mathematical Physics
publication date: 1994-03-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 46L85
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 202412
zbMATH Keywords: quantum groups
zbMATH Keywords: bicovariant differential calculus
zbMATH Keywords: exterior Hopf algebra of differential forms
zbMATH Keywords: quasitriangular quantum group
zbMATH Keywords: quantum plane
reviewed by: Andreas Cap
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 202412