Hopf bimodules are modules
type: Item
instance of: scholarly article
title: Hopf bimodules are modules
zbMATH Open document ID: 0934.16035
DOI: 10.1016/S0022-4049(97)00060-1
author: Claude Cibils
author: Marc Rosso
published in: Journal of Pure and Applied Algebra
publication date: 2000-01-30T00:00:00Z
review text: A Hopf bimodule over a finite dimensional Hopf algebra \(A\) is a vector space \(M\) which is an \(A\)-bimodule and an \(A\)-bicomodule such that the comodule structure maps \(\delta_L\colon M\to A\otimes M\) and \(\delta_R\colon M\to M\otimes A\) are \(A\)-bimodule maps. The category of all Hopf \(A\)-bimodules is equivalent to the category of left modules over the Drinfel'd double \({\mathcal D}(A)\) [\textit{V. G. Drinfel'd}, Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798-820 (1987; Zbl 0667.16003)], see the papers by \textit{M. Rosso} [C. R. Acad. Sci., Paris, Sér. I 320, No. 2, 145-148 (1995; Zbl 0929.17005)] and by \textit{P. Schauenburg} [J. Algebra 169, No. 3, 874-890 (1994; Zbl 0810.16037)]. This equivalence does not preserve underlying vector spaces. The authors construct an algebra \(X\) which acts on Hopf \(A\)-bimodules in such a way that the category of Hopf \(A\)-bimodules becomes equivalent to that of left \(X\)-modules, and this equivalence of categories preserves the underlying vector spaces. In Proposition 3.3, the authors prove, by using the classification of Hopf bimodules and Morita theory, that \(X\) is isomorphic to \({\mathcal H}(A)\otimes{\mathcal D}(A)^{op}\), where \({\mathcal H}(A)\) denotes the Heisenberg double of \(A\).
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16D20
Mathematics Subject Classification ID: 18D10
zbMATH DE Number: 1308183
zbMATH Keywords: Hopf bimodules
zbMATH Keywords: Morita equivalences
zbMATH Keywords: finite dimensional Hopf algebras
zbMATH Keywords: categories of left modules
zbMATH Keywords: Heisenberg doubles
zbMATH Keywords: Drinfel'd doubles
reviewed by: José Gómez-Torrecillas
MaRDI profile type: Publication
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full work available at URL: s0022-4049(97)00060-1
OpenAlex ID: W1987517168
description: scientific article; zbMATH DE number 1308183