Hopf bimodules are modules
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 \(A\).
scientific article; zbMATH DE number 1308183