2013/03/13 by Bojana Femić, Femić, Bojana
Mathematics · #16T05 #18D10 #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA #msc:16T05 #msc:18D10
paper · pdf · doi:10.48550/arxiv.1303.3070
42 pages; this is a second version of a paper from September 2010
arxiv created 2013/11/11 · arxiv updated 2013/11/12
We study versions of the categories of Yetter-Drinfel'd modules over a Hopf algebra H in a braided monoidal category \C. Contrarywise to Bespalov's approach, all our structures live in \C. This forces H to be transparent or equivalently to lie in Müger's center \Z2(\C) of \C. We prove that versions of the categories of Yetter-Drinfel'd modules in \C are braided monoidally isomorphic to the categories of (left/right) modules over the Drinfel'd double D(H)∈\C for H finite. We obtain that these categories polarize into two disjoint groups of mutually isomorphic braided monoidal categories. We conclude that if H∈\Z2(\C), then D(H)\C embeds as a subcategory into the braided center category \Z1(H\C) of the category H\C of left H-modules in \C. For \C braided, rigid and cocomplete and a quasitriangular Hopf algebra H such that H∈\Z2(\C) we prove that the whole center category of H\C is monoidally isomorphic to the category of left modules over \Aut(H\C)\rtimes H - the bosonization of the braided Hopf algebra \Aut(H\C) which is the coend in H\C. A family of examples of a transparent Hopf algebras is discussed.