Transparency condition in the categories of Yetter-Drinfel’d modules over Hopf algebras in braided categories

Abstract

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 M¨uger’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(HC) of the category HC 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 HC is monoidally isomorphic to the category of left modules over Aut(HC) ⋊ H - the bosonization of the braided Hopf algebra Aut(HC) which is the coend in HC. A family of examples of a transparent Hopf algebras is discussed.