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 Müger's center $\Z_2(\C)$ 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 $D(H)\in\C$ for $H$ finite. We obtain that these categories polarize into two disjoint groups of mutually isomorphic braided monoidal categories. We conclude that if $H\in\Z_2(\C)$, then ${}_{D(H)}\C$ embeds as a subcategory into the braided center category $\Z_1({}_H\C)$ 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 $H\in\Z_2(\C)$ 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.