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

We study versions of the categories of Yetter-Drinfel’d modules over a Hopf algebra $H$ in a braided monoidal category $\text{C}$. Contrarywise to Bespalov’s approach, all our structures live in $\text{C}$. This forces $H$ to be transparent or equivalently to lie in M"uger’s center ${\mathbb{Z}}_2(\text{C})$ of $\text{C}$. We prove that versions of the categories of Yetter-Drinfel’d modules in $\text{C}$ are braided monoidally isomorphic to the categories of (left/right) modules over the Drinfel’d double $D(H)\in \text{C}$ for $H$ finite. We obtain that these categories polarize into two disjoint groups of mutually isomorphic braided monoidal categories. We conclude that if $H\in {\mathbb{Z}}_2(\text{C})$, then ${}_{D(H)}\text{C}$ embeds as a subcategory into the braided center category ${\mathbb{Z}}_1({}_H\text{C})$ of the category ${}_H\text{C}$ of left $H$-modules in $\text{C}$. For $\text{C}$ braided, rigid and cocomplete and a quasitriangular Hopf algebra $H$ such that $H\in {\mathbb{Z}}_2(\text{C})$ we prove that the whole center category of ${}_H\text{C}$ is monoidally isomorphic to the category of left modules over $\text{Aut}({}_H\text{C})⋊H$ - the bosonization of the braided Hopf algebra $\text{Aut}({}_H\text{C})$ which is the coend in ${}_H\text{C}$. A family of examples of a transparent Hopf algebras is discussed.