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 \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 \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.Comment: 42 pages; this is a second version of a paper from September 201

Two-dimensional Kolmogorov-type Goodness-of-fit Tests Based on Characterizations and their Asymptotic Efficiencies

In this paper new two-dimensional goodness of fit tests are proposed. They are of supremum-type and are based on different types of characterizations. For the first time a characterization based on independence of two statistics is used for goodness-of-fit testing. The asymptotics of the statistics is studied and Bahadur efficiencies of the tests against some close alternatives are calculated. In the process a theorem on large deviations of Kolmogorov-type statistics has been extended to the multidimensional case