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

TL;DR

This paper explores the structure of Yetter-Drinfel'd modules over Hopf algebras within braided categories, revealing conditions for their equivalence to module categories over the Drinfel'd double and analyzing the implications of transparency.

Contribution

It establishes that in braided categories, Yetter-Drinfel'd modules are braided monoidally isomorphic to Drinfel'd double modules when the Hopf algebra is transparent, extending understanding of their categorical relationships.

Findings

Yetter-Drinfel'd categories are isomorphic to Drinfel'd double module categories for finite H.

Categories polarize into two disjoint isomorphic braided monoidal groups.

If H is in the M"uger's center, it embeds into the braided center of its module category.

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 $\Z_2(\C)$ 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.