On the Structure of Irreducible Yetter-Drinfeld Modules over Quasi-Triangular Hopf Algebras

Let $\left( H,R\right)$ be a finite dimensional semisimple and cosemisimple quasi-triangular Hopf algebra over a field $k$. In this paper, we give the structure of irreducible objects of the Yetter-Drinfeld module category ${} {}_{H}^{H}\mathcal{YD}.$ Let $H_{R}$ be the Majid's transmuted braided group of $\left( H,R\right) ,$ we show that $H_{R}$ is cosemisimple. As a coalgebra, let $H_{R}=D_{1}\oplus\cdots\oplus D_{r}$ be the sum of minimal $H$-adjoint-stable subcoalgebras. For each $i$ $\left( 1\leq i\leq r\right)$, we choose a minimal left coideal $W_{i}$ of $D_{i}$, and we can define the $R$-adjoint-stable algebra $N_{W_{i}}$ of $W_{i}$. Using Ostrik's theorem on characterizing module categories over monoidal categories, we prove that $V\in{}_{H}^{H}\mathcal{YD}$ is irreducible if and only if there exists an $i$ $\left( 1\leq i\leq r\right)$ and an irreducible right $N_{W_{i}}$-module $U_{i}$, such that $V\cong U_{i}\otimes_{N_{W_{i}}}\left( H\otimes W_{i}\right)$. Our structure theorem generalizes the results of Dijkgraaf-Pasquier-Roche and Gould on Yetter-Drinfeld modules over finite group algebras. If $k$ is an algebraically closed field of characteristic, we stress that the $R$-adjoint-stable algebra $N_{W_{i}}$ is an algebra over which the dimension of each irreducible right module divides its dimension.