Structures of Adjoint-Stable Algebras over Factorizable Hopf Algebras

Abstract

For a quasi-triangular Hopf algebra $\left(H,R\right)$, there is a notion of transmuted braided group $H_R$ of $H$ introduced by Majid. The transmuted braided group $H_R$ is a Hopf algebra in the braided category ${}_H\mathcal{M}$. The $R$-adjoint-stable algebra associated with any simple left $H_R$-comodule is defined by the authors, and is used to characterize the structure of all irreducible Yetter-Drinfeld modules in ${}_H^H\mathcal{YD}$. In this note, we prove for a semisimple factorizable Hopf algebra $\left(H,R\right)$ that any simple subcoalgebra of $H_R$ is $H$-stable and the $R$-adjoint-stable algebra for any simple left $H_R$-comodule is anti-isomorphic to $H$. As an application, we characterize all irreducible Yetter-Drinfeld modules.