Structures of adjoint-stable algebras over factorizable Hopf algebras

Abstract

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