On the structure of irreducible Yetter-Drinfeld modules over quasi-triangular Hopf algebras (Q2326448)

Let \(H\) be a semisimple and cosemisimple quasi-triangular Hopf algebra. It is first proved that Majid's transmuted braided group \(H_R\) associated to \(H\) is cosemisimple, and that any Yetter-Drinfeld submodule of \(H\) corresponds to a subcoalgebra of \(H_R\). Then, it is shown how to obtain all the irreducible Yetter-Drinfeld modules over \(H\) from a decomposition of \(H\) as a direct sum of minimal \(H\)-adjoint-stable subcoalgebras of \(H_R\). This includes the case of Hopf algebras of finite groups, thus this results is a generalization of Dijkgraaf-Pasquier-Roche and Gould's structure theorems for Yetter-Drinfeld modules over a finite group.