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

Abstract

Let (H,R) 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 YDHH. Let HR be the Majid's transmuted braided group of (H,R), we show that HR is cosemisimple. As a coalgebra, let HR=D1⊕⋯⊕Dr be the sum of minimal H-adjoint-stable subcoalgebras. For each i(1≤i≤r), we choose a minimal left coideal Wi of Di, and we can define the R-adjoint-stable algebra NWi of Wi. Using Ostrik's theorem on characterizing module categories over monoidal categories, we prove that V∈HHYD is irreducible if and only if there exists an i(1≤i≤r) and an irreducible right NWi-module Ui, such that V≅Ui⊗NWi(H⊗Wi).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 0, we stress that the R-adjoint-stable algebra NWi is an algebra over which the dimension of each irreducible right module divides its dimension.