2018/11/14 by Zhimin Liu, Liu, Zhimin, Shenglin Zhu +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1811.05593
openalex publication_date 2018/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 HHYD. 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 N_Wi 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 N_Wi-module Ui, such that V≅ Ui⊗_N_Wi( 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, we stress that the R-adjoint-stable algebra N_Wi is an algebra over which the dimension of each irreducible right module divides its dimension.