Structures of Adjoint-Stable Algebras over Factorizable Hopf Algebras

2022/08/20 by Zhimin Liu, Liu, Zhimin, Shenglin Zhu +1
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.2208.09670

openalex publication_date 2022/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a quasi-triangular Hopf algebra ( H,R) , 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 _HM. 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 YD. In this note, we prove for a semisimple factorizable Hopf algebra ( H,R) 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.

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