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 H_R of H introduced by Majid. The transmuted braided group H_R is a Hopf algebra in the braided category ${}_H\mathcal{M}$. 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 ${}_H^H\mathcal{Y}\mathcal{D}$. 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.

Keyword:

• Adjoint-stable algebras
• factorizable Hopf algebras
• Yetter-Drinfeld modules

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16T05

Additional information

Funding

Project funded by China Postdoctoral Science Foundation grant 2019M661327.