New braided monoidal categories over monoidal Hom-Hopf algebras
Volume 146 / 2017
Colloquium Mathematicum 146 (2017), 77-97
MSC: 16A10, 16W30.
DOI: 10.4064/cm6706-11-2015
Published online: 25 August 2016
Abstract
Let $(H,\alpha )$ and $(B,\beta )$ be two monoidal Hom-Hopf algebras. We introduce the notion of a generalized Hom-Long dimodule and show that the category $^{B}_{H}\mathcal {L}$ of generalized Hom-Long dimodules is an autonomous category. We prove that $^{B}_{H}\mathcal {L}$ is a braided monoidal category if $(H,\alpha )$ is quasitriangular and $(B,\beta )$ is coquasitriangular, and we show that $^{B}_{H}\mathcal {L}$ is a subcategory of the Hom-Yetter–Drinfeld category $^{H\otimes B}_{H\otimes B}\mathcal {HYD}$. Moreover, we prove that the category of Hom-modules (resp., Hom-comodules) over a triangular (resp., cotriangular) Hom-Hopf algebra contains a symmetric generalized Hom-Long dimodule category.
