New braided monoidal categories over monoidal Hom-Hopf algebras

Shengxiang Wang¹ ; Nanqing Ding²

¹ Department of Mathematics Nanjing University Nanjing, 210093, P.R. China and School of Mathematics and Finance Chuzhou University Chuzhou, 239000, P.R. China
² Department of Mathematics Nanjing University Nanjing, 210093, P.R. China

Colloquium Mathematicum, Tome 146 (2017) no. 1, pp. 77-97.

Résumé

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.

Détail

DOI : 10.4064/cm6706-11-2015

Keywords: alpha beta monoidal hom hopf algebras introduce notion generalized hom long dimodule category mathcal generalized hom long dimodules autonomous category prove mathcal braided monoidal category alpha quasitriangular beta coquasitriangular mathcal subcategory hom yetter drinfeld category otimes otimes mathcal hyd moreover prove category hom modules resp hom comodules triangular resp cotriangular hom hopf algebra contains symmetric generalized hom long dimodule category