Yetter-Drinfeld-Long bimodules are modules

Let H be a finite-dimensional bialgebra. In this paper, we prove that the category ℒR(H) of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category YH⊗H*H⊗H*D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{H \otimes H*}^{H \otimes H*}YD$$\end{document} over the tensor product bialgebra H ⊗ H* as monoidal categories. Moreover if H is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.