Yetter-Drinfeld-Long bimodules are modules

Abstract

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 ${}_{H \otimes H*}^{H \otimes H*}YD$ 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.