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 \(\begin{array}{*{20}{c}} {H \otimes H*} {H \otimes H*} \end{array}YD\) over the tensor product bialgebra \(H \otimes H*\) 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.
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