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.

The aim of this paper is to construct a new braided T-category via the generalized Yetter-Drinfel’d modules and Drinfel’d codouble over a Hopf algebra, an approach different from that proposed by Panaite and Staic [1]. Moreover, in the case of finite dimensional, we will show that this category coincides with the corepresentation of a certain coquasitriangular Turaev group algebra which we construct. Finally we apply our theory to the case of group algebra.

We mainly construct a bicrossproduct for a finite-dimensional monoidal Hom-Hopf algebra $(H,\alpha )$, generalizing Majid’s bicrossproduct. Naturally, the Hom-type bicrossproduct leads to the Drinfel’d double $(H^{\rm op}\bowtie H^{\ast },\alpha \otimes (

Let H-4 and H-8 be the Sweedler's and Kac-Paljutkin Hopf algebras, respectively. We prove that any Hopf algebra which factorizes through H-8 and H-4 (equivalently, any bicrossed product between the Hopf algebras H-8 and H-4) must be isomorphic to one of the following four Hopf algebras: H-8 circle times H-4,H-32,H-1,H-32,H-2,H-32,H-3. The set of all matched pairs (H-4,H-4, (sic), (sic)) is explicitly described, and then the associated bicrossed product is given by generators and relations.