Some Topics On Braided Hopf Algebras And Galois Extension in Braided Tensor Categories

Braided Hopf algebras have attracted much attention in both mathematics and mathematical physics (see e.g. [2][4][14][16][17][18][20][23]). The classification of finite dimensional Hopf algebras is interesting and important for their applications (see [1] [24]). Braided Hopf algebras play an important role in the classification of finite-dimensional pointed Hopf algebras (e.g. [1][2] [21]). The Hopf Galois (H-Galois) extension has its roots in the work of Chase-Harrison-Rosenberg [8] and Chase-Sweedler [9]. The general definition about H-Galois extension appeared in [15] and the relation between crossed product and H-Galois extension was obtained in [3][10][12] for ordinary Hopf algebras. See books [19][11] for reviews about the main results on this topic. In this paper, we construct new (finite dimensional) Hopf algebras using (finite dimensional) braided Hopf algebras in Yetter-Drinfeld categories by means of bosinization and Drinfeld double. We show that for any braided Hopf algebra B in HYD the bosonization B#H of B and H is also a braided Hopf algebra in HYD. Consequently, we can obtain other Hopf algebras such as (B#H)#H and so on. In [22], the braided Drinfeld double D(B) was constructed for finite dimensional braided Hopf algebra B in HYD with symmetric braiding on B. We give the relation between crossed product and H-Galois extension in braided tensor category C with equivalisers and coequivalisers. That is, we show that if there exist an equivaliser and a coequivaliser for any two morphisms in C (e.g. Yetter-Drinfeld categories), then A = B#σH is a crossed product algebra in C if and only if the extension A/B is Galois, the canonical epic q : A⊗A → A⊗B A is split and A is isomorphic as left B-modules and right H-comodules to B ⊗H in C. Let k be a field. We use the Sweedler’s sigma notations for coalgebras and comodules: ∆(x) = ∑ x1 ⊗ x2, δ (x) = ∑ x(−1) ⊗ x(0).