Classification Of Bicrossed Products Of Hopf Algebras Of Dimension 32 And A Conjecture

According to the theory of bicrossed product of Hopf algebras, a Hopf algebra E factorizes through A and H if and only if E is isomorphic to some bicrossed product of A and H [1]. In this paper, we describe and classify the bicrossed product of Hopf algebras of dimension 32. Finally, up to an isomorphism of Hopf algebra, there are only two Hopf algebras that factorize through H4 and H8, namely, H4 (?) H8 and (?)32,i,j· In the same way, we calculate (H8(?)H4) (?) k [C2],H8 (?) (H4(?)k [C2]), H4 (?) k [C2] is the result which have already got in the reference [1]. Then compare it with the new Hopf algebra. Consequently, except ordinary matched pairs, we can’t always find matched pairs (?)2’,(?)2’ and (?)1’,(?)1’, such that the canonical map (H8(?)H4) (?) k [C2]→H8(?)1’ (H4 (?)2’k [C2]) is an isomorphism of Hopf algebras.