Classifying bicrossed products of two Taft algebras

We classify all Hopf algebras which factor through two Taft algebras Tn2(q‾) and respectively Tm2(q). To start with, all possible matched pairs between the two Taft algebras are described: if q‾≠qn−1 then the matched pairs are in bijection with the group of d-th roots of unity in k, where d=(m,n) while if q‾=qn−1 then besides the matched pairs above we obtain an additional family of matched pairs indexed by k⁎. The corresponding bicrossed products (double cross product in Majid's terminology) are explicitly described by generators and relations and classified. As a consequence of our approach, we are able to compute the number of isomorphism types of these bicrossed products as well as to describe their automorphism groups.