Classifying bicrossed products of two Taft algebras

We classify all Hopf algebras which factor through two Taft algebras T n 2 (q‾) and respectively T m 2 (q). To start with, all possible matched pairs between the two Taft algebras are described: if q‾≠q n−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‾=q n−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.