Hopf Algebras which Factorize through the Taft Algebra Tm2(q) and the Group Hopf Algebra K[Cn]

We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra T-m2(q) and the group Hopf algebra K[C-n]: they are nm(2)-dimensional quantum groups T-nm2(omega)(q) associated to an n-th root of unity omega. Furthermore, using Dirichlet's prime number theorem we are able to count the number of isomorphism types of such Hopf algebras. More precisely, if d = gcd(m, nu(n)) and nu(n)/d = p(1)(alpha 1)...p(r)(alpha r) is the prime decomposition of nu(n)/d then the number of types of Hopf algebras that factorize through T-m2(q) and K[C-n] is equal to (alpha(1) + 1)(alpha(2) + 1)...(alpha(r) + 1), where nu(n) is the order of the group of n-th roots of unity in K. As a consequence of our approach, the automorphism groups of these Hopf algebras are described as well.