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

Abstract

We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra Tm²(q) and the group Hopf algebra K[Cn]: they are nm²-dimensional quantum groups Tωnm²(q) associated to an n-th root of unity ω. 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,ν(n)) and ν(n)/d=p₁α₁⋯prαr is the prime decomposition of ν(n)/d then the number of types of Hopf algebras that factorize through Tm²(q) and K[Cn] is equal to (α1+1)(α2+1)⋯(αr+1), where ν(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.