Hopf Algebras which Factorize through the Taft Algebra $T_{m^2}(q)$ and the Group Hopf Algebra $K[C_n]$

Abstract

We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra $T_{m^2}(q)$ and the group Hopf algebra $K[C_n]$: they are $n{m}^2$-dimensional quantum groups $T_{n{m}^2}^{\omega }(q)$ 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 $\frac{\nu (n)}{d}=p_1^{{\alpha }_1}\cdots p_r^{{\alpha }_r}$ is the prime decomposition of $\frac{\nu (n)}{d}$ then the number of types of Hopf algebras that factorize through $T_{m^2}(q)$ and $K[C_n]$ is equal to $({\alpha }_1+1)({\alpha }_2+1)\cdots ({\alpha }_r+1)$, 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.

Continues arXiv:1205.6110, arXiv:1205.6564, arXiv:1603.01854; restates preliminaries and definitions for sake of clarity

References in corpus (2)

• Classification of pointed rank one Hopf algebras
• Classifying bicrossed products of two Taft algebras

Cited by in corpus (2)

• The bicrossed products of $H_4$ and $H_8$
• Bicrossed products with the Taft algebra