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

TL;DR

This paper classifies all Hopf algebras that factor through the Taft algebra and group algebra, describing their structure, counting their types using number theory, and analyzing their automorphism groups.

Contribution

It provides a complete classification and enumeration of Hopf algebras factorizing through specific algebras, introducing new quantum groups $T_{nm^{2}}^ { ext{omega}}(q)$.

Findings

Classified all such Hopf algebras via generators and relations.

Counted isomorphism types using Dirichlet's prime number theorem.

Described automorphism groups of these Hopf algebras.

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 $nm^{2}$-dimensional quantum groups $T_{nm^{2}}^ {\omega}(q)$ 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 $d = {\rm gcd}(m,\nu(n))$ 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.