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This article is cited in 3 scientific papers (total in 3 papers)
Hopf Algebras which Factorize through the Taft Algebra $T_{m^{2}}(q)$ and the Group Hopf Algebra $K[C_{n}]$
Ana-Loredana Agoreab a Faculty of Engineering, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium
b “Simion Stoilow” Institute of Mathematics of the Romanian Academy,
P.O. Box 1-764, 014700 Bucharest, Romania
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.
Keywords:
bicrossed product; the factorization problem; classification of Hopf algebras.
Received: August 28, 2017; in final form March 20, 2018; Published online March 25, 2018
Citation:
Ana-Loredana Agore, “Hopf Algebras which Factorize through the Taft Algebra $T_{m^{2}}(q)$ and the Group Hopf Algebra $K[C_{n}]$”, SIGMA, 14 (2018), 027, 14 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1326 https://www.mathnet.ru/eng/sigma/v14/p27
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