Hopf Algebras which Factorize through the Taft Algebra $T_{m^{2}}(q)$ and the Group Hopf Algebra $K[C_{n}]$
| Autor: | Agore, Ana-Loredana |
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| Rok vydání: | 2016 |
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| Zdroj: | SIGMA 14 (2018), 027, 14 pages |
| Druh dokumentu: | Working Paper |
| DOI: | 10.3842/SIGMA.2018.027 |
| Popis: | 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. Comment: Continues arXiv:1205.6110, arXiv:1205.6564, arXiv:1603.01854; restates preliminaries and definitions for sake of clarity |
| Databáze: | arXiv |
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