| Title:
|
Classifying bicrossed products of two Sweedler's Hopf algebras (English) |
| Author:
|
Bontea, Costel-Gabriel |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
419-431 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We continue the study started recently by Agore, Bontea and Militaru in ``Classifying bicrossed products of Hopf algebras'' (2014), by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler's Hopf algebras. Equivalently, we classify all bicrossed products $H_4 \bowtie H_4$. There are three steps in our approach. First, we explicitly describe the set of all matched pairs $(H_4, H_4, \triangleright , \triangleleft )$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field $k$. Then, for any $\lambda \in k$, we describe by generators and relations the associated bicrossed product, $\mathcal {H}_{16, \lambda }$. This is a $16$-dimensional, pointed, unimodular and non-semisimple Hopf algebra. A Hopf algebra $E$ factorizes through $H_4$ and $H_4$ if and only if $ E \cong H_4 \otimes H_4$ or $E \cong {\mathcal H}_{16, \lambda }$. In the last step we classify these quantum groups by proving that there are only three isomorphism classes represented by: $H_4 \otimes H_4$, ${\mathcal H}_{16, 0}$ and ${\mathcal H}_{16, 1} \cong D(H_4)$, the Drinfel'd double of $H_4$. The automorphism group of these objects is also computed: in particular, we prove that ${\rm Aut}_{\rm Hopf}( D(H_4))$ is isomorphic to a semidirect product of groups, $k^{\times } \rtimes \mathbb {Z}_2$. (English) |
| Keyword:
|
bicrossed product of Hopf algebras |
| Keyword:
|
Sweedler's Hopf algebra |
| Keyword:
|
Drinfel'd double |
| MSC:
|
16S40 |
| MSC:
|
16T05 |
| MSC:
|
16T10 |
| idZBL:
|
Zbl 06391502 |
| idMR:
|
MR3277744 |
| DOI:
|
10.1007/s10587-014-0109-6 |
| . |
| Date available:
|
2014-11-10T09:42:49Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144006 |
| . |
| Reference:
|
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| Reference:
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