Geodesic
Classifying bicrossed products of two Sweedler's Hopf algebras
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 419-431 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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, $k^{\times } \rtimes \mathbb {Z}_2$.
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, $k^{\times } \rtimes \mathbb {Z}_2$.
DOI : 10.1007/s10587-014-0109-6
Classification : 16S40, 16T05, 16T10
Keywords: bicrossed product of Hopf algebras; Sweedler's Hopf algebra; Drinfel'd double 
@article{10_1007_s10587_014_0109_6,
     author = {Bontea, Costel-Gabriel},
     title = {Classifying bicrossed products of two {Sweedler's} {Hopf} algebras},
     journal = {Czechoslovak Mathematical Journal},
     pages = {419--431},
     year = {2014},
     volume = {64},
     number = {2},
     doi = {10.1007/s10587-014-0109-6},
     mrnumber = {3277744},
     zbl = {06391502},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0109-6/}
}
TY  - JOUR
AU  - Bontea, Costel-Gabriel
TI  - Classifying bicrossed products of two Sweedler's Hopf algebras
JO  - Czechoslovak Mathematical Journal
PY  - 2014
SP  - 419
EP  - 431
VL  - 64
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0109-6/
DO  - 10.1007/s10587-014-0109-6
LA  - en
ID  - 10_1007_s10587_014_0109_6
ER  - 
%0 Journal Article
%A Bontea, Costel-Gabriel
%T Classifying bicrossed products of two Sweedler's Hopf algebras
%J Czechoslovak Mathematical Journal
%D 2014
%P 419-431
%V 64
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0109-6/
%R 10.1007/s10587-014-0109-6
%G en
%F 10_1007_s10587_014_0109_6
Bontea, Costel-Gabriel. Classifying bicrossed products of two Sweedler's Hopf algebras. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 419-431. doi: 10.1007/s10587-014-0109-6

[1] Agore, A. L., Bontea, C. G., Militaru, G.: Classifying bicrossed products of Hopf algebras. Algebr. Represent. Theory 17 (2014), 227-264. | DOI | MR

[2] Andruskiewitsch, N., Devoto, J.: Extensions of Hopf algebras. St. Petersbg. Math. J. 7 17-52 (1996), translation from Algebra Anal. 7 22-61 (1995). | MR | Zbl

[3] Andruskiewitsch, N., Schneider, H.-J.: Lifting of quantum linear spaces and pointed Hopf algebras of order $p^{3}$. J. Algebra 209 658-691 (1998). | DOI | MR | Zbl

[4] Andruskiewitsch, N., Schneider, H.-J.: Finite quantum groups and Cartan matrices. Adv. Math. 154 1-45 (2000). | DOI | MR | Zbl

[5] Andruskiewitsch, N., Schneider, H.-J.: On the classification of finite-dimensional pointed Hopf algebras. Ann. Math. (2) 171 375-417 (2010). | DOI | MR | Zbl

[6] Caenepeel, S., Dăscălescu, S., Raianu, Ş.: Classifying pointed Hopf algebras of dimension 16. Commun. Algebra 28 541-568 (2000). | DOI | MR | Zbl

[7] Doi, Y., Takeuchi, M.: Quaternion algebras and Hopf crossed products. Commun. Algebra 23 3291-3325 (1995). | DOI | MR | Zbl

[8] García, G. A., Vay, C.: Hopf algebras of dimension 16. Algebr. Represent. Theory 13 383-405 (2010). | DOI | MR | Zbl

[9] Kassel, C.: Quantum Groups, Graduate Texts in Mathematics. Graduate Texts in Mathematics 155 Springer, New York (1995). | MR

[10] Majid, S.: Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction. J. Algebra 130 17-64 (1990). | DOI | MR | Zbl

[11] Majid, S.: Foundations of Quantum Groups Theory. Cambridge University Press, Cambridge (1995).

[12] Majid, S.: More examples of bicrossproduct and double cross product Hopf algebras. Isr. J. Math. 72 133-148 (1990). | DOI | MR | Zbl

[13] Masuoka, A.: Cleft extensions for a Hopf algebra generated by a nearly primitive element. Commun. Algebra 22 4537-4559 (1994). | DOI | MR | Zbl

[14] Montgomery, S.: Hopf Algebras and Their Actions on Rings. Expanded version of ten authors lectures given at the CBMS Conference on Hopf algebras and their actions on rings, DePaul University in Chicago, USA, 1992. Regional Conference Series in Mathematics 82 AMS, Providence, RI (1993). | MR | Zbl

[15] Radford, D. E.: Minimal quasitriangular Hopf algebras. J. Algebra 157 285-315 (1993). | DOI | MR | Zbl

[16] Takeuchi, M.: Matched pairs of groups and bismash products of Hopf algebras. Commun. Algebra 9 841-882 (1981). | DOI | MR | Zbl

Cité par Sources :