### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 049, 12 pages      arXiv:1210.7700      https://doi.org/10.3842/SIGMA.2014.049
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

### The Classification of All Crossed Products $H_4 \# k[C_{n}]$

Ana-Loredana Agore a, b, Costel-Gabriel Bontea c, a and Gigel Militaru c
a) Faculty of Engineering, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium
b) Department of Applied Mathematics, Bucharest University of Economic Studies, Piata Romana 6, RO-010374 Bucharest 1, Romania
c) Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, RO-010014 Bucharest 1, Romania

Received November 18, 2013, in final form April 18, 2014; Published online April 23, 2014

Abstract
Using the computational approach introduced in [Agore A.L., Bontea C.G., Militaru G., J. Algebra Appl. 12 (2013), 1250227, 24 pages] we classify all coalgebra split extensions of $H_4$ by $k[C_n]$, where $C_n$ is the cyclic group of order $n$ and $H_4$ is Sweedler's $4$-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras $H_4 \# k[C_{n}]$ by explicitly computing two classifying objects: the cohomological 'group' ${\mathcal H}^{2} ( k[C_{n}], H_4)$ and $\text{CRP}( k[C_{n}], H_4):=$ the set of types of isomorphisms of all crossed products $H_4 \# k[C_{n}]$. More precisely, all crossed products $H_4 \# k[C_n]$ are described by generators and relations and classified: they are $4n$-dimensional quantum groups $H_{4n, \lambda, t}$, parameterized by the set of all pairs $(\lambda, t)$ consisting of an arbitrary unitary map $t : C_n \to C_2$ and an $n$-th root $\lambda$ of $\pm 1$. As an application, the group of Hopf algebra automorphisms of $H_{4n, \lambda, t}$ is explicitly described.

Key words: crossed product of Hopf algebras; split extension of Hopf algebras.

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References

1. Adem A., Milgram R.J., Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften, Vol. 309, 2nd ed., Springer-Verlag, Berlin, 2004.
2. Agore A.L., Bontea C.G., Militaru G., Classifying coalgebra split extensions of Hopf algebras, J. Algebra Appl. 12 (2013), 1250227, 24 pages, arXiv:1207.0411.
3. Agore A.L., Bontea C.G., Militaru G., Classifying bicrossed products of Hopf algebras, Algebr. Represent. Theory 17 (2014), 227-264, arXiv:1205.6110.
4. Agore A.L., Militaru G., Extending structures II: The quantum version, J. Algebra 336 (2011), 321-341, arXiv:1011.2174.
5. Andruskiewitsch N., Notes on extensions of Hopf algebras, Canad. J. Math. 48 (1996), 3-42.
6. Andruskiewitsch N., Devoto J., Extensions of Hopf algebras, St. Petersburg Math. J. 7 (1996), 17-52.
7. Andruskiewitsch N., Natale S., Examples of self-dual Hopf algebras, J. Math. Sci. Univ. Tokyo 6 (1999), 181-215.
8. Andruskiewitsch N., Schneider H.J., Lifting of quantum linear spaces and pointed Hopf algebras of order $p^3$, J. Algebra 209 (1998), 658-691, math.QA/9803058.
9. Brzeziński T., Hajac P.M., Coalgebra extensions and algebra coextensions of Galois type, Comm. Algebra 27 (1999), 1347-1367, q-alg/9708010.
10. García G.A., Vay C., Hopf algebras of dimension 16, Algebr. Represent. Theory 13 (2010), 383-405, arXiv:0712.0405.
11. Krop L., Classification of isomorphism types of Hopf algebras in a class of Abelian extensions, arXiv:1211.5621.
12. Masuoka A., Hopf algebra extensions and cohomology, in New Directions in Hopf Algebras, Math. Sci. Res. Inst. Publ., Vol. 43, Cambridge University Press, Cambridge, 2002, 167-209.
13. Montgomery S., Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, Vol. 82, Amer. Math. Soc., Providence, RI, 1993.
14. Sweedler M.E., Cohomology of algebras over Hopf algebras, Trans. Amer. Math. Soc. 133 (1968), 205-239.