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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.
pdf (414 kb)
tex (20 kb)
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