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

SIGMA 10 (2014), 065, 16 pages      arXiv:1312.4018

Bicrossed Products, Matched Pair Deformations and the Factorization Index for Lie Algebras

Ana-Loredana Agore a, b 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 January 20, 2014, in final form June 10, 2014; Published online June 16, 2014

For a perfect Lie algebra $\mathfrak{h}$ we classify all Lie algebras containing $\mathfrak{h}$ as a subalgebra of codimension $1$. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product $\mathfrak{h} \ltimes (k^* \times {\rm Aut}_{\rm Lie} (\mathfrak{h}))$. In the non-perfect case the classification of these Lie algebras is a difficult task. Let $\mathfrak{l} (2n+1, k)$ be the Lie algebra with the bracket $[E_i, G] = E_i$, $[G, F_i] = F_i$, for all $i = 1, \dots, n$. We explicitly describe all Lie algebras containing $\mathfrak{l} (2n+1, k)$ as a subalgebra of codimension $1$ by computing all possible bicrossed products $k \bowtie \mathfrak{l} (2n+1, k)$. They are parameterized by a set of matrices ${\rm M}_n (k)^4 \times k^{2n+2}$ which are explicitly determined. Several matched pair deformations of $\mathfrak{l} (2n+1, k)$ are described in order to compute the factorization index of some extensions of the type $k \subset k \bowtie \mathfrak{l} (2n+1, k)$. We provide an example of such extension having an infinite factorization index.

Key words: matched pairs of Lie algebras; bicrossed products; factorization index.

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