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

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 $\mathfrak{h}⋉(k^{\ast }\times {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{L}\mathrm{i}\mathrm{e}}(\mathfrak{h}))$. 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 $[E_i, G] = E_i$,$[G, F_i] = F_i$,forall$i = 1, \dots, n$. We explicitly describe all Lie algebras containing $\mathfrak{l}(2n+1,k)$ as a subalgebra of codimension $1$ by computing all possible bicrossed products $k \bowtie \mathfrak{l} (2n+1, k)$\mathrm{.}Theyareparameterizedbyasetofmatrices${\rm M}_n (k)^4 \times k^{2n+2}$ 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.