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

For a perfect Lie algebra h we classify all Lie algebras containing h as a subalgebra of codimension 1. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product h n (k∗ × AutLie(h)). In the non-perfect case the classification of these Lie algebras is a difficult task. Let l(2n + 1, k) be the Lie algebra with the bracket [Ei, G] = Ei, [G,Fi] = Fi, for all i = 1, . . . , n. We explicitly describe all Lie algebras containing l(2n + 1, k) as a subalgebra of codimension 1 by computing all possible bicrossed products k ./ l(2n + 1, k). They are parameterized by a set of matrices Mn(k) 4 × k which are explicitly determined. Several matched pair deformations of l(2n+ 1, k) are described in order to compute the factorization index of some extensions of the type k ⊂ k ./ l(2n+ 1, k). We provide an example of such extension having an infinite factorization index.