Extending structures for Lie algebras

Let $mathfrak{g }$ be a Lie algebra, $E$ a vector space containing $mathfrak{g }$ as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on $E$ such that $mathfrak{g }$ is a Lie subalgebra of $E$ . A general product, called the unified product, is introduced as a tool for our approach. Let $V$ be a complement of $mathfrak{g }$ in $E$ : the unified product $mathfrak{g } ,natural , V$ is associated to a system $(triangleleft , , triangleright , , f, {-, , -})$ consisting of two actions $triangleleft $ and $triangleright $ , a generalized cocycle $f$ and a twisted Jacobi bracket ${-, , -}$ on $V$ . There exists a Lie algebra structure $[-,-]$ on $E$ containing $mathfrak{g }$ as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras $(E, [-,-]) cong mathfrak{g } ,natural , V$ . All such Lie algebra structures on $E$ are classified by two cohomological type objects which are explicitly constructed. The first one $mathcal{H }^{2}_{mathfrak{g }} (V, mathfrak{g })$ will classify all Lie algebra structures on $E$ up to an isomorphism that stabilizes $mathfrak{g }$ while the second object $mathcal{H }^{2} (V, mathfrak{g })$ provides the classification from the view point of the extension problem. Several examples that compute both classifying objects $mathcal{H }^{2}_{mathfrak{g }} (V, mathfrak{g })$ and $mathcal{H }^{2} (V, mathfrak{g })$ are worked out in detail in the case of flag extending structures.