Extending structures for Lie algebras

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