Let $\mathfrak{ g}$ be a Lie algebra, $E$ a vector space containing $\mathfrak{ g}$ as a subspace. The paper is devoted to the \emph{ 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 ofthe 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.
Categories: math.RA math-ph math.DG math.MP