Extending structures for Lie algebras

Abstract

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.