The global extension problem, co-flag and metabelian Leibniz algebras

Abstract

Let $\mathfrak{L}$ be a Leibniz algebra, $E$ a vector space and $\pi : E \to \mathfrak{L}$ an epimorphism of vector spaces with $ \mathfrak{g} = {\rm Ker} (\pi)$. The global extension problem asks for the classification of all Leibniz algebra structures that can be defined on $E$ such that $\pi : E \to \mathfrak{L}$ is a morphism of Leibniz algebras: from a geometrical viewpoint this means to give the decomposition of the groupoid of all such structures in its connected components and to indicate a point in each component. All such Leibniz algebra structures on $E$ are classified by a global cohomological object ${\mathbb G} {\mathbb H} {\mathbb L}^{2} \, (\mathfrak{L}, \, \mathfrak{g})$ which is explicitly constructed. It is shown that ${\mathbb G} {\mathbb H} {\mathbb L}^{2} \, (\mathfrak{L}, \, \mathfrak{g})$ is the coproduct of all local cohomological objects $ {\mathbb H} {\mathbb L}^{2} \, \, (\mathfrak{L}, \, (\mathfrak{g}, [-,-]_{\mathfrak{g}}))$ that are classifying sets for all extensions of $\mathfrak{L}$ by all Leibniz algebra structures $(\mathfrak{g}, [-,-]_{\mathfrak{g}})$ on $\mathfrak{g}$. The second cohomology group ${\rm HL}^2 \, (\mathfrak{L}, \, \mathfrak{g})$ of Loday and Pirashvili appears as the most elementary piece among all components of ${\mathbb G} {\mathbb H} {\mathbb L}^{2} \, (\mathfrak{L}, \, \mathfrak{g})$. Several examples are worked out in details for co-flag Leibniz algebras over $\mathfrak{L}$, i.e. Leibniz algebras $\mathfrak{h}$ that have a finite chain of epimorphisms of Leibniz algebras $\mathfrak{L}_n : = \mathfrak{h} \stackrel{\pi_{n}}{\longrightarrow} \mathfrak{L}_{n-1} \, \cdots \, \mathfrak{L}_1 \stackrel{\pi_{1}} {\longrightarrow} \mathfrak{L}_{0} := \mathfrak{L}$ such that ${\rm dim} ({\rm Ker} (\pi_{i})) = 1$, for all $i = 1, \cdots, n$.