Unified products for Leibniz algebras. Applications

Abstract

Let $\mathfrak{g}$ be a Leibniz algebra and $E$ a vector space containing $\mathfrak{g}$ as a subspace. All Leibniz algebra structures on $E$ containing $\mathfrak{g}$ as a subalgebra are explicitly described and classified by two non-abelian cohomological type objects: ${\mathcal H}{\mathcal L}^{2}_{\mathfrak{g}} \, (V, \, \mathfrak{g})$ provides the classification up to an isomorphism that stabilizes $\mathfrak{g}$ and ${\mathcal H}{\mathcal L}^{2} \, (V, \, \mathfrak{g})$ will classify all such structures from the view point of the extension problem - here $V$ is a complement of $\mathfrak{g}$ in $E$. A general product, called the unified product, is introduced as a tool for our approach. The crossed (resp. bicrossed) products between two Leibniz algebras are introduced as special cases of the unified product: the first one is responsible for the extension problem while the bicrossed product is responsible for the factorization problem. The description and the classification of all complements of a given extension $\mathfrak{g} \subseteq \mathfrak{E} $ of Leibniz algebras are given as a converse of the factorization problem. They are classified by another cohomological object denoted by ${\mathcal H}{\mathcal A}^{2}(\mathfrak{h}, \mathfrak{g} \, | \, (\triangleright, \triangleleft, \leftharpoonup, \rightharpoonup))$, where $(\triangleright, \triangleleft, \leftharpoonup, \rightharpoonup)$ is the canonical matched pair associated to a given complement $\mathfrak{h}$. Several examples are worked out in details.