Unified products for Leibniz algebras. Applications

Abstract

Let g be a Leibniz algebra and E a vector space containing g as a subspace. All Leibniz algebra structures on E containing g as a subalgebra are explicitly described and classified by two non-abelian cohomological type objects: HLg2(V,g) provides the classification up to an isomorphism that stabilizes g and HL2(V,g) will classify all such structures from the view point of the extension problem – here V is a complement of 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 g⊆E of Leibniz algebras are given as a converse of the factorization problem. They are classified by another cohomological object denoted by HA2(h,g|(▹,◃,↼,⇀)), where (▹,◃,↼,⇀) is the canonical matched pair associated to a given complement h. Several examples are worked out in details.