Search

We prove that the celebrated Itô’s theorem for groups remains valid at the level of Leibniz algebras: ifis a Leibniz algebra such that, for two abelian subalgebrasand, thenis metabelian, i.e.. A structure-type theorem for metabelian Leibniz/Lie algebras is proved. All metabelian Leibniz algebras having the derived algebra of dimensionare described, classified and their automorphisms groups are explicitly determined as subgroups of a semidirect product of groupsassociated to any vector space. [ABSTRACT FROM PUBLISHER]

Letbe a Leibniz algebra,be a vector space andbe an epimorphism of vector spaces with. The global extension problem asks for the classification of all Leibniz algebra structures that can be defined onsuch thatis a morphism of Leibniz algebras: from a geometrical viewpoint this is 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 onare classified by a global cohomological objectwhich is explicitly constructed. It is shown thatis the coproduct of all local cohomological objectswhich are classifying sets for all extensions ofby all Leibniz algebra structureson. The second cohomology groupof Loday and Pirashvili appears as the most elementary piece among all components of. Several examples are worked out in details for co-flag Leibniz algebras over, i.e. Leibniz algebrasthat have a finite chain of epimorphisms of Leibniz algebrassuch that, for all. Metabelian Leibniz algebras are introduced, described and classified using pure linear algebra tools. [ABSTRACT FROM AUTHOR]