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

Abstract

Let be a Leibniz algebra, be a vector space and be an epimorphism of vector spaces with . The global extension problem asks for the classification of all Leibniz algebra structures that can be defined on such that is 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 on are classified by a global cohomological object which is explicitly constructed. It is shown that is the coproduct of all local cohomological objects which are classifying sets for all extensions of by all Leibniz algebra structures on . The second cohomology group of 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 algebras that have a finite chain of epimorphisms of Leibniz algebras such that , for all . Metabelian Leibniz algebras are introduced, described and classified using pure linear algebra tools.

Keywords:

• Leibniz algebras: the extension problem
• non-abelian cohomology
• metabelian and co-flag structures

AMS Subject Classifications:

• 17A32
• 17B05
• 17B56

Notes

1 The next proposition can be also proven directly by testing the Leibniz law for the bracket (33 ) in all points of the form and , for all , .

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, [grant number 88/05.10.2011].