Classification of Leibniz algebras with a given nilradical and with some corresponding Lie algebra

Resumo

The classification problem of finite-dimensional solvable Leibniz algebras is fundamental and a very difficult problem. It is split into two parts: (1) classification of nilpotent Leibniz algebras; (2) description of solvable Leibniz algebras with given nilradical. The first problem is most complicated, even in Lie algebras case. For the second problem Mubarakzjanov developed a method for solvable Lie algebras. Namely, due to this method any solvable Lie algebra can be represented as an algebraic sum of the nilradical and its complimentary vector space, whose dimension does not exceed the number of nil-independent derivations of the nilradical. Using this method the classifications of solvable Lie algebras with filiform, quasi-filiform, abelian, etc., nilradicals are obtained. It is planned to adapt Mubarakzjanov’s method for Leibniz algebras and using the adapted method we plan to classify solvable Leibniz algebras with given nilradicals. Another line of planned research is a description of Leibniz algebras whose quotient Lie algebra by the ideal spanned by squares of elements of the algebra (called corresponding Lie algebra) is a given Lie algebra. Note that each non-Lie Leibniz algebra contains a non-trivial ideal, which is the subspace spanned by squares of elements (denoted by I). In particular, we will investigate Leibniz algebras with corresponding Lie algebras such as filiform, Diamond Lie algebras, etc., and with the ideal I defined as Leibniz module over the mentioned Lie algebras.