Indecomposable Lie algebras with nontrivial Levi decomposition cannot have filiform radical

Abstract

It is shown that a semidirect sum g = s − → ⊕ R r of a semisimple Lie algebra s and a solvable Lie algebra r with respect to a representation of s which does not decompose into a direct sum of ideals cannot have a radical r associated to a filiform Lie algebra. This proves that this class of nilpotent Lie algebras has none interest for the structure theory of nonsolvable Lie algebras. Mathematics Subject Classification: 17B05, 17B10

Classification of solvable Leibniz algebras with naturally graded filiform nilradical

“…in a similar way as in the case of Lie algebras, the description of Leibniz algebras is reduced to the description of the solvable ones. The analysis of works devoted to the study of solvable Lie algebras (for example [1,3,11,13,15], where solvable Lie algebras with various types of nilradical were studied, such as naturally graded filiform and quasi-filiform algebras, abelian, triangular, etc.) shows that we can also apply similar methods to solvable Leibniz algebras with a given nilradical.…”

On solvable Lie and Leibniz superalgebras with maximal codimension of nilradical

“…He noted that the codimension of nilradical does not exceed the number of nil-independent derivations and the number of generators of the nilradical. Since then the classification of solvable Lie algebras with the Abelian, Heisenberg, filiform, quasi-filiform nilradicals are obtained, see for instance [5,6,22,26,27,25]. Recently, the topic of research has drawn a lot of attention and in particular in [13] the authors extended Mubarakzjanov's method to Leibniz algebras.…”

Maximal solvable extensions of a pure non-characteristically nilpotent Lie algebra

“…The research of solvable Lie algebras with some special types of nilradical comes from different problems in Physics and was the subject of various papers [3,4,5,6,16,20,21,23,24] and many other references given therein. For example, the cases where the nilradical of a solvable Lie algebra is filiform, quasi-filiform and Abelian were considered in [4,6,16,24].…”