Solvable extensions of the naturally graded quasi-filiform Leibniz algebra of second type ℒ2

For a sequence of the naturally graded quasi-filiform Leibniz algebra of second type ?² introduced by Camacho, Gómez, González and Omirov, all possible right and left solvable indecomposable extensions over the field ? are constructed so that the algebra serves as the nilradical of the corresponding solvable Leibniz algebras we find in the paper.     

Right and left solvable extensions of an associative Leibniz algebra

A sequence of nilpotent Leibniz algebras denoted by N_n,18 is introduced. Here n denotes the dimension of the algebra defined for n≥4; the first term in the sequence is ?₁₈ in the list of four-dimensional nilpotent Leibniz algebras introduced by Albeverio et al. [4 Albeverio, S., Omirov, B. A., Rakhimov, I. S. (2006). Classification of 4-dimensional nilpotent complex Leibniz algebras. Extr. Math. 21(3):197–210. [Google Scholar]]. Then all possible right and left solvable indecomposable extensions over the field ? are constructed so that N_n,18 serves as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program established to classify solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

Solvable extensions of naturally graded quasi-filiform Leibniz algebras of second type ℒ1 and ℒ3

For sequences of naturally graded quasi-filiform Leibniz algebras of second type ?¹ and ?³ introduced by Camacho et al., all possible right and left solvable indecomposable extensions over the field ? are constructed so that these algebras serve as the nilradicals of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program established to classify solvable Lie algebras using special properties rather than trying to extend one dimension at a time.