On some solvable Leibniz algebras and their completeness

Abstract

The paper is devoted studying solvable Leibniz algebras with a nilradical possessing the codimension equals the number of its generators. We describe this class in non-split nilradical case. Then the case of split nilradical is worked out. We show that the results obtained earlier on this class of Leibniz algebras come as particular cases of the results of this paper. It is shown that such a solvable extension is unique. Finally, we prove that the solvable Leibniz algebras considered are complete.

Leibniz algebras whose solvable ideal is the maximal extension of the nilradical

“…Furthermore, by τ we denote the nilindex of N , that is, N τ −1 = 0 and N τ = 0. The description of solvable Leibniz algebras for which the codimension of nilradical is equal to the number of generators of the nilradical is given in [1].…”

“…It should be noted that the solvable Leibniz algebra R is given in Theorem 3.3 is also complete [1].…”

“…Also, K.K. Abdurasulov [1] proved the completeness of a solvable Leibniz algebra with the nilradical of codimension equals the number of generators of the nilradical.…”