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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.

In the present paper we prove that a complex maximal extension of a complex finitedimensional nilpotent Lie algebra is isomorphic to a semidirect sum of nilradical and its maximal torus with a product of natural action of the torus. We prove that such solvable Lie algebras are complete. In fact, we prove that among all complex solvable Lie algebras only maximal extensions of nilpotent algebras have not outer derivations. In addition, we allocate a subclass of maximal solvable extensions of nilpotent Lie algebras that have trivial cohomology group. The comparisons with a several existing results are given, as well.

The paper is devoted to the study of pro-solvable Lie algebras whose maximal pronilpotent ideal is either m 0 or m 2 . Namely, we describe such Lie algebras and establish their completeness. Triviality of the second cohomology group for one of the obtained algebra is established.

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