On the uniqueness of maximal solvable extensions of nilpotent Lie superalgebras

Abstract: In the present paper under certain conditions the description of maximal solvable extension of nilpotent Lie superalgebras is obtained. Namely, we establish that under the condition which ensures the fulfillment of an analogue of Lie's theorem, a maximal solvable extension of a $d$-locally diagonalizable nilpotent Lie (super)algebra is decomposed into a semidirect sum of its nilradical and a maximal torus of the nilradical. In the case of Lie algebras this result gives an answer to \v{S}nobl's conjecture. We present an alternative method of describing of the aforementioned solvable Lie (super)algebras. In addition, a criterion of the completeness of solvable Lie algebras is established and it is proved that the first group of cohomologies with coefficient itself for a solvable Lie algebra, vanishes if and only if it is a maximal solvable extension of a d-locally diagonalizable nilpotent Lie algebra. Finally, we present an example to illustrate that the result obtained for the descriptions of maximal solvable extensions of nilpotent Lie (super)algebras does not hold true for Leibniz algebras.