On the uniqueness of maximal solvable extensions of nilpotent Leibniz superalgebras

Abstract

In the present paper under certain conditions the description of the maximal solvable extension of complex finite-dimensional nilpotent Leibniz superalgebras is obtained. Specifically, we establish that under the condition ensuring the fulfillment of Lie's theorem for a maximal solvable extension of a special kind of nilpotent Leibniz superalgebra (which is consistent and d-locally diagonalizable), it is decomposed into a semidirect sum of a nilpotent Leibniz superalgebra and a maximal torus on it. In other words, under certain conditions the direct sum of the nilpotent superalgebra and its torus (as a vector spaces), admits a solvable Leibniz superalgebra structure. In addition, for the left-side action of a maximal torus on nilpotent Leibniz superalgebra, which does not admit $C^p$ as a direct summand and is diagonalizable, we prove the uniqueness of the maximal extension. Along with the answer to Šnobl's conjecture for Lie algebras this result covers several already known results for Lie (super)algebras and Leibniz algebras.