On maximal solvable extensions of a pure non-characteristically nilpotent Lie algebra

TL;DR

This paper introduces a new class of Lie algebras called pure non-characteristically nilpotent and characterizes their maximal solvable extensions, showing they are semidirect sums with specific properties.

Contribution

It defines pure non-characteristically nilpotent Lie algebras and proves their maximal extensions are semidirect sums, also identifying subclasses with trivial cohomology.

Findings

Maximal extensions are isomorphic to semidirect sums of nilradical and torus

Such solvable Lie algebras are complete

Certain subclasses have trivial cohomology group

Abstract

In this paper we introduce the notion of pure non-characteristically nilpotent Lie algebra and under a condition we prove that a complex maximal extension of a finite-dimensional pure non-characteristically nilpotent Lie algebra is isomorphic to a semidirect sum of the nilradical and its maximal torus. We also prove that such solvable Lie algebras are complete and we specify a subclass of the maximal solvable extensions of pure non-characteristically nilpotent Lie algebras that have trivial cohomology group. Some comparisons with the results obtained earlier are given.