Solvable Lie algebras with Borel nilradicals

This paper is part of a research program the aim of which is to find all indecomposable solvable extensions of a given class of nilpotent Lie algebras. Specifically in this paper, we consider a nilpotent Lie algebra n that is isomorphic to the nilradical of the Borel subalgebra of a complex simple Lie algebra or of its split real form. We treat all the classical and exceptional simple Lie algebras in a uniform manner. We identify the nilpotent Lie algebra n as the one that consists of all positive root spaces. We present general structural properties of all solvable extensions of n. In particular, we study the extension by one non-nilpotent element and by the maximal number of such elements. We show that the extension of maximal dimension is always unique and isomorphic to the Borel subalgebra of the corresponding simple Lie algebra.