Solvable Lie algebras with Borel nilradicals

TL;DR

This paper classifies all solvable Lie algebra extensions with Borel nilradicals, showing that the maximal extension is unique and corresponds to the Borel subalgebra of the simple Lie algebra.

Contribution

It provides a uniform analysis of solvable extensions of nilpotent Lie algebras associated with Borel subalgebras across all simple Lie algebras, identifying the maximal extension.

Findings

Maximal solvable extension is unique and isomorphic to the Borel subalgebra.

Structural properties of all solvable extensions are characterized.

The approach applies uniformly to classical and exceptional Lie algebras.

Abstract

The present article 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 article 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 classical and exceptional simple Lie algebras in a uniform manner. We identify the nilpotent Lie algebra n as the one consisting of all positive root spaces. We present general structural properties of all solvable extensions of n. In particular, we study the extension by one nonnilpotent 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.