# Nilpotent Decomposition of Solvable Lie Algebras

## TL;DR

This paper introduces a novel method for decomposing solvable Lie algebras into two nilpotent parts, providing new insights into their structure and classification, especially when these parts are Abelian or Heisenberg algebras.

## Contribution

It proposes a new approach to analyze solvable Lie algebras by decomposing them into two nilpotent subalgebras, enhancing understanding of their structure and isomorphism classes.

## Key findings

- Solvable algebras decompose if left nilpotent is Abelian of dimension >1 and right nilpotent is Abelian of dimension 1.
- All solvable algebras with fixed left nilpotent as a Heisenberg algebra and right as Abelian are isomorphic.
- Decomposition helps classify solvable Lie algebras based on properties of their nilpotent components.

## Abstract

Semisimple Lie algebras have been completely classified by Cartan and Killing. The Levi theorem states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of its largest solvable ideal and a semisimple Lie algebra. These focus the classification of solvable Lie algebras as one of the main challenges of Lie algebra research. One approach towards this task is to take a class of nilpotent Lie algebras and construct all extensions of these algebras to solvable ones. In this paper, we propose another approach, i.e., to decompose a solvable nonnilpotent Lie algebra to two nilpotent Lie algebras which are called the left and right nilpotent algebras of the solvable algebra. The right nilpotent algebra is the smallest ideal of the lower central series of the solvable algebra, while the left nilpotent algebra is the factor algebra of the solvable algebra and its right nilpotent algebra. We show that the solvable algebras are decomposable if its left nilpotent algebra is an Abelian algebra of dimension higher than one and its right algebra is an Abelian algebra of dimension one. We further show that all the solvable algebras are isomorphic if their left nilpotent algebras are Heisenberg algebras of fixed dimension and their right algebras are Abelian algebras of dimension one.