Solvable quadratic Lie algebras in low dimensions

TL;DR

This paper classifies all solvable quadratic Lie algebras up to dimension 6, identifying indecomposable examples using decomposition and extension methods, thereby expanding the understanding of their structure.

Contribution

It provides a complete classification of solvable quadratic Lie algebras in low dimensions, introducing new families and applying advanced decomposition techniques.

Findings

Identified two non-Abelian indecomposable solvable quadratic Lie algebras below dimension 6.

Classified three families of indecomposable solvable quadratic Lie algebras in dimension 6.

Extended the classification framework using double extension methods.

Abstract

In this paper, we classify solvable quadratic Lie algebras up to dimension 6. In dimensions smaller than 6, we use the Witt decomposition given in \cite{Bou59} and a result in \cite{PU07} to obtain two non-Abelian indecomposable solvable quadratic Lie algebras. In the case of dimension 6, by applying the method of double extension given in \cite{Kac85} and \cite{MR85} and the classification result of singular quadratic Lie algebras in \cite{DPU}, we have three families of solvable quadratic Lie algebras which are indecomposable and not isomorphic.