# Minimal Representations of Lie Algebras With Non-Trivial Levi Decomposition

## TL;DR

This paper constructs minimal dimension matrix representations for certain low-dimensional Lie algebras with non-trivial Levi decompositions, using subspace techniques related to semi-simple Lie algebra representations.

## Contribution

It provides explicit minimal matrix representations for specific low-dimensional Lie algebras with Levi decomposition, advancing understanding of their structure.

## Key findings

- Minimal representations for 5-8 dimensional Lie algebras obtained
- Uses subspace techniques related to semi-simple Lie algebra representations
- Provides explicit construction methods for these representations

## Abstract

We obtain minimal dimension matrix representations for each of the Lie algebras of dimension five, six, seven, and eight obtained by Turkowski that have a non-trivial Levi decomposition. The Key technique involves using subspace associated to a particular representation of semi-simple Lie algebra to help in the construction of the radical in the putative Levi decomposition.