Identifying the simple finite-dimensional Lie algebras over $\mathbb{C}$ by means of simple sequences

TL;DR

The paper introduces a new method to identify simple finite-dimensional Lie algebras over complex numbers using sequences of minors of Cartan matrices, simplifying the classification process.

Contribution

It provides a necessary and sufficient condition based on minors of Cartan matrices, eliminating the need for explicit Lie algebra construction for each Dynkin diagram.

Findings

The method accurately classifies all simple finite-dimensional Lie algebras over ield.

It reduces computational complexity by using minors of Cartan matrices.

The approach streamlines the identification process of Dynkin diagrams.

Abstract

A novel method of determining which Dynkin diagrams represent simple finite-dimensional Lie algebras over $\mathbb{C}$ is presented. It is based on a condition that is both necessary and sufficient for a suitably defined Cartan matrix to be expressible by scalar products in a Euclidean vector space. The sufficiency of this condition makes unnecessary subsequent verification of the existence of a Lie algebra or root system corresponding to each Dynkin diagram by explicit construction. The Dynkin diagrams are selected by examination of an easily calculated sequence of minors of a symmetrised Cartan matrix. These minors are mostly integers.