Digitalization of exceptional simple Lie algebras into matrices over complex numbers

TL;DR

This paper digitally represents exceptional simple Lie algebras as matrices over complex numbers using Maxima, verifying their algebraic properties and identifying their types through root system calculations.

Contribution

It introduces a method to digitalize exceptional Lie algebras into matrices and verifies their structure and classification computationally.

Findings

Digitalized matrices form simple Lie algebras under Lie brackets.

Root systems are computed to identify algebra types.

Examples of classical subalgebras are provided.

Abstract

We give the images of the adjoint representations of exceptional simple Lie algebras by matrices over complex numbers. Next, we digitalize these matrices by the use of Maxima, which is a computer algebra system. These digitalized matrices are provided by using Maxima's function. We prove that these digitalized matrices are closed by Lie bracket operations and make up simple Lie algebras. Moreover, to prove the type of the exceptional simple Lie algebra, we calculate the root system using Maxima for Lie bracket operations as matrix calculations. We show some examples of classical Lie subalgebras of these digitalized matrices.