Digitalization of exceptional simple Lie algebras into matrices over complex numbers
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.
Keywords
Cite
@article{arxiv.2010.06696,
title = {Digitalization of exceptional simple Lie algebras into matrices over complex numbers},
author = {Takao Imai},
journal= {arXiv preprint arXiv:2010.06696},
year = {2022}
}
Comments
TeXworks v0.6.6(Debian),379pages with 5 figures and 90 pages of matrix table. I added the following digitalization in the version3: An exceptional Lie algebra of type F4 into 27x27 matrices, an exceptional Lie algebra of type E6 into 27x27 matrices, an exceptional Lie algebra of type G2 into 8x8 matrices, and a subalgebra sp(4) of r8C