Working with E8+++

I wanted to confirm some work being done with E11 (or more specifically the Extended E8+++), so I used my “VisibLie” notebook (which includes the “SuperLie” package for analyzing Lie Algebras) to get the following information:

I first created the Dynkin diagram which produces the Cartan Matrix:

This allows the evaluation of the EigenSystem of the Cartan matrix as follows:

As well as using the SuperLie package to evaluate the Positive Roots, Weights and Heights of the E8+++ (The first 80 are shown. For the full list in .pdf click here or click on the image below):

The 4870 positive roots up to height 47 generate a Hasse diagram as follows (with 3 sections zoomed in):

I am working on replication of the Petrie projection by A.G. Lisi based on his prescription using his basis vectors for 2D projection and a Simple Roots Matrix:

Adjusting this E11 Simple Roots Matrix (srmE11) to match the Dynkin diagram above and then verifying it creates the same Cartan matrix, we have:

To get the 9740 E8+++ vertices, we simply use the dot product of the transpose of srmE11 against the positive and negative root vectors (first 50 shown):

A first cut at plotting these involves projecting them to a 2D using the dot product of the 2 basis vectors provided, which gives the following ListPlot:

The limited results from SuperLie are consistent with the image given by Lisi on the Bee’s BackReaction blog, after eliminating the vertical “levels” in my raw plot.

Email me if you want more detail, see errors, or would like to help.


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