{"id":5161,"date":"2018-11-24T08:59:09","date_gmt":"2018-11-24T15:59:09","guid":{"rendered":"http:\/\/theoryofeverything.org\/theToE\/?p=5161"},"modified":"2018-12-08T15:47:28","modified_gmt":"2018-12-08T22:47:28","slug":"working-with-e8","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2018\/11\/24\/working-with-e8\/","title":{"rendered":"Working with E8+++"},"content":{"rendered":"

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<\/a> for analyzing Lie Algebras) to get the following information:<\/p>\n

I first created the Dynkin diagram which produces the Cartan Matrix:
\n\"\"<\/a><\/p>\n

This allows the evaluation of the EigenSystem of the Cartan matrix as follows:
\n\"\"<\/a><\/p>\n

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<\/a> or click on the image below):
\n\"\"<\/a><\/p>\n

The 4870 positive roots up to height 47 generate a Hasse diagram as follows (with 3 sections zoomed in):
\n\"\"<\/a><\/p>\n

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:
\n\"\"<\/a><\/p>\n

Adjusting this E11 Simple Roots Matrix (srmE11) to match the Dynkin diagram above and then verifying it creates the same Cartan matrix, we have:
\n\"\"<\/a><\/p>\n

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):
\n\"\"<\/a><\/p>\n

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:
\n\"\"<\/a><\/p>\n

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

Email me if you want more detail, see errors, or would like to help.<\/p>\n

Greg
\nJGMoxness@theoryofeverything.org<\/p>\n","protected":false},"excerpt":{"rendered":"

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 … Continue reading Working with E8+++<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[11,4],"class_list":["post-5161","post","type-post","status-publish","format-standard","hentry","category-physics","tag-2d","tag-e8"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/5161","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/comments?post=5161"}],"version-history":[{"count":5,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/5161\/revisions"}],"predecessor-version":[{"id":5220,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/5161\/revisions\/5220"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=5161"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=5161"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=5161"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}