{"id":2023,"date":"2014-10-29T10:05:49","date_gmt":"2014-10-29T17:05:49","guid":{"rendered":"http:\/\/theoryofeverything.org\/MyToE\/?p=2023"},"modified":"2014-10-29T10:05:49","modified_gmt":"2014-10-29T17:05:49","slug":"hyperbolic-dynkin-136-detail","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2014\/10\/29\/hyperbolic-dynkin-136-detail\/","title":{"rendered":"Hyperbolic Dynkin #136 Detail"},"content":{"rendered":"

In followup to John Baez’ G+<\/a> thread on Hyperbolic Dynkin diagrams<\/a>, specifically on the only rank 4 compact symmetrizable diagram 136, I used my “VisibLie” notebook<\/a> (which includes the “SuperLie” package<\/a> for analyzing Lie Algebras) to get the following information:\ufeff<\/p>\n

\"H136_detail3\"<\/a><\/p>\n

Using the “SimpLie” Google Code OpenSource software<\/a>, we get Hasse diagram of:
\n\"Projection<\/a>
\nSymmetrized Cartan matrix:
\n 1 -1 0 -1\/2
\n-1 2 -1 0
\n 0 -1 2 -1
\n-1\/2 0 -1 1<\/p>\n

One possible set of basis vectors for this is:
\nsrH136={{0,2,0,0},{0,-1,1,0},{0,0,-1,1},{Sqrt[-7],-1,-1,-3}};<\/p>\n

with 4D angles between each node of:
\n{{1->2,135.},{2->3,120.},{3->4,135.},{4->1,120.},{1->3,90.},{2->4,90.}}<\/p>\n

Norm’d Length between nodes is:
\n{2, Sqrt[2], Sqrt[2], 2}<\/p>\n

as visualized in 2D and 3D, with 42 vertices and 26 edges of Norm’d 4D length of 2 and 94 at Sqrt[2]:
\n\"H136_Lowe_projection7\"<\/a><\/p>\n

\"H136_Lowe_projection8\"<\/a><\/p>\n

\"JGM_ToE-now-21d\"<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

In followup to John Baez’ G+ thread on Hyperbolic Dynkin diagrams, specifically on the only rank 4 compact symmetrizable diagram 136, I used my “VisibLie” notebook (which includes the “SuperLie” package for analyzing Lie Algebras) to get the following information:\ufeff Using the “SimpLie” Google Code OpenSource software, we get Hasse diagram of: Symmetrized Cartan matrix: … Continue reading Hyperbolic Dynkin #136 Detail<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[18,26],"class_list":["post-2023","post","type-post","status-publish","format-standard","hentry","category-physics","tag-dynkin","tag-hasse"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/2023","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=2023"}],"version-history":[{"count":1,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/2023\/revisions"}],"predecessor-version":[{"id":5180,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/2023\/revisions\/5180"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=2023"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=2023"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=2023"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}