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Advances in Applied Clifford Algebras

, Volume 27, Issue 1, pp 397–421 | Cite as

The E 8 Geometry from a Clifford Perspective

  • Pierre-Philippe Dechant
Open Access
Article

Abstract

This paper considers the geometry of E 8 from a Clifford point of view in three complementary ways. Firstly, in earlier work, I had shown how to construct the four-dimensional exceptional root systems from the 3D root systems using Clifford techniques, by constructing them in the 4D even subalgebra of the 3D Clifford algebra; for instance the icosahedral root system H 3 gives rise to the largest (and therefore exceptional) non-crystallographic root system H 4. Arnold’s trinities and the McKay correspondence then hint that there might be an indirect connection between the icosahedron and E 8. Secondly, in a related construction, I have now made this connection explicit for the first time: in the 8D Clifford algebra of 3D space the 120 elements of the icosahedral group H 3 are doubly covered by 240 8-component objects, which endowed with a ‘reduced inner product’ are exactly the E 8 root system. It was previously known that E 8 splits into H 4-invariant subspaces, and we discuss the folding construction relating the two pictures. This folding is a partial version of the one used for the construction of the Coxeter plane, so thirdly we discuss the geometry of the Coxeter plane in a Clifford algebra framework. We advocate the complete factorisation of the Coxeter versor in the Clifford algebra into exponentials of bivectors describing rotations in orthogonal planes with the rotation angle giving the correct exponents, which gives much more geometric insight than the usual approach of complexification and search for complex eigenvalues. In particular, we explicitly find these factorisations for the 2D, 3D and 4D root systems, D 6 as well as E 8, whose Coxeter versor factorises as \({W = \exp(\frac{\pi}{30}B_C)\exp(\frac{11\pi}{30}B_2)\exp(\frac{7\pi}{30}B_3)\exp(\frac{13\pi}{30}B_4)}\) . This explicitly describes 30-fold rotations in 4 orthogonal planes with the correct exponents \({\{1, 7, 11, 13, 17, 19, 23, 29\}}\) arising completely algebraically from the factorisation.

Mathematics Subject Classification

52B10 52B12 52B15 15A66 20F55 17B22 14E16 

Keywords

E8 Exceptional phenomena Clifford algebras Icosahedral symmetry Coxeter groups Root systems Spinors Coxeter plane Lie algebras Lie groups Representation theory Quantum algebras Trinities McKay correspondence 

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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Departments of Mathematics and Biology, York Centre for Complex Systems AnalysisUniversity of YorkHeslingtonUK

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