Activities per year
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.
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.
| Original language | English |
|---|---|
| Pages (from-to) | 397–421 |
| Number of pages | 25 |
| Journal | Advances in Applied Clifford Algebras |
| Volume | 27 |
| Issue number | 1 |
| Early online date | 28 Apr 2016 |
| DOIs | |
| Publication status | Published - 8 Mar 2017 |
Keywords
- e8
- exceptional
- root system
- clifford algebra
- lie algebra
- lie group
- coxeter group
- coxeter plane
- invariants
- degrees
- exponents
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Yau Institute: Tsinghua Summer Workshop in Geometry and Physics 2017
Pierre-Philippe Dechant (Keynote/plenary speaker)
6 Aug 2017 → 12 Aug 2017Activity: Talk or presentation › Workshop
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Chern Institute: Nankai Symposium on Physics, Geometry and Number Theory
Pierre-Philippe Dechant (Keynote/plenary speaker)
30 Jul 2017 → 5 Aug 2017Activity: Talk or presentation › Symposium