E8: An Exceptionally Beautiful Piece of Mathematics

Classification of simple Lie algebras

The classification of semisimple complex Lie algebras is regarded by many as a jewel of mathematics with such great contributors as Killing, Cartan, and Dynkin (see e.g. [1]).

In a nutshell, every semisimple Lie algebra is a direct sum simple Lie algebras, which by definition are non-Abelian Lie algebras whose only ideals are ${\{0\}}$ and themselves. It turns out, that over the complex numbers these simple building blocks are either a member of the four infinite family of the so-called classical Lie algebras, labeled by ${A_n}$ , ${B_n}$ , ${C_n}$ , and ${D_n}$ ( ${n\in\mathbb{N}}$ ) or one of the five exceptional Lie algebras, which are denoted by ${E_6}$ , ${E_7}$ , ${E_8}$ , ${F_4}$ , and ${G_2}$ . The concept of root systems plays an important role in the classification:

A finite set of non-zero vectors ${R}$ in some real (finite-dimensional) Euclidean space ${E,(.,.)}$ is called a root system if the following conditions are met:

(R1) ${R}$ spans ${E}$ ;
(R2) the only multiples of ${\alpha\in R}$ which belong to ${R}$ are ${\pm\alpha}$ ;
(R3) ${R}$ is closed under the reflections through hyperplanes perpendicular to its elements;
(R4) ${\langle\beta,\alpha\rangle:=2(\beta,\alpha)/(\alpha,\alpha)\in{\mathbb Z}}$ , for any ${\alpha,\beta\in R}$ .

The dimension of ${E}$ is called the rank of the root system ${R}$ .

It can be shown that for any root system ${R}$ there exists a subset of roots ${B\subset R}$ called base such that:

(B1) ${B}$ forms a basis of ${E}$ ;
(B2) every root ${\beta\in R}$ can be written as ${\beta=\sum_{\alpha\in B}k_\alpha \alpha}$ with integral coefficients ${k_\alpha}$ all non-negative or all non-positive.

The elements of such a base ${B}$ are called simple roots. For example, there are four rank ${2}$ root systems (up to isomorphism):

Dynkin’s great insight was the invention of simple roots and that all information about a simple Lie algebra can be condensed into a connected graph called Dynkin diagram: Its vertices represent simple roots of a root system of the Lie algebra, and the vertices corresponding to ${\alpha_j}$ and ${\alpha_k}$ are joined by ${\langle\alpha_j,\alpha_k\rangle\langle\alpha_k,\alpha_j\rangle\in\{0,1,2,3\}}$ edges. More than one edges between two vertices indicate that the corresponding roots have different lengths. If this is the case an arrow pointing towards the vertex with the longer root is used.

The classification theorem states that the Dynkin diagram of any simple complex Lie algebra must be one of the following:

Classification of simple complex Lie algebras.

Note: The main goal of the (admittedly crude) summary above is to give a short description of the classification theorem. For a detailed exposition see Humphreys’ book [2] or Terry Tao’s blogpost [3].

E8 root system

Among the exceptional Lie algebras, ${E_8}$ is the largest one. Its root system lives in ${8}$ -dimensional space and can be described as follows. Let ${e_1,\dots,e_8}$ denote the standard basis in ${{\mathbb R}^8}$ , that is

$\displaystyle e_1=(1,0,\dots,0,0),\quad e_2=(0,1,\dots,0,0),\quad\dots,\quad e_8=(0,0,\dots,0,1).$

The ${E_8}$ root system consists of ${240}$ vectors. It has ${112}$ vectors of the form

$\displaystyle \pm e_j\pm e_k\quad (j\neq k)$

with every possible indices and signs. By the way these vectors constitute the ${D_8}$ root system. The remaining ${128}$ vectors can be written as

$\displaystyle \frac{1}{2}(\pm e_1\pm e_2\pm e_3\pm e_4\pm e_5\pm e_6\pm e_7\pm e_8)$

where the number of minus signs must be even.

Notice that these ${240}$ points are on the surface of the ${7}$ -sphere of radius ${\sqrt{2}}$ centered at the origin. In addition, each root has exactly ${56}$ closest neighbours (at distance ${\sqrt{2}}$ ). If we put spheres with radius ${\sqrt{2}/2}$ around every point and the origin, we get a very tightly packed configuration. The sphere at the origin is in contact with all ${240}$ spheres. It turns out that this is the solution of the kissing number problem in ${8}$ dimensions.

The number of symmetries of the ${E_8}$ root system is immense. Its symmetry group (Weyl group) is of order ${696,729,600(=4!\times 6!\times 8!)}$ . Compare this with the ${8}$ -cube which has ‘only’ ${10,321,920(=2^8\times 8!)}$ symmetries.

Unfortunately, we can’t visualize ${E_8}$ in its ${8}$ -dimensional glory, but there is something we can do. We can connect the points closest to each other and look at its “shadow” on the so-called Coxeter plane to get something like this:

This particular projection is “as symmetric as possible” and all of the points are distinct, although many edges overlap. To my knowledge this beautiful projection was first drawn by hand(!) by P. McMullen in the 60’s, and then by J. Stembridge (along with many other projections) using a computer around 2007 [4]. A few month ago I saw a Spanish blogpost from 2010 [5], where it was made by hand using strings. Inspired by this, me and my girlfriend made our own ${E_8}$ about two month ago. Here is a gallery of the process:

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As already mentioned the root system ${D_8}$ is a subset of ${E_8}$ . It also contains other root systems, such as ${A_8}$ , ${E_6}$ , and ${E_7}$ . Some of these are nicely demonstrated in a movie made by T. Nutma [6], where “the camera” rotates through a selection of Coxeter planes in succession:

References

[1] A.J. Coleman, The greatest mathematical paper of all time, The Mathematical Intelligencer 11(3) 29-38, 1989.

[2] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9, Springer, 1972.

[3] T. Tao, Notes on the classification of complex Lie algebras, blogpost, April 27, 2013.

[4] J. Stembridge, Coxeter Planes, April 9, 2007.

[5] J.L. Rodríguez Blancas, Hilorama de E8, blogpost, October 10, 2010.

[6] T. Nutma, The Exceptional Symmetry of E8, video, September 8, 2014.

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Tamás Görbe

A blog about maths & physics