---
tag: MKT1263M0A
kind: example
name: Exceptional Lie group E8
msc: ["22E15", "17B25", "22E46"]
instance_of: [AM7YJP8AR6]
uses: [YX5FFSCR8P]
sources:
  - url: https://en.wikipedia.org/wiki/E8_(mathematics)
---

## Example

The compact Lie group $E_8$ is the compact, connected and simply connected [[simple Lie group|AM7YJP8AR6]] whose Lie algebra has the root system of type $E_8$.
The root system $E_8$ lives in $\R^8$ and consists of the $240$ vectors of the following two kinds:

1. the $112$ vectors with two entries $\pm 1$ and six entries $0$;
2. the $128$ vectors $(\pm \tfrac12, \dots, \pm \tfrac12)$ with an even number of minus signs.

Every root has squared length $2$.
Check of the count: there are $\binom{8}{2} \cdot 4 = 112$ vectors of the first kind and $2^8 / 2 = 128$ of the second kind.

The dimension of $E_8$ is the rank plus the number of roots, $8 + 240 = 248$.

## Remarks

$E_8$ is the largest of the five exceptional compact simple Lie groups $G_2, F_4, E_6, E_7, E_8$, see [[G2|YX5FFSCR8P]] for the smallest.
Its Dynkin diagram is a tree with $8$ nodes and branch lengths $1$, $2$ and $4$ from the branch point.
The compact group $E_8$ has trivial centre, so it is at the same time simply connected and adjoint.
Its smallest nontrivial representation is the adjoint representation, of dimension $248$.
The $240$ roots are the shortest nonzero vectors of the [[E8 lattice|2CXH65EKY7]], which is spanned by them.
The Weyl group of $E_8$ has order $696\,729\,600$.
The name $E_8$ also denotes the complex simple Lie group of this type and its real forms.

## Explanations

### L1 human

$E_8$ is the largest and most intricate of the exceptional Lie groups, the ones that do not belong to the infinite families of matrix groups.
It has dimension $248$, yet it is built from a very symmetric arrangement of $240$ vectors in eight-dimensional space.
The lattice that these vectors span gives the densest packing of equal balls in eight dimensions.
The group appears in many places, from the classification of lattices to string theory.