# nLab E8

Contents

## Philosophy

group theory

### Cohomology and Extensions

#### Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Idea

The Lie group called $E_8$ is the largest-dimensional one of the five exceptional Lie groups.

## Properties

### As part of the ADE pattern

Dynkin diagram/
Dynkin quiver
dihedron,
Platonic solid
finite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
$A_{n \geq 1}$cyclic group
$\mathbb{Z}_{n+1}$
cyclic group
$\mathbb{Z}_{n+1}$
special unitary group
$SU(n+1)$
A1cyclic group of order 2
$\mathbb{Z}_2$
cyclic group of order 2
$\mathbb{Z}_2$
SU(2)
A2cyclic group of order 3
$\mathbb{Z}_3$
cyclic group of order 3
$\mathbb{Z}_3$
SU(3)
A3
=
D3
cyclic group of order 4
$\mathbb{Z}_4$
cyclic group of order 4
$2 D_2 \simeq \mathbb{Z}_4$
SU(4)
$\simeq$
Spin(6)
D4dihedron on
bigon
Klein four-group
$D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$
quaternion group
$2 D_4 \simeq$ Q8
SO(8), Spin(8)
D5dihedron on
triangle
dihedral group of order 6
$D_6$
binary dihedral group of order 12
$2 D_6$
SO(10), Spin(10)
D6dihedron on
square
dihedral group of order 8
$D_8$
binary dihedral group of order 16
$2 D_{8}$
SO(12), Spin(12)
$D_{n \geq 4}$dihedron,
hosohedron
dihedral group
$D_{2(n-2)}$
binary dihedral group
$2 D_{2(n-2)}$
special orthogonal group, spin group
$SO(2n)$, $Spin(2n)$
$E_6$tetrahedrontetrahedral group
$T$
binary tetrahedral group
$2T$
E6
$E_7$cube,
octahedron
octahedral group
$O$
binary octahedral group
$2O$
E7
$E_8$dodecahedron,
icosahedron
icosahedral group
$I$
binary icosahedral group
$2I$
E8

### Homotopy groups

The first nontrivial homotopy group of the topological space underlying $E_8$ is

$\pi_3(E_8) \simeq \mathbb{Z}$

as for any compact Lie group. Then the next nontrivial homotopy group is

$\pi_{15}(E_8) \simeq \mathbb{Z} \,.$

This means that all the way up to the 15 coskeleton the group $E_8$ looks, homotopy theoretically like the Eilenberg-MacLane space $K(\mathbb{Z},3) \simeq B^3 \mathbb{Z} \simeq B^2 U(1) \simeq B \mathbb{C}P^\infty$.

### Subgroups

The subgroup of the exceptional Lie group E8 which corresponds to the Lie algebra-inclusion $\mathfrak{so}(16) \hookrightarrow \mathfrak{e}_8$ is the semi-spin group SemiSpin(16)

$SemiSpin(16) \;\subset\; E_8$

On the other hand, the special orthogonal group $SO(16)$ is not a subgroup of $E_8$ (e.g. McInnes 99a, p. 11).

### Invariant polynomials

By the above discussion of homotopy groups, it follows (by Chern-Weil theory) that the first invariant polynomials on the Lie algebra $\mathfrak{e}_8$ are the quadratic Killing form and then next an octic polynomial. That is described in (Cederwall-Palmkvist).

### As U-duality of 3d SuGra

$E_8$ is the U-duality group (see there) of 11-dimensional supergravity compactified to 3 dimensions.

supergravity gauge group (split real form)T-duality group (via toroidal KK-compactification)U-dualitymaximal gauged supergravity
$SL(2,\mathbb{R})$1$SL(2,\mathbb{Z})$ S-duality10d type IIB supergravity
SL$(2,\mathbb{R}) \times$ O(1,1)$\mathbb{Z}_2$$SL(2,\mathbb{Z}) \times \mathbb{Z}_2$9d supergravity
SU(3)$\times$ SU(2)SL$(3,\mathbb{R}) \times SL(2,\mathbb{R})$$O(2,2;\mathbb{Z})$$SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})$8d supergravity
SU(5)$SL(5,\mathbb{R})$$O(3,3;\mathbb{Z})$$SL(5,\mathbb{Z})$7d supergravity
Spin(10)$Spin(5,5)$$O(4,4;\mathbb{Z})$$O(5,5,\mathbb{Z})$6d supergravity
E6$E_{6(6)}$$O(5,5;\mathbb{Z})$$E_{6(6)}(\mathbb{Z})$5d supergravity
E7$E_{7(7)}$$O(6,6;\mathbb{Z})$$E_{7(7)}(\mathbb{Z})$4d supergravity
E8$E_{8(8)}$$O(7,7;\mathbb{Z})$$E_{8(8)}(\mathbb{Z})$3d supergravity
E9$E_{9(9)}$$O(8,8;\mathbb{Z})$$E_{9(9)}(\mathbb{Z})$2d supergravityE8-equivariant elliptic cohomology
E10$E_{10(10)}$$O(9,9;\mathbb{Z})$$E_{10(10)}(\mathbb{Z})$
E11$E_{11(11)}$$O(10,10;\mathbb{Z})$$E_{11(11)}(\mathbb{Z})$

The group $E_8$ plays a role in some exceptional differential geometry/differential cohomology. See for instance

• G2, F4,

E6, E7, E8, E9, E10, E11, $\cdots$

## References

### General

Surveys:

An introductory survey with an eye towards the relation to the octonions is given in section 4.6 of

### Homotopy groups

The lower homotopy groups of $E_8$ are a classical result due to

• Raoul Bott and H. Samelson, Application of the theory of Morse to symmetric spaces , Amer.

J. Math., 80 (1958), 964-1029.

The higher homotopy groups are discussed in

• Hideyuki Kachi, Homotopy groups of compact Lie groups $E_6$, $E_7$ and $E_8$ Nagoya Math. J. Volume 32 (1968), 109-139. (project EUCLID)

The octic invariant polynomial of $E_8$ is discussed in
On string bordism of the classifying space of $E_8$:
• Michael Hill, The $String$ bordism of $B E_8$ and $B E_8 \times B E_8$ through dimension 14, Ill. J. Math. 53 1 (2009) 183-196 [doi:10.1215/ijm/1264170845]