nLab E₈

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

ADE classification and McKay correspondence

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 $SU(n+1)$
A1		cyclic group of order 2 $\mathbb{Z}_2$	cyclic group of order 2 $\mathbb{Z}_2$	SU(2)
A2		cyclic 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)
D4	dihedron 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)
D5	dihedron on triangle	dihedral group of order 6 $D_6$	binary dihedral group of order 12 $2 D_6$	SO(10), Spin(10)
D6	dihedron 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 $SO(2n)$, $Spin(2n)$
$E_6$	tetrahedron	tetrahedral 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).

Under the inclusion of the maximal compact subgroup, the fundamental representation of $E_{8(8)}$ branches as

(e.g. Hohm & Samtleben 2014 p 4)

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-duality	maximal gauged supergravity
	$SL(2,\mathbb{R})$	1	$SL(2,\mathbb{Z})$ S-duality	D=10 type IIB supergravity
	SL$(2,\mathbb{R}) \times$ O(1,1)	$\mathbb{Z}_2$	$SL(2,\mathbb{Z})$ $\times \mathbb{Z}_2$	D=9 supergravity
SU(3)$\times$ SU(2)	SL$(3,\mathbb{R}) \times SL(2,\mathbb{R})$	$O(2,2;\mathbb{Z})$	$SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})$	D=8 supergravity
SU(5)	$SL(5,\mathbb{R})$	$O(3,3;\mathbb{Z})$	$SL(5,\mathbb{Z})$	D=7 supergravity
Spin(10)	$Spin(5,5)$	$O(4,4;\mathbb{Z})$	$O(5,5,\mathbb{Z})$	D=6 supergravity
E₆	$E_{6(6)}$	$O(5,5;\mathbb{Z})$	$E_{6(6)}(\mathbb{Z})$	D=5 supergravity
E₇	$E_{7(7)}$	$O(6,6;\mathbb{Z})$	$E_{7(7)}(\mathbb{Z})$	D=4 supergravity
E₈	$E_{8(8)}$	$O(7,7;\mathbb{Z})$	$E_{8(8)}(\mathbb{Z})$	D=3 supergravity
E₉	$E_{9(9)}$	$O(8,8;\mathbb{Z})$	$E_{9(9)}(\mathbb{Z})$	D=2 supergravity	E₈-equivariant elliptic cohomology
E₁₀	$E_{10(10)}$	$O(9,9;\mathbb{Z})$	$E_{10(10)}(\mathbb{Z})$
E₁₁	$E_{11(11)}$	$O(10,10;\mathbb{Z})$	$E_{11(11)}(\mathbb{Z})$

(Hull-Townsend 94, table 1, table 2)

Related entries

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

• exceptional generalized geometry, supergravity C-field, Hořava-Witten theory, heterotic string theory

• G₂, F₄,

  E₆, E₇, E₈, E₉, E₁₀, E₁₁, $\cdots$

References

General

Surveys:

• Skip Garibaldi, $E_8$, the most exceptional group (arXiv:1605.01721)

• Wikipedia, E₈

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

• John Baez, The Octonions (web)

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)

See also

• André Henriques, MO comment

Invariant polynomials

The octic invariant polynomial of $E_8$ is discussed in

• Martin Cederwall, Jakob Palmkvist, The octic $E_8$ invariant J.Math.Phys.48:073505 (2007) (arXiv:hep-th/0702024)

More

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]

On $E_8$-exceptional field theory-formulation of D=11 supergravity:

• Olaf Hohm, Henning Samtleben: Exceptional Field Theory III: $E_{8(8)}$, Phys. Rev. D 90 066002 (2014) [arXiv:1406.3348, doi:10.1103/PhysRevD.90.066002]