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E8

Context

Group Theory

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

The Lie group called E 8E_8 is the largest-dimensional one of the five exceptional Lie groups.

Properties

As part of the ADE pattern

ADE classification

Dynkin diagram via
McKay correspondence
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A nA_ncyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
D n+4D_{n+4}dihedron,
hosohedron
dihedral group
D n+2D_{n+2}
binary dihedral group
2D n+22 D_{n+2}
special orthogonal group
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
E8

Homotopy groups

The first nontrivial homotopy group of the topological space underlying E 8E_8 is

π 3(E 8) \pi_3(E_8) \simeq \mathbb{Z}

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

π 15(E 8). \pi_{15}(E_8) \simeq \mathbb{Z} \,.

This means that all the way up to the 15 coskeleton the group E 8E_8 looks, homotopy theoretically like the Eilenberg-MacLane space K(,3)B 3B 2U(1)BP K(\mathbb{Z},3) \simeq B^3 \mathbb{Z} \simeq B^2 U(1) \simeq B \mathbb{C}P^\infty.

Invariant polynomials

By the above discussion of homotopy groups, it follows (by Chern-Weil theory) that the first invariant polynomials on the Lie algebra 𝔢 8\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 8E_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,)SL(2,\mathbb{R})1SL(2,)SL(2,\mathbb{Z}) S-duality10d type IIB supergravity
SL(2,)×(2,\mathbb{R}) \times O(1,1) 2\mathbb{Z}_2SL(2,)× 2SL(2,\mathbb{Z}) \times \mathbb{Z}_29d supergravity
SU(3)×\times SU(2)SL(3,)×SL(2,)(3,\mathbb{R}) \times SL(2,\mathbb{R})O(2,2;)O(2,2;\mathbb{Z})SL(3,)×SL(2,)SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})8d supergravity
SU(5)SL(5,)SL(5,\mathbb{R})O(3,3;)O(3,3;\mathbb{Z})SL(5,)SL(5,\mathbb{Z})7d supergravity
Spin(10)Spin(5,5)Spin(5,5)O(4,4;)O(4,4;\mathbb{Z})O(5,5,)O(5,5,\mathbb{Z})6d supergravity
E6E 6(6)E_{6(6)}O(5,5;)O(5,5;\mathbb{Z})E 6(6)()E_{6(6)}(\mathbb{Z})5d supergravity
E7E 7(7)E_{7(7)}O(6,6;)O(6,6;\mathbb{Z})E 7(7)()E_{7(7)}(\mathbb{Z})4d supergravity
E8E 8(8)E_{8(8)}O(7,7;)O(7,7;\mathbb{Z})E 8(8)()E_{8(8)}(\mathbb{Z})3d supergravity
E9E 9(9)E_{9(9)}O(8,8;)O(8,8;\mathbb{Z})E 9(9)()E_{9(9)}(\mathbb{Z})2d supergravityE8-equivariant elliptic cohomology
E10E 10(10)E_{10(10)}O(9,9;)O(9,9;\mathbb{Z})E 10(10)()E_{10(10)}(\mathbb{Z})
E11E 11(11)E_{11(11)}O(10,10;)O(10,10;\mathbb{Z})E 11(11)()E_{11(11)}(\mathbb{Z})

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

The group E 8E_8 plays a role in some exceptional differential geometry/differential cohomology. See for instance

References

General

Surveys include

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 8E_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 6E_6, E 7E_7 and E 8E_8 Nagoya Math. J. Volume 32 (1968), 109-139. (project EUCLID)

See also

Invariant polynomials

The octic invariant polynomial is discussed in

Last revised on May 10, 2016 at 06:18:42. See the history of this page for a list of all contributions to it.