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E8

Context

Group Theory

Lie theory

∞-Lie theory

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

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
Spin(10)SL(5,)SL(5,\mathbb{R})O(3,3;)O(3,3;\mathbb{Z})SL(5,)SL(5,\mathbb{Z})7d supergravity
SU(5)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)

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

References

General

Surveys include

  • wikipedia, E8

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

Revised on May 16, 2014 21:56:56 by Tim Porter (2.26.24.117)