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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).

The gorup 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 April 15, 2013 07:37:12 by David Roberts (192.43.227.18)