group theory

∞-Lie theory

# Contents

## Idea

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

## Properties

### Homotopy groups

The first nontrivial homotopy group of the topological space underlying $E_8$ 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 $K(\mathbb{Z},3) \simeq B^3 \mathbb{Z} \simeq B^2 U(1) \simeq B \mathbb{C}P^\infty$.

### 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).

The gorup $E_8$ 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 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 $E_8$ 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)