smallest simplest $E_8$ -module - MathOverflow

smallest simplest $E_8$ -module

2013-01-19T12:40:49Z

What is the smallest simplest(non-trivial) $E_8$ -module ?

Answer by Robert Bryant for smallest simplest $E_8$ -module

2013-01-19T13:46:47Z

Cartan showed that the lowest dimensional (nontrivial) $E_8$-module is ${\frak{e}}_8$ itself, i.e., the adjoint representation, which has dimension $248$. The next smallest nontrivial irreducible module is considerably larger dimension, $3875$, and I think that the next one after that has dimension $30380$.

At http://www-math.univ-poitiers.fr/~maavl/LiE/form.html, you can check these online using LiE.

Answer by Dietrich Burde for smallest simplest $E_8$ -module

2013-04-08T20:44:39Z

Let $\mu(L)$ denote the minimal dimension of a faithful module of $L$. The complex simple Lie algebra $E_8$ satisfies $\mu(L)=\dim (L)$, as Cartan showed. Indeed, $E_8$ is the only complex simple Lie algebra with this property. There are more results in this direction (which are perhaps interesting): suppose that $L$ is a complex semisimple Lie algebra satisfying $\mu(L)=\dim (L)$. Then $L\simeq E_8\oplus \cdots \oplus E_8$. Even more general, let $L$ be a Lie algebra, where the solvable radical $rad(L)$ is abelian. Then always $\mu(L)\le \dim (L)$, and equality holds if and only if $L$ is abelian of dimension less than $5$, or $L$ is isomorphic to $E_8\oplus \cdots \oplus E_8$. (For references see arXiv:1006.2062).