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