smallest simplest $E_8$ -module - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T06:17:57Z http://mathoverflow.net/feeds/question/119325 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119325/smallest-simplest-e-8-module smallest simplest $E_8$ -module Hassan Jolany 2013-01-19T12:40:49Z 2013-04-09T18:51:16Z <p>What is the smallest simplest(non-trivial) $E_8$ -module ?</p> http://mathoverflow.net/questions/119325/smallest-simplest-e-8-module/119330#119330 Answer by Robert Bryant for smallest simplest $E_8$ -module Robert Bryant 2013-01-19T13:46:47Z 2013-01-19T13:46:47Z <p>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 $30380$. </p> <p>At <a href="http://www-math.univ-poitiers.fr/~maavl/LiE/form.html" rel="nofollow">http://www-math.univ-poitiers.fr/~maavl/LiE/form.html</a>, you can check these online using LiE.</p> http://mathoverflow.net/questions/119325/smallest-simplest-e-8-module/126901#126901 Answer by Dietrich Burde for smallest simplest $E_8$ -module Dietrich Burde 2013-04-08T20:44:39Z 2013-04-08T20:44:39Z <p>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 <em>only</em> 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).</p>