= Prime-dimensional faithful irreducible module forces simplicity
{title2=$\dim V\text{ prime},\quad V\text{ faithful irreducible}\ \Longrightarrow\ L\text{ simple}$}
Let a semisimple complex <Lie algebra> $L=\bigoplus_iL_i$ act faithfully and irreducibly on a space $V$ of prime <dimension>. Complete reducibility for $L_1$ splits $V$ into isotypic components. The commuting other <ideals of a Lie algebra>[ideals] preserve these components, so irreducibility leaves just one, of the form $W_1\otimes\operatorname{Hom}_{L_1}(W_1,V)$. Repeating with the other factors gives $V\cong\bigotimes_iW_i$, with each $W_i$ an <irreducible module> for its simple <ideal of a Lie algebra>[ideal].
Faithfulness makes every factor nontrivial. A nonabelian <simple Lie algebra> has only the trivial one-dimensional <module>, so $\dim W_i\ge2$. Their product is the prime $\dim V$, forcing just one simple <ideal of a Lie algebra>[ideal]. Hence $L$ is simple. This is a <representation>-<dimension> obstruction, independent of the classification of <root of a root system>[root] systems.
Back to article page
