= Simple root
{title2=$\alpha_i$}
{wiki}
A simple-root system is a <fundamental system of a root system>: a <basis> $\Delta$ consisting of <roots> such that every <root> has coordinates all nonnegative or all nonpositive. Its nonnegative <linear combinations> select the associated <positive roots>. For a reduced <crystallographic root system>, these simple roots are precisely the <positive roots> that cannot be written as the sum of two <positive roots>; every <root> is then an <integer> combination of <simple roots> with coefficients of one sign.
For the <A2 root system>, take $\alpha_1=e_1-e_2$ and $\alpha_2=e_2-e_3$. The third <positive root> is $\alpha_1+\alpha_2$, so it is not simple. In a complex <semisimple Lie algebra>, the <simple roots> select raising and lowering generators through their <root vectors>, while their <coroots> define the <Dynkin labels> of a <highest weight>.
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