Root space question

Question

Do the roots of a root space decomposition have a kernel? Since it is the duel space to the cartan subalgebra ,evaluation of the roots on a non-equal index cartan basis element should be zero.

Thanks

Answer 1

If I understand your question correctly, then yes, they do. And this has nothing to do with roots. For any linear function $f\in V^*$ the set $\{v\in V\mid <f,v>=0\}$ is a subspace of $V$ of codimension $1$ (if $f\neq 0$).