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## Main Question or Discussion Point

I'm reading "lie algebras and particle physics" by Georgi and on I'm up top where he is creating the simple algebras from simple roots and there is something I am not getting here.

On page 108 he seems to be making the claim that any simple root φ had the property that any lowering operator of any SU(2) subalgebra containing it must annihilate it. I don't see why this must be the case.

Let's assume it's false, then there exists roots α, β such that α+β =φ. If α, β are both negative, then this is impossible since φ being simple means it must be positive. If α, β are positive then φ isn't simple, which we assumed it was. But why can't we have α positive and β negative or vice versa?

In fact I would naively imagine that that's exactly what would happen. Consider some spin j SU(2) subalgebra, I would imagine that root corresponding to spin -j would be negative and it would be "raised" up to positive roots and eventually end up at some positive weight corresponding to +j. Which would mean that any simple root, since it must be positive, cannot be a lowest state and consequentially any simple root can always be lowered.

Am I wrong? Am I missing something?

On page 108 he seems to be making the claim that any simple root φ had the property that any lowering operator of any SU(2) subalgebra containing it must annihilate it. I don't see why this must be the case.

Let's assume it's false, then there exists roots α, β such that α+β =φ. If α, β are both negative, then this is impossible since φ being simple means it must be positive. If α, β are positive then φ isn't simple, which we assumed it was. But why can't we have α positive and β negative or vice versa?

In fact I would naively imagine that that's exactly what would happen. Consider some spin j SU(2) subalgebra, I would imagine that root corresponding to spin -j would be negative and it would be "raised" up to positive roots and eventually end up at some positive weight corresponding to +j. Which would mean that any simple root, since it must be positive, cannot be a lowest state and consequentially any simple root can always be lowered.

Am I wrong? Am I missing something?