Q about constricting simple lie algebras from simple roots

[hideelo]

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?

 

[jvicens]

Thank you for your question. I understand your confusion and I will try my best to clarify the concept for you.

Firstly, let me explain the concept of simple roots and their relation to SU(2) subalgebras. In the context of Lie algebras and particle physics, simple roots are a set of fundamental vectors that can be used to generate all the other roots in the algebra. These simple roots are typically associated with an SU(2) subalgebra, which is a subgroup of the larger Lie algebra. The lowering and raising operators you mentioned are part of the representation theory of these SU(2) subalgebras.

Now, let's address your question about the claim made by Georgi on page 108. What he is saying is that any simple root must be annihilated by all the lowering operators of its corresponding SU(2) subalgebra. This is a consequence of the definition of a simple root and the properties of these SU(2) subalgebras. Essentially, a simple root is the lowest weight state in its SU(2) subalgebra, and as such, it cannot be lowered any further. If it could be lowered, then it would not be the lowest weight state and thus, not a simple root.

To address your example of a spin j SU(2) subalgebra, the root corresponding to spin -j would indeed be negative and would be "raised" up to positive roots. However, as I mentioned before, the simple root is the lowest weight state and cannot be lowered any further. Therefore, it cannot be raised up to a positive weight state. This is why any simple root must be annihilated by all the lowering operators of its SU(2) subalgebra.

I hope this explanation helps to clarify the concept for you. Please let me know if you have any further questions or if there is anything else I can assist you with.Scientist in Lie Algebras and Particle Physics