Length of roots in su(2) su(3) and other Lie algebras.

[naima]

Hi all

I found these equalities from Gordon Brown (1963).
He uses the killing form to measure the length of the roots in a semi simple algebra.
First and second equalities are quite obvious and come from the definition.
Could you help me for the last one which prove that we have a projector?
It is Ʃ_g g(h_a) g(h_b )=Ʃ_g (a,g)(g,b)

He writes that the sum of the squared lengths is the dimension of the Cartan subalgebra.
If we take the su(2) weak interaction, we have two ladder bosons W- W+ (corresponding to opposite roots) adding or substracting one unit to isospin and dim H = 1.
What is the relation with the length 1/√2 he finds for each root?

 

[AndyW]

Hello,

Thank you for sharing these equalities from Gordon Brown's work. It is always interesting to see how different mathematicians approach and solve problems in their respective fields.

To address your question about the last equality, it is indeed a proof that we have a projector. The left-hand side of the equality is the sum of the squared lengths of the roots, while the right-hand side is the sum of the scalar products of the roots with the fundamental weights. This can be interpreted as a projection of the roots onto the fundamental weights, which is why it is referred to as a "projector". This equality is a fundamental result in the study of semi-simple algebras and is often used in the classification of Lie algebras.

As for the relation between the length 1/√2 and the ladder bosons in the su(2) weak interaction, it is not immediately clear without more context and information. However, it is possible that the length 1/√2 is related to the properties of the ladder bosons and their interactions in the su(2) algebra. It would be helpful to look at the specific equations and calculations used by Gordon Brown to better understand this relation.

I hope this helps. Keep exploring and asking questions!