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

1. Apr 12, 2012

### 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?