Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Apr 12, 2012 #1


    User Avatar
    Gold Member

    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?
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted