- #1
naima
Gold Member
- 938
- 54
Happy new year from France.
I am reading books on elementary particle and i see that their
gauge bosons may be neutral or have opposite charge. They live
in semisimple Lie algebras. So I searched in math books how to prove
that in a semisimple Lie algebra if α is a root so is -α.
I found that it is related to the fact that the killing form is not degenerate.
Could you comment this:
If α is a root the space gα is not nul and orthogonal to Ʃλ ≠- α g λ. Since the Killing form is non degenerate g-α must be ≠ 0 then -α is a root.
Here g λ = {x ∈ g |∃n ∈ N ∀h ∈ h , (π(h) − λ(h))n (x) = 0}.
I am reading books on elementary particle and i see that their
gauge bosons may be neutral or have opposite charge. They live
in semisimple Lie algebras. So I searched in math books how to prove
that in a semisimple Lie algebra if α is a root so is -α.
I found that it is related to the fact that the killing form is not degenerate.
Could you comment this:
If α is a root the space gα is not nul and orthogonal to Ʃλ ≠- α g λ. Since the Killing form is non degenerate g-α must be ≠ 0 then -α is a root.
Here g λ = {x ∈ g |∃n ∈ N ∀h ∈ h , (π(h) − λ(h))n (x) = 0}.