- #1
naima
Gold Member
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I am readin Belinte's book about Lie algebras (I have also the Cahn) .
And I try to understand this. He writes
"Each basic weight is invariant under all but one of the simple Weyl reflections since w_i l_j = l_j for i<>j while w_i l_i = l_i - alpha_i
(alpha_i is simple by definition of simple reflections). Hence th Dynkin indices m' of the reflected weights w_j mu are related to the indices m_i of mu by m'_i = m_i - A_ij m_j"
Could you, please, tell me how to prove that w_i l_j = l_j for i<>j
thanks
And I try to understand this. He writes
"Each basic weight is invariant under all but one of the simple Weyl reflections since w_i l_j = l_j for i<>j while w_i l_i = l_i - alpha_i
(alpha_i is simple by definition of simple reflections). Hence th Dynkin indices m' of the reflected weights w_j mu are related to the indices m_i of mu by m'_i = m_i - A_ij m_j"
Could you, please, tell me how to prove that w_i l_j = l_j for i<>j
thanks