A question about simple Weyl reflections

[naima]

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

 

[morphism]

In the future it would be helpful if you defined all your terms. I had to google the phrase "Each basic weight is invariant under all but one" to find the book you're reading just so I could understand what it is you're asking.

Anyway, this pretty much follows right from the definition of w_i: $$w_i \lambda_j = \lambda_j - 2\frac{\langle \alpha_i, \lambda_j \rangle}{\langle \alpha_i, \alpha_i \rangle} \alpha_i = \lambda_j - \delta_{ij} \alpha_i. $$

 

[naima]

Great.

Thank you Morphism.

Being more often on the physics forums, I sometimes ignore what are the terms to be specified in mathematics.
You were very efficient (and useful).