A question about simple Weyl reflections

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In summary, the conversation discusses understanding Lie algebras and the proof that each basic weight is invariant under all but one of the simple Weyl reflections. The relationship between the Dynkin indices of the reflected weights and the indices of the original weight is also discussed. The conversation ends with a thank you to the expert for their efficient and useful response.
  • #1
naima
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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
 
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  • #2
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: $$w_i \lambda_j = \lambda_j - 2\frac{\langle \alpha_i, \lambda_j \rangle}{\langle \alpha_i, \alpha_i \rangle} \alpha_i = \lambda_j - \delta_{ij} \alpha_i. $$
 
  • #3
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).
 

FAQ: A question about simple Weyl reflections

1. What is a simple Weyl reflection?

A simple Weyl reflection is a type of geometric transformation that is commonly used in mathematical group theory. It involves reflecting an object or point across a hyperplane or line of symmetry in a specific way.

2. How do simple Weyl reflections relate to Lie groups?

Simple Weyl reflections are closely related to Lie groups, which are mathematical structures used to study continuous symmetries. In fact, simple Weyl reflections are an important tool for understanding the structure of Lie groups.

3. What is the significance of simple Weyl reflections in mathematics?

Simple Weyl reflections play a crucial role in various areas of mathematics, such as group theory, representation theory, and algebraic geometry. They are also used in the study of root systems and Lie algebras.

4. Can simple Weyl reflections be applied in other fields besides mathematics?

Yes, simple Weyl reflections have applications in other fields as well. For example, they are used in crystallography to describe the symmetries of crystals and in physics to study quantum field theory.

5. How do I perform a simple Weyl reflection?

To perform a simple Weyl reflection, you need to first choose a hyperplane or line of symmetry and then reflect the point or object across that hyperplane or line in a specific way. The exact process may vary depending on the context and application.

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