How to compute the norm of a root from the Cartan matrix?

[pscplaton]

Hello!

As far as I understand, the Cartan matrix is associated with a unique semi simple algebra. How can we compute the norm of a root α from it since its components are invariant under rescaling (if all the simple roots are multiplied by the same constant, the Cartan matrix remains unchanged) ?

The square norm of the root is $α(H_α)$ where $H_α$ is such that $Tr(ad (H_α) ad (h))=α(h)$ for any h in the cartan subalgebra, so it seems to me that there is no freedom for choosing the norm

Thanks!

 

[jvicens]

Hello there,

You are correct that the Cartan matrix is associated with a unique semi-simple algebra. The Cartan matrix is a key tool in the study of Lie algebras, as it encodes the information about the algebra's structure and representation theory.

To compute the norm of a root α, we first need to understand the concept of a Cartan subalgebra. A Cartan subalgebra is a maximal abelian subalgebra of the Lie algebra, and it is generated by a set of elements called Cartan generators. These generators are chosen in such a way that they commute with each other and with all other elements of the Lie algebra.

Now, the Cartan matrix is defined in terms of the Cartan generators and the simple roots of the algebra. The simple roots are a set of linearly independent roots that span the root space of the algebra. The Cartan matrix contains information about the inner product of these simple roots, which is used to define the norm of a root.

To compute the norm of a root α, we need to find the corresponding Cartan generator H_α. This can be done by solving the equation Tr(ad (H_α) ad (h))=α(h) for any h in the Cartan subalgebra. Once we have the Cartan generator, we can calculate the norm of the root as α(H_α).

You are correct that the Cartan matrix remains unchanged under rescaling of the simple roots. This is because the Cartan matrix is defined in terms of the inner product of the simple roots, which is invariant under rescaling. However, the norm of a root is not affected by rescaling, as it is defined in terms of the Cartan generator and the inner product with the root.

I hope this helps clarify how the norm of a root can be computed from the Cartan matrix. Let me know if you have any other questions.