2
$\begingroup$

There are four usual bases one can use to express the roots and weights of a given algebra.

  • The $\alpha$-basis, where we write the roots and weights in terms of the simple roots $\alpha_i$.
  • The $\omega$-basis, where we write the roots and weights in terms of the fundamental weights $\omega_i$. The coefficients in this basis are often called Dynkin labels.
  • The orthogonal-basis where one embeds the root/weight-space into a bigger Euclidean space. (See this question)
  • The $H$-Basis where the coefficients for each weight or root correspond to the eigenvalues of the Cartan generators $H_i$

While I'm able to find list of the simple-roots in the $\alpha$-, the $\omega$- and orthogonal bases in almost any book, I'm struggeling for two days now to find a list of simple roots for groups like $A_4=SU(5)$ in the $H$-basis.

Does a list of this kind exist somewhere? Any book, paper or lecture note suggestion would be awesome!

$\endgroup$
1
$\begingroup$

The $H$-basis is based off the generalized Gell-Mann Matrices. In $N$ dimensions, there are $N-1$ diagonal matrices, called Cartan Generators. They are generalized as \begin{equation} [H_m]_{ij} = \frac{1}{\sqrt{2m(m+1)}}\left(\sum^m_{k=1}\delta_{ik}\delta_{jk}-m\delta_{i,m+1}\delta_{j,m+1} \right) .\end{equation} Note that in this equation $i$ and $j$ are set based on your $N$ while $m=1,2,3, ... N-1$. The weights are the eigenvalues of these matrices with $N-1$ dimension. They can be generalized as well to \begin{equation} [w_m]^{j} = \frac{1}{\sqrt{2m(m+1)}}\left(\sum^m_{k=1}\delta_{jk}-m\delta_{j,m+1} \right) .\end{equation} Take notice that $[w^j]_m=[H_m]_{jj}$. Now finally the simple roots can be built:\begin{equation} \alpha^i=w^i-w^{i+1} \end{equation} for $i=1...N-1$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.