# List of simple roots in the H-basis for various Lie algebras?

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!

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 $$[H_m]_{ij} = \frac{1}{\sqrt{2m(m+1)}}\left(\sum^m_{k=1}\delta_{ik}\delta_{jk}-m\delta_{i,m+1}\delta_{j,m+1} \right) .$$ 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 $$[w_m]^{j} = \frac{1}{\sqrt{2m(m+1)}}\left(\sum^m_{k=1}\delta_{jk}-m\delta_{j,m+1} \right) .$$ Take notice that $[w^j]_m=[H_m]_{jj}$. Now finally the simple roots can be built:$$\alpha^i=w^i-w^{i+1}$$ for $i=1...N-1$