I want to develop a theory for the representation of sl(5,C).
I am asked to identify the Cartan Subalgebra of the algebra, which we can easily construct considering the matrices:
\begin{equation} diag(a_1,a_2,a_3,a_4,a_5): a_1+a_2+a_3+a_4+a_5=0 \end{equation} So we obtain 4 matrices of the form: ($\delta_i^i, -\delta_{i+1}^{1+1}$). The next step is to define a basis of the dual space h$^*$. So I deploy the isomorphic structure of the algebra and its dual (Because the killing form here is non-degenerate). I determine my dual basis as follows: \begin{equation} H_1^*=\frac{4}{5}H_1+\frac{3}{5}H_2+\frac{2}{5}H_3+\frac{1}{5}H_4 \\ H_2^*=\frac{3}{5}H_1+\frac{6}{5}H_2+\frac{4}{5}H_3+\frac{2}{5}H_4 \\ H_3^*=\frac{2}{5}H_1+\frac{4}{5}H_2+\frac{6}{5}H_3+\frac{3}{5}H_4 \\ H_4^*=\frac{1}{5}H_1+\frac{2}{5}H_2+\frac{3}{5}H_3+\frac{4}{5}H_4 \\ \end{equation}
I hope I have not made any calculation mistakes here. Then I am asked to define the roots of the system. We have all the information needed in the Cartan Matrix:
\begin{pmatrix} 2&-1& 0&0&\\ -1&2& -1&0\\ 0&-1& 2&-1\\ 0&0& -1&2 \\ \end{pmatrix}
And I am now asked to describe the roots in the basis $h^*$.
My question is: I am supposed simply to evaluate $a_i(H_j^*) = a_i(\frac{4}{5}H_1+\frac{3}{5}H_2+\frac{2}{5}H_3+\frac{1}{5}H_4 )$ for instance and essentially redefine "cartan matrix" on the new basis which should be the inverse of the original one?
Thanks in advance