Express fundamental coweights using coroots.

Question

In type $A_2$ root system, we have $\alpha_1 = 2 \omega_1 - \omega_2$, $\alpha_2 = - \omega_1 + 2 \omega_2$. How to express fundamental coweights $\omega_1^{\vee}, \omega_2^{\vee}$ using coroots $\alpha_1^{\vee}, \alpha_2^{\vee}$? Any help will be greatly appreciated!

Answer 1

The fundamental roots are dual to the fundamental coweights and the fundamental weights are dual to the fundamental coroots.

$$ \alpha_i(\omega_j^\vee) = \delta_{ij} \qquad \qquad \omega_i(\alpha_j^\vee) = \delta_{ij}$$

Using these relationships you can find the relationship between these bases. The Cartan matrix gives a change of basis matrix from weights to roots. So the transpose of the Cartan matrix gives a change of basis between the coroots to the coweights -- that's where I have to double check my own thinking to make sure I didn't get it backwards! :)

Let $a_{ij}$ be the $(i,j)$-entry of our Cartan matrix. Then...

$$\alpha_i(\alpha_j^\vee) = \sum\limits_{k} a_{ik}\omega_k(\alpha_j^\vee) = \sum\limits_{k} a_{ik}\delta_{kj}=a_{ij}$$

Actually $\alpha_i(\alpha_j^\vee)=a_{ij}$ may (essentially) be your definition of the entries of the Cartan matrix (depending on where your starting point is).

Now suppose that $c_{ij}$ is the $(i,j)$-entry of your change of basis matrix from the coweights to the coroots. This means that $\alpha_j^\vee = \sum\limits_{k} c_{jk}\omega_k^\vee$. Then...

$$a_{ij}=\alpha_i(\alpha_j^\vee)=\alpha_i\left(\sum\limits_{k} c_{jk}\omega_k^\vee\right)=\sum\limits_k c_{jk} \alpha_i(\omega_k^\vee) = \sum\limits_k c_{jk}\delta_{ik} = c_{ji}$$

Thus the change of basis matrix is just the transpose. [There's nothing special about roots and weights here. This is just a general linear algebra fact about switching between dual bases.]

Now your Cartan matrix is $A=\begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}$ is symmetric. So you get...

$$\alpha^\vee = 2\omega_1^\vee-\omega_2^\vee \qquad \qquad \alpha_2^\vee = -\omega_1^\vee+2\omega_2^\vee$$

For simple Lie algebras of type ADE (simply laced algebras) the same thing happens (since their Cartan matrices are symmetric). If you play around with the non-simply laced types, be mindful of the transpose.