Write imaginary roots as linear combination of fundamental weights on a Kac-Moody Lie algebra

Question

Let $\mathfrak{g}$ be a Kac-Moody algebra of type $\widehat{A_2}$. I want to write $\delta$, the basic imaginary root, as linear combination of the fundamental weights.

How can I do it? Any help would be appreciated.

Answer 1

Let's do the case of a general affine algebra. We have simple roots $\Pi=\{\alpha_0,\alpha_1,\dots,\alpha_\ell\}$.

The best resource for this that I know is Kac's brilliant 1994 book Infinite Dimensional Lie Algebras.

In general, for an affine algebra the imaginary roots are $$ \Delta_{\text{im}} = \{n\delta|n\in\mathbb Z\backslash0\}, $$ where $$ \delta:=\sum_{i=0}^\ell a_i\alpha_i. $$ So what are the $a_i$? Well they are listed in table Aff in Kac's book. Computing them really follows from working through the definition of imgainary roots. I highly recommend working through the book I mentioned in detail, especially chapter 5 for this. For $\hat A$-type they are all 1, so the imaginary roots are just non-zero integer multiples of the sum of all simple roots. So in the case of $\hat A_2$, the answer is $$ \Delta_{\text{im}}(\hat A_2) = \{n(\alpha_0+\alpha_1+\alpha_2)|n\in\mathbb Z\backslash0\}. $$

I'm sorry, I read this again and I realised you want the basis to be the fundamental weights, not the simple roots. The fundamental weights are linearly independent to the space spanned by the imaginary roots.