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Let $\mathfrak{g}$ be an affine lie algebra with a canonical center element $c$. I.e. if the cartan matrix is $A$ and $c_i$ are positive without a common factor such that $A (c_0,\ldots,c_n)=0$ then $c=\sum_{i=0}^n c_i h_i$.

Express the highest root $\theta$ as $\sum_{i=0}^n a_i \alpha_i$ where $\alpha_i$ are the simple roots. I am having trouble with a step on page 44 of these notes

that

the coroot corresponding to $\theta$ is $h_\theta=c-h_0$, i.e. that

$$2\frac{(\alpha, \theta)}{(\theta,\theta)}= \sum_{i >0} c_i \alpha(h_i).$$

Here $(,)$ denotes an invariant inner product normalized so that the inner product of $\theta$ with itself is $2$.

My trouble is that I don't see the relation between the canonical central element and the highest root. I feel like I am not using all the information that is available and I would appreciate if you could tell me what I am missing.

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  • $\begingroup$ If it is easier, I would also be very happy to have an answer to my question for the first case where there is a nontrivial cartan matrix - ie. for $\mathfrak{g}=central extension of sl_3 \otimes \mathbb{C}[t^{\pm1}]$. $\endgroup$ – user062295 Mar 23 '17 at 22:47

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