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Consider the root system of a Kac-Moody algebra. Denote by $\alpha_i$ the simple root associated with node $i$ by for $i \in \{1, \ldots, n-1\}$ and by $\beta$ the simple root associated with $n$.

The Dynkin diagram for $\widetilde{E}_8$ is \begin{align} \circ - \circ - & \circ - \circ - \circ - \circ - \circ - \circ \\ & \ | \\ & \ \bullet \end{align} where $\bullet$ corresponds to the simple root $\beta$. The degree of a root is the coefficient of the root at $\beta$.

Are there some reference of imaginary roots of type $\widetilde{E}_8$? I only find one imaginary root $\gamma=3 \beta + 2 \alpha_1 + 4 \alpha_2 + 6 \alpha_3 + 5 \alpha_4 + 4 \alpha_5 + 3 \alpha_6 + 2 \alpha_7 + \alpha_8$. The root $\gamma$ has degree $3$.

Thank you very much.

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    $\begingroup$ In an (untwisted?) affine root system, aren't all imaginary roots $\mathbb{Z}$-multiples of some fundamental imaginary root $\delta$ (which I think is what you denote by $\gamma$)? $\endgroup$
    – Sam Hopkins
    Commented Dec 4, 2019 at 17:25
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    $\begingroup$ Yes, that's right (twisted or not). If you have found one imaginary root $\delta$ that is primitive, then the imaginary roots are exactly the integer multiples of $\delta$. A good reference is Chapters 4 and 5 of Kac' book "Infinite-dimensional Lie algebras" or MacDonald's "Affine root systems and Dedekind's $\eta $-function". $\endgroup$
    – Nathan Reading
    Commented Dec 5, 2019 at 2:09
  • $\begingroup$ Specifically, you want Theorem 5.6 on page 64 of the Kac book (which refers to tables on pages 54 and 55). $\endgroup$
    – Nathan Reading
    Commented Dec 5, 2019 at 2:16

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