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I am reading the paper Demazure modules and graded limits of minimal affinizations. On page 4, line 3 from bottom, it is said that if $\alpha=\sum_{i\in I} n_i\alpha_i^{\vee}$, then $\alpha^{\vee} = \sum_i\frac{(\alpha_i, \alpha_i)}{(\alpha, \alpha)} n_i\alpha_i^{\vee}$. How to prove this?

On page 5, it is said that if $\alpha = \beta + s\delta$ is a positive root, then they define $\alpha^{\vee} = \beta^{\vee} + \frac{2s}{(\beta, \beta)}K$. Is this definition compatible with the usual definition of ${}^\vee$? Thank you very much.

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It would be good to provide more details in the question. Apparently this is about loop algebras. The $\alpha_i$ are simple roots, I suppose. –  Marc van Leeuwen Nov 9 '12 at 12:34
And what is $K$? –  Marc van Leeuwen Nov 9 '12 at 12:48
I think that $\alpha^{\vee} = \frac{2\alpha}{(\alpha, \alpha)} = \sum_{i}\frac{2}{(\alpha, \alpha)} n_i\alpha_i^{\vee}$. –  LJR Nov 10 '12 at 1:19

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