I am reading section 8.4 in Humphreys' book Introduction to Lie Algebras and Representation Theory. He is showing that the only scalar multiples of a root are 1 and -1, but I have trouble understanding his reasoning: He considers the direct sum $M=\bigoplus_{c\in\mathbb{F}} L_{c\alpha}$ for some fixed root $\alpha$. He shows that the only even weights of $h_\alpha$ on $M$ are 0, 2 and -2. Then, he says that this proves that twice a root is never a root. Why is that true? The function $\frac 12\alpha$ could still be a root.
Question about 8.4 in Humphreys
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linear-algebra
representation-theory
lie-algebras
1 Answers
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If $\alpha$ and $\beta=\alpha/2$ were both roots, then we would have a situation, where $2\beta$ and $\beta$ are both roots. Thus $h_\beta$ would have even weights $0,\pm2,\pm4$ on $M=\bigoplus_{c\in F}L_{c\beta}$ contradicting the fact just proven.
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0My question was **about** the proof that twice a root is never a root. I don't understand why it works. – 2012-08-27
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0@user38830: Clearer now? – 2012-08-27
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0no =/, see my edit – 2012-08-27
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0I will check my copy of Humphreys when I get home. Fairly sure that reversing the roles of $\alpha$ and $\beta$ is the way to go. – 2012-08-27
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2No, it's ok, I get it now. Thanks! =) – 2012-08-27