If $\alpha$ is a root, then the only multiples of α which are roots are $\alpha, -\alpha, 0$. Here $\alpha$ is a root of a simple lie algebra. How do I prove this?
If $\alpha$ is a root of a simple lie algebra, then prove that the only multiples of $\alpha$ which are roots are $\alpha, -\alpha,0$
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$\begingroup$
abstract-algebra
representation-theory
lie-algebras
1 Answers
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An argument showing this is given on page 38 of Humphreys' Introduction to Lie Algebras and Representation Theory, #9 in Springer GTM series. It is based on knowledge of $sl_2$-theory and some other basic properties.
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0hey, thanx, but is there any other way of proving it. I am reading semi simple lie algebras and representations by cahn, and have not covered sl2 theory. – 2012-03-06