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Let $\Phi$ be a root system of ADE type, $L$ is the corresponding root lattice, show that $\Phi=\{\alpha\in L:(\alpha,\alpha)=2\}$, where $(,)$ is the normalized non-degenerate symmetric bilinear form

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    Please don't give orders and tell us what you've tried.2012-08-19
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    @Rasmus i've tried but didn't solve it completely, it's a homework given by one professor. Even some proof sketch is appreciated.2012-08-20
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    Ok, so which cases did you manage to prove? Or did you only manage to prove some parts of the claim? Or what do you mean by *not completely*?2012-08-20
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    My guess is that somewhere in your course material a description of the root lattice has been given (in terms of an orthonormal basis of a suitably ambient copy of $\mathbb{R}^n$). Then you can do it case by case (which may be the easy way) or (the IMHO harder way, but one that also sheds additional light) show that the set of those vectors satistfies the axioms of a root system, and then verify that the root system is of the prescribed type (e.g. by identifying a basis).2012-08-20
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    I know how to prove $\Phi\subset...$, but for the converse, i don't know if there is some easy way. Once, i planned to check case by case, but i didn't for i guess there may be some abstract way.2012-08-20
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    Try case-by-case!2012-08-20
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    @JyrkiLahtonen Ok, i'll try it, many thanks.2012-08-21
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    Good luck! Report your progress here, so that we can give pointers, criticism and upvotes!2012-08-21
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    @JyrkiLahtonen Ok, but maybe not very fast. These days I'm a little busy.2012-08-21

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