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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

Rasmus
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user18537
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    Please don't give orders and tell us what you've tried. – Rasmus Aug 19 '12 at 16:48
  • @Rasmus i've tried but didn't solve it completely, it's a homework given by one professor. Even some proof sketch is appreciated. – user18537 Aug 20 '12 at 01:53
  • 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? – Jyrki Lahtonen Aug 20 '12 at 11:55
  • 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). – Jyrki Lahtonen Aug 20 '12 at 12:02
  • 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. – user18537 Aug 20 '12 at 12:38
  • Try case-by-case! – Jyrki Lahtonen Aug 20 '12 at 16:59
  • @JyrkiLahtonen Ok, i'll try it, many thanks. – user18537 Aug 21 '12 at 03:18
  • Good luck! Report your progress here, so that we can give pointers, criticism and upvotes! – Jyrki Lahtonen Aug 21 '12 at 07:30
  • @JyrkiLahtonen Ok, but maybe not very fast. These days I'm a little busy. – user18537 Aug 21 '12 at 14:18

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