# Recover the root system from a root lattice

I am new to Lie algebra and have a maybe naive question about root system.

If we have a root system $$\Phi$$ we can associate it with a lattice $$\Lambda(\Phi)$$. I want to know how to recover the $$\Phi$$ from $$\Lambda(\Phi)$$? In particular I want to know that for type ADE, is $$\Phi$$ just the vectors in $$\Lambda$$ with minimal norm?

Edit

I note that my original question does not make much sense because non-isomorphic root systems may give the same lattice, for example $$A_1\times A_1$$ and $$B_2$$. So let's only consider the case of type ADE.

• What do you consider the root lattice to be? An abstract $\mathbb{Z}$-module, a subset of a Euclidean space, ... Commented Oct 1, 2018 at 13:36
• @Joppy A subset of a Euclidean space, or equivalently, a $\mathbb Z$-module together with the inner product. Sorry for did not make myself clear, but I mentioned "minimal norm" so I assumed you could guess so :) Commented Oct 1, 2018 at 14:19
• Related: mathoverflow.net/q/293756/27465 Commented Oct 3, 2018 at 21:59
• It is indeed true. The vectors in $\Lambda$ of minimal norm ($>0$) are usually normalized to have norm $\sqrt{2}$ and give you back precisely the root system $\Phi$ as it is proved case by case for type $\mathsf{ADE}$ in [the Wikipedia page on root systems][1]. [1]: en.wikipedia.org/wiki/Root_system Commented Oct 14, 2018 at 11:51
• @user213008 That's very nice. But I didn't see the statement on the wiki page? Commented Oct 14, 2018 at 22:03

The vectors in $$\Lambda$$ of minimal norm ($$>0$$) are usually normalized to have norm $$\sqrt{2}$$ and give you back precisely the root system $$\Phi$$ as it is proved case by case for type $$\mathsf{ADE}$$ in the Wikipedia page on root systems.
It is not explicitly stated like this. But look under the section "Explicit construction of the irreducible root systems", read those subsection for type $$\mathsf{ADE}$$. Each time the root system is defined upon the root lattice by vectors of length $$\sqrt{2}$$.