2
$\begingroup$

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.

$\endgroup$
7
  • 1
    $\begingroup$ What do you consider the root lattice to be? An abstract $\mathbb{Z}$-module, a subset of a Euclidean space, ... $\endgroup$
    – Joppy
    Commented Oct 1, 2018 at 13:36
  • 1
    $\begingroup$ @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 :) $\endgroup$
    – Akatsuki
    Commented Oct 1, 2018 at 14:19
  • 1
    $\begingroup$ Related: mathoverflow.net/q/293756/27465 $\endgroup$ Commented Oct 3, 2018 at 21:59
  • $\begingroup$ 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 $\endgroup$ Commented Oct 14, 2018 at 11:51
  • 2
    $\begingroup$ @user213008 That's very nice. But I didn't see the statement on the wiki page? $\endgroup$
    – Akatsuki
    Commented Oct 14, 2018 at 22:03

1 Answer 1

1
$\begingroup$

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 $\sqrt{2}$.

(These are the comments of mine above made into an answer.)

$\endgroup$
1

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .