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Let $L$ be a semisimple Lie Algebra with root system $\Phi$ and base $\Delta$ of $\Phi$. Let $V$ be a finit dimensional, faithful $L$-modul with weights $\Pi(V)$.
I am trying to show that the $\mathbb{Z}$-span of $\Pi{(V)}$ (denoted $\Lambda(V)$) lies between $\Lambda$, the lattic of integral linear functions, and $\Lambda_{r}$, the root lattic.
The inclusion $\Lambda(V) \subseteq \Lambda$ is obvious, since all weights are integral linear functions and $\mu(h_{i})=\left<\mu,\alpha_{i} \right>$, $\alpha_{i}\in\Delta$ is linear in the first variable.
I am struggeling with the second inclusion $\Lambda_{r} \subseteq \Lambda(V)$.
I tried the following:
First of all let $\alpha\in\Phi$ be arbitrary. It suffices to show that either $\alpha$ or $-\alpha$ lie in $\Lambda(V)$, because then does the whole $\mathbb{Z}$-span of them. To do so I chose $\mu\in\Pi(V)$ and took a look at the $\alpha$-string throught $\mu$.
If $\mu+\alpha$ is a weights it is $\mu+\alpha-\mu=\alpha\in\Lambda(V)$ since $-\mu\in\Lambda(V)$.
Otherwise I did the same with $\mu-\alpha$.
This would show, that all $\stackrel{+}{-}\alpha$ lie in $\Lambda(V)$ and so does $\Lambda_{r}$, but I am not sure if this is anywhere near correct and I have no clue where the fact that $V$ is faithful should come in to play.
Thank you for helping me.

Idun E.
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1 Answers1

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$V$ being a faithful ${\mathfrak g}$-module means that the structure homomorphism $\rho: {\mathfrak g}\to{\mathfrak g}{\mathfrak l}(V)$ is injective. By definition, $\rho$ is a homomorphism of Lie algebras, but we may also view it as a morphism of ${\mathfrak g}$-modules if we equip ${\mathfrak g}$ with the adjoint action and ${\mathfrak g}{\mathfrak l}(V)$ with the ${\mathfrak g}$-action given by $X.- := [\rho(X),-]$.

Now your claim follows from the following observations: Firstly, by the injectivity of ${\mathfrak g}\to {\mathfrak g}{\mathfrak l}(V)$ any root of ${\mathfrak g}$ (i.e. any weight of the adjoint representation of ${\mathfrak g}$) is a weight of ${\mathfrak g}{\mathfrak l}(V)$. Secondly, ${\mathfrak g}{\mathfrak l}(V)\cong V\otimes_{\mathbb k} V^{\ast}$ as ${\mathfrak g}$-modules, and weights add upon tensoring. Can you fill in the details yourself?

Hanno
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  • I think I got the first part, but I am not sure if I understand the second. The injectivity yields, that the roots are weights of $\mathfrak{gl}(V)$, so in order to show, that those are in fact weights of $V$ one uses $\mathfrak{gl}(V)\cong V\otimes V^{*}$. I get what you mean by weights add upon tensoring, but I don´t know how exactly to use that to show the point. – Idun E. Jul 20 '15 at 13:43
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    The weights of ${\mathfrak g}{\mathfrak l}(V)=V\otimes V^{\ast}$ are precisely the differences of two weights from $V$, so knowing that any root is a weight of ${\mathfrak g}{\mathfrak l}(V)$ shows that they are in particular contained in the ${\mathbb Z}$-span of $\Lambda(V)$. – Hanno Jul 20 '15 at 14:57
  • I got it now! Thank you! – Idun E. Jul 20 '15 at 15:28
  • You're welcome :) – Hanno Jul 20 '15 at 15:33
  • @Hanno Could you expand a bit on what the weights of gl(V) look like? And what you mean by add upon tensoring? I'm trying to solve the same exercise now. – Auclair Mar 05 '18 at 05:03