0

$\newcommand{\g}{\mathfrak{g}}$ I have a finite question:

Let $\g$ be a finite dimensional semisimple lie algebra and let a choice of a weyl chamber be made. If $\gamma>0$ is a particular positive root, then is the set $R_\gamma$ of positive roots that can't be written as $\gamma+\alpha$ for some $\alpha>0$ just $\gamma$ and all of the simple roots?

Reason for asking:

In the case when $\g$ is a Kac-Moody lie algebra of a given cartan matrix, I am trying to see what the expression $\sum_{i, \alpha \in R_\gamma} [e_{-\alpha}^i,x]e_\alpha^i$ is, where I am taking a sum over products of dual basis elements of root spaces.

More motivation(added 3/10 12:57 pm): I calculated that this sum is one part of the casimir commutated with an element $x=e_\gamma^i$. Namely $[\sum_{\alpha>0} e_{-\alpha}^ie_{\alpha}^i ,x]=\sum_{i, \alpha \in R_\gamma} [e_{-\alpha}^i,x]e_\alpha^i$

user062295
  • 1,343
  • When I look at a different argument in Kac's book, he claims that this thing is true because of the fact that

    Given two distinct positive root $\alpha$ and $\beta$, $\alpha + \mathbb{Z} \beta$ is a positive root whenever it is a root.

    I am having trouble seeing how this fact implies my claim.

     – user062295 Mar 11 '17 at 01:48
  • I will give anyone who answers my question a 100 point bounty award. – user062295 Mar 11 '17 at 02:12

0 Answers0