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Let $a,b$ be roots ($a\ne \pm b$) of a Lie algebra $g$ of type $X$, where $X$ can be classic or exceptional $(A,B,C,D,E,F,G)$. It is well known that the length of an $a$-string through $b$ is at most 4.

What are all types of $g$ such that:

1) $a+b$ can be a root and $a+2b$ is not a root?

2) $a+2b$ can be a root and $a+3b$ is not a root?

3) $a+3b$ can be a root?

Thanks!

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    $\begingroup$ You asked the same in Math.SE, and if I recall correctly, you even got an answer---I cannot find that question now, though. Can you elaborate on why the answer you got did not satisfy you? $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Dec 29, 2010 at 0:07
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    $\begingroup$ On the other hand, you ignored the (reasonable) request to clarify what "scenery" means. Until you say that, you won't get a good answer, because no one knows what it means. I'm downvoting for lack of clarity; I might change my mind if you define "scenery." $\endgroup$
    – Ben Webster ♦
    Commented Dec 29, 2010 at 1:15
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    $\begingroup$ So, what you're saying is that you posted this without reading the comments on the Math.SE post (the relevant comment was written 2 hours before you posted this)? That's even worse. $\endgroup$
    – Ben Webster ♦
    Commented Dec 29, 2010 at 2:39
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    $\begingroup$ And the question is still ill-defined. Do you want all g where some such pair exists? $\endgroup$
    – Ben Webster ♦
    Commented Dec 29, 2010 at 2:57
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    $\begingroup$ No, it's not clear in the first part of of the question. If you ask "for which g does this property of a and b hold," you have to specify if you mean some pair of a and b, all pairs of a and b, or something else. $\endgroup$
    – Ben Webster ♦
    Commented Dec 29, 2010 at 8:21

1 Answer 1

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3) is only possible if $a$ is 3 times longer than $b$, so it only happens in $G_2$.

2) is only possible if $a$ is 2 times longer than $b$ (or if $a=b+a'$ for $a'$ a root 3 times longer), so it happens in all non-simply laced types ($B,C,F$ and $G$).

1) happens in every simple root system other than $A_1$, since this happens whenever $a$ and $b$ are the same length and not orthogonal.

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  • $\begingroup$ if b and c are the short and long simple roots of G_2, and you take a=b+c, then 2) is satisfied. What's imprecise about that? $\endgroup$
    – Ben Webster ♦
    Commented Dec 29, 2010 at 8:24
  • $\begingroup$ @Chris: Pick $a$ and $b$ such that $a+3b$ is a root. Now, do a substitution, setting new $a$ to old $a+b$. You never specified that $a$ and $b$ had to be simple. $\endgroup$
    – S. Carnahan ♦
    Commented Dec 29, 2010 at 16:52

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