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I'm reading Kac's Infinite Dimensional Lie Algebras. In Chapter 4, he classifies the affine root systems. Bourbaki classified the affine Coxeter groups, but multiple root systems can give the same Coxeter group. Kac denotes the elements of his classification by $X^{(r)}$, where $X$ is a symbol denoting a finite root system (e.g. $G_2$) and $r$ is $1$, $2$ or $3$.

When $r=1$, everything makes sense to me. $X^{(r)}$ is a root system which gives rise to the affine Coxeter group which Bourbaki would consider to be of type $X$.

When $r$ is $2$ or $3$, I believe I understand the object Kac is defining. However, I do not understand how he chooses its name. For example, look at the root system he calls $A_{2 \ell-1}^{(2)}$. The root system is of rank $\ell+1$, not $(2 \ell-1)+1$ as you would expect. The corresponding Coxeter group is the affine Coxeter group of type $B_{\ell}$. If you look at Macdonald's very readable paper Affine root systems and Dedkind's $\eta$-function, Macdonald denotes this root system by $B_{\ell}^{\vee}$, which makes much more sense to me.

Similar issues apply to every entry in table 'Aff 2'.

I have two questions:

Why does Kac choose the notation he does?

Also,

Is Kac's notation so established that I have to use it? Is there a mainstream alternative? I find Macdonald's much more reasonable, but I'm pretty sure that hasn't caught on.

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1 Answer 1

up vote 8 down vote accepted

I think the notation might be explained by the explicit construction of the twisted affine Lie algebras as fixed points of automorphisms of the untwisted ones: the $r$ indicates the order of the chosen automorphism of the extended Dynkin diagram corresponding to $X$ and twised affine Lie algebra is a subalgebra of the affine Lie algebra corresponding to $X$. See Chapter 8 of Kac's book.

This notation is fairly well established, given that Kac's book is the standard reference, but I'm not an expert.

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