# The Unapologetic Mathematician

## The Category of Root Systems

As with so many of the objects we study, root systems form a category. If $\Phi$ is a root system in the inner product space $V$, and $\Phi'$ is a root system in the inner product space $V'$, then a morphism from $\Phi$ to $\Phi'$ will be a linear map $\tau:V\rightarrow V'$ so that if $\alpha\in\Phi$ then $\tau(\alpha)\in\Phi'$. Further, we’ll require that $\tau(\beta)\rtimes\tau(\alpha)=\beta\rtimes\alpha$ for all roots $\alpha,\beta\in\Phi$.

Immediately from this, we find that the Weyl group $\mathcal{W}$ of $\Phi$ not only acts on $\Phi$ itself, but on $\Phi'$. Indeed, $\tau$ induces a homomorphism $\mathcal{W}\rightarrow\mathcal{W}'$ that sends the generator $\sigma_\alpha$ to the generator $\sigma_{\tau(\alpha)}$. Even better, $\tau$ actually intertwines these actions! That is, $\sigma_{\tau(\alpha)}(\tau(\beta))=\tau(\sigma_\alpha(\beta))$. Indeed, we can calculate

\displaystyle\begin{aligned}\sigma_{\tau(\alpha)}(\tau(\beta))&=\tau(\beta)-(\tau(\beta)\rtimes\tau(\alpha))\tau(\alpha)\\&=\tau(\beta-(\tau(\beta)\rtimes\tau(\alpha))\alpha)\\&=\tau(\beta-(\beta\rtimes\alpha)\alpha)\\&=\tau(\sigma_\alpha(\beta))\end{aligned}

In particular, we can say that two root systems are isomorphic if there’s an invertible linear transformation $\tau$ sending $\Phi$ to $\Phi'$, and whose inverse $\tau^{-1}$ sends $\Phi'$ back onto $\Phi$. In this case, the intertwining property can be written as an isomorphism of Weyl groups sending $\sigma\in\mathcal{W}$ to $\tau\sigma\tau^{-1}\in\mathcal{W}'$.

Even more particularly, an automorphism of $\Phi$ is an isomorphism from $\Phi$ to itself. That is, it’s an invertible linear transformation from $V$ to itself that leaves $\Phi$ invariant. And so we see that $\mathcal{W}$ itself is a subgroup of $\mathrm{Aut}(\Phi)$. In fact, the Weyl group is a normal subgroup of the automorphism group. That is, given an element $\sigma$ of $\mathcal{W}$ and an automorphism $\tau$ of $\Phi$, the conjugation $\tau\sigma\tau^{-1}$ is again in the Weyl group. And this is exactly what we proved last time!

We can now revise our goal: we want to classify all possible root systems up to isomorphism.

January 22, 2010