The Unapologetic Mathematician

Mathematics for the interested outsider

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 Posted by | Algebra, Geometry, Group Actions, Root Systems | 9 Comments

   

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