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 John Armstrong | Algebra, Geometry, Group Actions, Root Systems

9 Comments »

I love how deep you have gotten us!

Comment by Jonathan Vos Post | January 23, 2010 | Reply
[…] Root Systems We should also note that the category of root systems has binary (and thus finite) coproducts. They both start the same way: given root […]

Pingback by Coproduct Root Systems « The Unapologetic Mathematician | January 25, 2010 | Reply
[…] bunch of irreducible root systems. Now all we have to do is classify the irreducible root systems (up to isomorphism) and we’re done! Possibly related posts: (automatically generated)Coproduct Root SystemsDual […]

Pingback by Irreducible Root Systems « The Unapologetic Mathematician | January 27, 2010 | Reply
[…] Root Systems I Now we can turn towards the project of classifying irreducible root systems up to isomorphism. And we start with some properties of irreducible root […]

Pingback by Properties of Irreducible Root Systems I « The Unapologetic Mathematician | February 10, 2010 | Reply
[…] The matrix we get, depends on the particular ordering of the base we chose, of course, so the Cartan matrix isn’t quite uniquely determined by the root system. This is relatively unimportant, actually. More to the point is the other direction: the Cartan matrix determines the root system up to isomorphism! […]

Pingback by Cartan Matrices « The Unapologetic Mathematician | February 16, 2010 | Reply
[…] Root Systems At long last, we can state the classification of irreducible root systems up to isomorphism. We’ve shown that for each such root system we can construct a connected Dynkin diagram, […]

Pingback by The Classification of (Possible) Root Systems « The Unapologetic Mathematician | February 19, 2010 | Reply
[…] Now, just like we saw when we showed that the Cartan matrix determines the root system up to isomorphism, we can extend to a map from the root system generated by to the root system generated by . That is, a transformation of Dynkin diagrams gives rise to a morphism of root systems. […]

Pingback by Transformations of Dynkin Diagrams « The Unapologetic Mathematician | March 5, 2010 | Reply
[…] able to determine the automorphism group of our root systems. That is, given an object in the category of root systems, the morphisms from that root system back to itself (as usual) form a group, and it’s […]

Pingback by The Automorphism Group of a Root System « The Unapologetic Mathematician | March 11, 2010 | Reply
[…] get a perspective on the classification, we defined the category of root systems. In particular, this leads us to the idea of decomposing a root system into irreducible root […]

Pingback by Root Systems Recap « The Unapologetic Mathematician | March 12, 2010 | Reply

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