The Unapologetic Mathematician

Mathematics for the interested outsider

Coproduct 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 systems \Phi and \Phi' in inner-product spaces V and V', we take the direct sum V\oplus V' of the vector spaces, which makes vectors from each vector space orthogonal to vectors from the other one.

The coproduct \Phi\amalg\Phi' root system consists of the vectors of the form (\alpha,0) for \alpha\in\Phi and (0,\alpha') for \alpha'\in\Phi'. Indeed, this collection is finite, spans V\oplus V', and does not contain (0,0). The only multiples of any given vector in \Phi\amalg\Phi' are that vector and its negative. The reflection \sigma_{(\alpha,0)} sends vectors coming from \Phi to each other, and leaves vectors coming from \Phi' fixed, and similarly for the reflection \sigma_{(0,\alpha')}. Finally,

\displaystyle\begin{aligned}(\beta,0)\rtimes(\alpha,0)=\beta\rtimes\alpha&\in\mathbb{Z}\\(0,\beta')\rtimes(0,\alpha')=\beta'\rtimes\alpha'&\in\mathbb{Z}\\(\beta,0)\rtimes(0,\alpha')=(0,\beta')\rtimes(\alpha,0)=0&\in\mathbb{Z}\end{aligned}

All this goes to show that \Phi\amalg\Phi' actually is a root system. As a set, it’s the disjoint union of the two sets of roots.

As a coproduct, we do have the inclusion morphisms \iota_1:\Phi\rightarrow\Phi\amalg\Phi' and \iota_2:\Phi'\rightarrow\Phi\amalg\Phi', which are inherited from the direct sum of V and V'. This satisfies the universal condition of a coproduct, since the direct sum does. Indeed, if \Psi\subseteq U is another root system, and if f:V\rightarrow U and f':V'\rightarrow U are linear transformations sending \Phi and \Phi' into \Psi, respectively, then (a,b)\mapsto f(a)+g(b) sends \Phi\amalg\Phi' into \Psi, and is the unique such transformation compatible with the inclusions.

Interestingly, the Weyl group of the coproduct is the product \mathcal{W}\times\mathcal{W}' of the Weyl groups. Indeed, for every generator \sigma_\alpha of \mathcal{W} and every generator \sigma_{\alpha'} of \mathcal{W}' we get a generator \sigma_{(\alpha,0)}. And the two families of generators commute with each other, because each one only acts on the one summand.

On the other hand, there are no product root systems in general! There is only one natural candidate for \Phi\times\Phi' that would be compatible with the projections \pi_1:V\oplus V'\rightarrow V and \pi_2:V\oplus V'\rightarrow V'. It’s made up of the points (\alpha,\alpha') for \alpha\in\Phi and \alpha'\in\Phi'. But now we must consider how the projections interact with reflections, and it isn’t very pretty.

The projections should act as intertwinors. Specifically, we should have

\displaystyle\pi_1(\sigma_{(\alpha,\alpha')}(\beta,\beta'))=\sigma_{\pi_1(\alpha,\alpha')}(\pi_1(\beta,\beta'))=\sigma_\alpha(\beta)

and similarly for the other projection. In other words

\displaystyle\sigma_{(\alpha,\alpha')}(\beta,\beta')=(\sigma_\alpha(\beta),\sigma_{\alpha'}(\beta'))

But this isn’t a reflection! Indeed, each reflection has determinant -1, and this is the composition of two reflections (one for each component) so it has determinant {1}. Thus it cannot be a reflection, and everything comes crashing down.

That all said, the Weyl group of the coproduct root system is the product of the two Weyl groups, and many people are mostly concerned with the Weyl group of symmetries anyway. And besides, the direct sum is just as much a product as it is a coproduct. And so people will often write \Phi\times\Phi' even though it’s really not a product. I won’t write it like that here, but be warned that that notation is out there, lurking.

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January 25, 2010 Posted by | Algebra, Category theory, Geometry, Root Systems | 6 Comments