The Category of Root Systems
As with so many of the objects we study, root systems form a category. If is a root system in the inner product space , and is a root system in the inner product space , then a morphism from to will be a linear map so that if then . Further, we’ll require that for all roots .
Immediately from this, we find that the Weyl group of not only acts on itself, but on . Indeed, induces a homomorphism that sends the generator to the generator . Even better, actually intertwines these actions! That is, . Indeed, we can calculate
In particular, we can say that two root systems are isomorphic if there’s an invertible linear transformation sending to , and whose inverse sends back onto . In this case, the intertwining property can be written as an isomorphism of Weyl groups sending to .
Even more particularly, an automorphism of is an isomorphism from to itself. That is, it’s an invertible linear transformation from to itself that leaves invariant. And so we see that itself is a subgroup of . In fact, the Weyl group is a normal subgroup of the automorphism group. That is, given an element of and an automorphism of , the conjugation 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.