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.
January 22, 2010
Posted by John Armstrong |
Algebra, Geometry, Group Actions, Root Systems |
9 Comments
