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 
and 
in inner-product spaces 
and 
, we take the direct sum 
of the vector spaces, which makes vectors from each vector space orthogonal to vectors from the other one.
The coproduct 
root system consists of the vectors of the form 
for 
and 
for 
. Indeed, this collection is finite, spans , and does not contain 
. The only multiples of any given vector in are that vector and its negative. The reflection 
sends vectors coming from to each other, and leaves vectors coming from fixed, and similarly for the reflection 
. Finally,
All this goes to show that 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 
and 
, which are inherited from the direct sum of and . This satisfies the universal condition of a coproduct, since the direct sum does. Indeed, if 
is another root system, and if 
and 
are linear transformations sending and into 
, respectively, then 
sends into , and is the unique such transformation compatible with the inclusions.
Interestingly, the Weyl group of the coproduct is the product 
of the Weyl groups. Indeed, for every generator 
of 
and every generator 
of 
we get a generator . 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 
that would be compatible with the projections 
and 
. It’s made up of the points 
for and . 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
and similarly for the other projection. In other words
But this isn’t a reflection! Indeed, each reflection has determinant 
, and this is the composition of two reflections (one for each component) so it has determinant 
. 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 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.
January 25, 2010
Posted by John Armstrong |
Algebra, Category theory, Geometry, Root Systems |
6 Comments
