Images and Kernels
A nice quick one today. Let’s take two –modules
and
. We’ll write
for the vector space of intertwinors from
to
. This is pretty appropriate because these are the morphisms in the category of
-modules. It turns out that this category has kernels and has images. Those two references are pretty technical, so we’ll talk in more down-to-earth terms.
Any intertwinor is first and foremost a linear map
. And as usual the kernel of
is the subspace
of vectors
for which
. I say that this isn’t just a subspace of
, but it’s a submodule as well. That is,
is an invariant subspace of
. Indeed, we check that if
and
is any element of
, then
, so
as well.
Similarly, as usual the image of is the subspace
of vectors
for which there’s some
with
. And again I say that this is an invariant subspace. Indeed, if
and
is any element of
, then
as well.
Thus these images and kernels are not just subspaces of the vector spaces and
, but submodules to boot. That is, they can act as images and kernels in the category of
-modules just like they do in the category of complex vector spaces.