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.
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September 29, 2010 -
Posted by John Armstrong |
Algebra, Group theory, Representation Theory
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