## 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.

[…] Now that we know that images and kernels of -morphisms between -modules are -modules as well, we can bring in a very general […]

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[…] A couple days ago we mentioned the vector space . Today, we specialize to the case , where we use the usual alternate name. We write and call it the “endomorphism algebra” of . Not only is it a vector space of -morphisms, but it has a multiplication from the fact that the source and target of each one are the same and so we can compose them. […]

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[…] of all, we can consider the kernel of a matrix representation . This is not the kernel we’ve talked about recently, which is the kernel of a -morphism. This is the kernel of a group homomorphism. In this context, […]

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