# The Unapologetic Mathematician

## Images and Kernels

A nice quick one today. Let’s take two $G$modules $V$ and $W$. We’ll write $\hom_G(V,W)$ for the vector space of intertwinors from $V$ to $W$. This is pretty appropriate because these are the morphisms in the category of $G$-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 $f\in\hom_G(V,W)$ is first and foremost a linear map $f:V\to W$. And as usual the kernel of $f$ is the subspace $\mathrm{Ker}(f)\subseteq V$ of vectors $v$ for which $f(v)=0$. I say that this isn’t just a subspace of $V$, but it’s a submodule as well. That is, $\mathrm{Ker}(f)$ is an invariant subspace of $V$. Indeed, we check that if $v\in\mathrm{Ker}(f)$ and $g$ is any element of $G$, then $f(gv)=gf(v)=g0=0$, so $gv\in\mathrm{Ker}(f)$ as well.

Similarly, as usual the image of $f$ is the subspace $\mathrm{Im}(f)\subseteq W$ of vectors $w$ for which there’s some $v\in V$ with $f(v)=w$. And again I say that this is an invariant subspace. Indeed, if $w=f(v)\in\mathrm{Im}(f)$ and $g$ is any element of $G$, then $gw=gf(v)=f(gv)\in\mathrm{Im}(f)$ as well.

Thus these images and kernels are not just subspaces of the vector spaces $V$ and $W$, but submodules to boot. That is, they can act as images and kernels in the category of $G$-modules just like they do in the category of complex vector spaces.

September 29, 2010 -