The Unapologetic Mathematician

Mathematics for the interested outsider

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.

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

3 Comments »

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

    Pingback by Schur’s Lemma « The Unapologetic Mathematician | September 30, 2010 | Reply

  2. [...] 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. [...]

    Pingback by Endomorphism and Commutant Algebras « The Unapologetic Mathematician | October 1, 2010 | Reply

  3. [...] 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, [...]

    Pingback by Lifting and Descending Representations « The Unapologetic Mathematician | October 29, 2010 | Reply


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