The Unapologetic Mathematician

Mathematics for the interested outsider

Hom-Space Additivity

Today I’d like to show that the space \hom_G(V,W) of homomorphisms between two G-modules is “additive”. That is, it satisfies the isomorphisms:

\displaystyle\begin{aligned}\hom_G(V_1\oplus V_2,W)&\cong\hom_G(V_1,W)\oplus\hom_G(V_2,W)\\\hom_G(V,W_1\oplus W_2)&\cong\hom_G(V,W_1)\oplus\hom_G(V,W_2)\end{aligned}

We should be careful here: the direct sums inside the \hom are direct sums of G-modules, while those outside are direct sums of vector spaces.

The second of these is actually the easier. If f:V\to W_1\oplus W_2 is a G-morphism, then we can write it as f=(f_1,f_2), where f_1:V\to W_1 and f_2:V\to W_2. Indeed, just take the projection \pi_i:W_1\oplus W_2\to W_i and compose it with f to get f_i=\pi_i\circ f. These projections are also G-morphisms, since W_1 and W_2 are G-submodules. Since every f can be uniquely decomposed, we get a linear map \hom_G(V,W_1\oplus W_2)\to\hom_G(V,W_1)\oplus\hom_G(V,W_2).

Then the general rules of direct sums tell us we can inject W_1 and W_2 back into W_1\oplus W_2, and write

\displaystyle f=I_{W_1\oplus W_2}\circ f=(\iota_1\circ\pi_1+\iota_2\circ\pi_2)\circ f=\iota_1\circ f_1+\iota_2\circ f_2

Thus given any G-morphisms f_1:V\to W_1 and f_2:V\to W_2 we can reconstruct an f:V\to W_1\oplus W_2. This gives us a map in the other direction — \hom_G(V,W_1)\oplus\hom_G(V,W_2)\to\hom_G(V,W_1\oplus W_2) — which is clearly the inverse of the first one, and thus establishes our isomorphism.

Now that we’ve established the second isomorphism, the first becomes clearer. Given a G-morphism h:V_1\oplus V_2\to W we need to find morphisms h_i:V_i\to W. Before we composed with projections, so this time let’s compose with injections! Indeed, \iota_i:V_i\to V_1\oplus V_2 composes with h to give h_i=h\circ\iota_i:V_i\to W. On the other hand, given morphisms h_i:V_i\to W, we can use the projections \pi_i:V_1\oplus V_2\to V_i and compose them with the h_i to get two morphisms h_i\circ\pi_i:V_1\oplus V_2\to W. Adding them together gives a single morphism, and if the h_i came from an h, then this reconstructs the original. Indeed:

\displaystyle h_1\circ\pi_1+h_2\circ\pi_2=h\circ\iota_1\circ\pi_1+h\circ\iota_2\circ\pi_2=h\circ(\iota_1\circ\pi_1+\iota_2\circ\pi_2)=h\circ I_{V_1\oplus V_2}=h

And so the first isomorphism holds as well.

We should note that these are not just isomorphisms, but “natural” isomorphisms. That the construction \hom_G(\underline{\hphantom{X}},\underline{\hphantom{X}}) is a functor is clear, and it’s straightforward to verify that these isomorphisms are natural for those who are interested in the category-theoretic details.

October 11, 2010 - Posted by | Algebra, Category theory, Group theory, Representation Theory


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