The Unapologetic Mathematician

The Category of Representations

Now let’s narrow back in to representations of algebras, and the special case of representations of groups, but with an eye to the categorical interpretation. So, representations are functors. And this immediately leads us to the category of such functors. The objects, recall, are functors, while the morphisms are natural transformations. Now let’s consider what, exactly, a natural transformation consists of in this case.

Let’s say we have representations $\rho:A:\rightarrow\hom_\mathbb{F}(V,V)$ and $\sigma:A\rightarrow\hom_\mathbb{F}(W,W)$. That is, we have functors $\rho$ and $\sigma$ with $\rho(*)=V$, $\sigma(*)=W$ — where $*$ is the single object of $A$, when it’s considered as a category — and the given actions on morphisms. We want to consider a natural transformation $\phi:\rho\rightarrow\sigma$.

Such a natural transformation consists of a list of morphisms indexed by the objects of the category $A$. But $A$ has only one object: $*$. Thus we only have one morphism, $\phi_*$, which we will just call $\phi$.

Now we must impose the naturality condition. For each arrow $a:*\rightarrow *$ in $A$ we ask that the diagram

$\displaystyle\begin{matrix}V&\xrightarrow{\phi}&W\\\downarrow^{\rho(a)}&&\downarrow^{\sigma(a)}\\V&\xrightarrow{\phi}&W\end{matrix}$

commute. That is, we want $\phi\circ\rho(a)=\sigma(a)\circ\phi$ for every algebra element $a$. We call such a transformation an “intertwiner” of the representations. These intertwiners are the morphisms in the category of $\mathbf{Rep}(A)$ of representations of $A$. If we want to be more particular about the base field, we might also write $\mathbf{Rep}_\mathbb{F}(A)$.

Here’s another way of putting it. Think of $\phi$ as a “translation” from $V$ to $W$. If $\phi$ is an isomorphism of vector spaces, for instance, it could be a change of basis. We want to take a transformation from the algebra $A$ and apply it, and we also want to translate. We could first apply the transformation in $V$, using the representation $\rho$, and then translate to $W$. Or we could first translate from $V$ to $W$ and then apply the transformation, now using the representation $\sigma$. Our condition is that either order gives the same result, no matter which element of $A$ we’re considering.

October 28, 2008