# 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.

## 8 Comments	»

1. [...] these representations don’t live in a vacuum. No, they’re just the objects of a whole category of representations. We need to consider the morphisms between representations [...]

Pingback by | October 30, 2008 | Reply

2. [...] and Quotient Representations Today we consider subobjects and quotient objects in the category of representations of an algebra . Since the objects are representations we call these [...]

Pingback by | December 5, 2008 | Reply

3. [...] grading day, another straightforward post. It should come as no surprise that the collection of intertwining maps between any two representations forms a vector [...]

Pingback by | December 11, 2008 | Reply

4. [...] Images of Intertwiners The next obvious things to consider are the kernel and the image of an intertwining map. So let’s say we’ve got a representation , a representation , and an intertwiner [...]

Pingback by | December 12, 2008 | Reply

5. [...] Category of Representations is Abelian We’ve been considering the category of representations of an algebra , and we’re just about done showing that is [...]

Pingback by | December 15, 2008 | Reply

6. [...] of . Then since the symmetrizer and antisymmetrizer are elements of the group algebra , they define intertwiners from to itself. The their images are not just subspaces on which the symmetric group acts nicely, [...]

Pingback by | December 22, 2008 | Reply

7. [...] The antisymmetrizer (for today) is an element of the group algebra , and thus defines an intertwiner from to itself. Its image is thus a subrepresentation of acting on [...]

Pingback by | December 23, 2008 | Reply

8. I love ending up in this blog every time I google something. It always answers my questions in such a terse and precise way.

Comment by Ebrahim | December 7, 2012 | Reply

« Previous | Next »

## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 34, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9224261045455933, "perplexity_flag": "head"}