The Unapologetic Mathematician

Mathematics for the interested outsider

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.

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October 28, 2008 - Posted by | Category theory, Group theory, Representation Theory, Ring theory

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 Representations of a Polynomial Algebra « The Unapologetic Mathematician | 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 Subrepresentations and Quotient Representations « The Unapologetic Mathematician | 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 Intertwiner Spaces « The Unapologetic Mathematician | 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 Kernels and Images of Intertwiners « The Unapologetic Mathematician | 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 The Category of Representations is Abelian « The Unapologetic Mathematician | 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 Symmetric Tensors « The Unapologetic Mathematician | 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 Antisymmetric Tensors « The Unapologetic Mathematician | 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


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