# The Unapologetic Mathematician

## Subrepresentations and Quotient Representations

Today we consider subobjects and quotient objects in the category of representations of an algebra $A$. Since the objects are representations we call these “subrepresentations” and “quotient representations”.

As in any category, a subobject $\sigma$ (on the vector space $W$) of a representation $\rho$ (on the vector space $V$) is a monomorphism $f:\sigma\rightarrow\rho$. This natural transformation is specified by a single linear map $f:W\rightarrow V$. It’s straightforward to show that if $f$ is to be left-cancellable as an intertwinor, it must be left-cancellable as a linear map. That is, it must be an injective linear transformation from $W$ to $V$.

Thus we can identify $f$ with its image subspace $f(W)\subseteq V$. Even better, the naturality condition means that we can identify the $\sigma$ action of $A$ on $W$ with the restriction of $\rho$ to this subspace. The result is that we can define a subrepresentation of $\rho:A\rightarrow\mathrm{End}(V)$ as a subspace of $V$ so that $\rho(a)$ actually sends $W$ into itself for every $a\in A$. That is, it’s a subspace which is fixed by the action $\rho$.

If $\sigma$ is a subrepresentation of $\rho$, then we can put the structure of a representation on the quotient space $V/W$. Indeed, note that any vector in the quotient space is the coset $v+W$ of a vector $v\in V$. We define the quotient action using the action on $V$: $\left[\left[\rho/\sigma\right](a)\right](v+W)=\left[\rho(a)\right](v)+W$. But what if $v'=v+w$ is another representative of the same coset? Then we calculate:

\displaystyle\begin{aligned}\left[\left[\rho/\sigma\right](a)\right](v'+W)=\left[\rho(a)\right](v')+W\\=\left[\rho(a)\right](v+w)+W\\=\left[\rho(a)\right](v)+\left[\rho(a)\right](w)+W\\=\left[\rho(a)\right](v)+W\\=\left[\left[\rho/\sigma\right](a)\right](v+W)\end{aligned}

because $\rho(a)$ sends the subspace $W$ back to itself.

December 5, 2008