The Unapologetic Mathematician

Mathematics for the interested outsider

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 Posted by | Algebra, Representation Theory | 7 Comments

   

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