Subrepresentations 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 “subrepresentations” and “quotient
representations”. As in any category, a subobject (on the vector
space ) of a representation (on the vector space ) is a
monomorphism . This natural transformation is specified by a
single linear map . It’s straightforward to show that if 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 to . Thus we can identify with its
image subspace . Even better, the naturality condition means
that we can identify the action of on with the restriction of to
this subspace. The result is that we can define a
subrepresentation of as a subspace of so that actually sends
into itself for every . That is, it’s a subspace which is fixed
by the action . If is a subrepresentation of , then we can put