Schur's lemma #
We first prove the part of Schur's Lemma that holds in any preadditive category with kernels, that any nonzero morphism between simple objects is an isomorphism.
Second, we prove Schur's lemma for
𝕜-linear categories with finite dimensional hom spaces,
over an algebraically closed field
the hom space
X ⟶ Y between simple objects
Y is at most one dimensional,
and is 1-dimensional iff
Y are isomorphic.
Future work #
It might be nice to provide a
division_ring instance on
End X when
X is simple.
This is an easy consequence of the results here,
but may take some care setting up usable instances.
The part of Schur's lemma that holds in any preadditive category with kernels: that a nonzero morphism between simple objects is an isomorphism.
As a corollary of Schur's lemma for preadditive categories, any morphism between simple objects is (exclusively) either an isomorphism or zero.
Part of Schur's lemma for
the hom space between two non-isomorphic simple objects is 0-dimensional.
An auxiliary lemma for Schur's lemma.
X ⟶ X is finite dimensional, and every nonzero endomorphism is invertible,
X ⟶ X is 1-dimensional.
Schur's lemma for endomorphisms in
Schur's lemma for
if hom spaces are finite dimensional, then the hom space between simples is at most 1-dimensional.
for the refinements when we know whether or not the simples are isomorphic.