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.
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.
In any preadditive category with kernels, the endomorphisms of a simple object form a division ring.
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
Endomorphisms of a simple object form a field if they are finite dimensional.
This can't be an instance as
𝕜 would be undetermined.
Schur's lemma for
if hom spaces are finite dimensional, then the hom space between simples is at most 1-dimensional.