Two related basic facts in representation theory bear the name of Schur: one general about categories of modules, and another specific to the case of complex numbers. If is a group, a -module is any -module with an action of , where is a fixed commutative unital ground ring.
Given a group and a linear map between two irreducible (= simple) -modules (linear maps intertwining the actions) then is either the zero morphism or an isomorphism. It follows that the endomorphisms of simple irreducible -module form a division ring.
Set , and suppose further that the ground ring is an algebraically closed field; then is a multiple of the identity operator. In other words, the nontrivial automorphisms of simple modules, a priori possible by (1), are ruled out over algebraically closed fields.
Part (1) is essentially category-theoretic and can be generalized in many ways, for example, by replacing by some -algebra and taking the representations compatible with the action of ; more generally, given an abelian category, the endomorphism ring of a simple object is a division ring. Schur’s lemma is one of the basic facts of representation theory. For (2), if the endomorphism rings of all objects in an abelian category are finite-dimensional over an algebraically closed ground field (as is the case for group representations), then the endomorphism ring of a simple object is itself.