nLab
Schur's lemma

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 GG is a group, a GG-module is any kk-module with an action of GG, where kk is a fixed commutative unital ground ring.

  1. Given a group GG and a linear map ϕ:MN\phi : M\to N between two irreducible (= simple) GG-modules (linear maps intertwining the actions) then ϕ\phi is either the zero morphism or an isomorphism. It follows that the endomorphisms of simple irreducible GG-module form a division ring.

  2. Set M=NM=N, and suppose further that the ground ring kk is an algebraically closed field; then ϕ\phi is a multiple λI\lambda I 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 GG by some kk-algebra and taking the representations compatible with the action of kk; 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 kk (as is the case for group representations), then the endomorphism ring of a simple object is kk itself.

Revised on October 17, 2009 20:12:57 by Toby Bartels (71.104.230.172)