Module theory MOC

Schur’s lemma

Schur’s lemma is most naturally stated in the language of modules. Let be simple modules over a ring . Then any nonzero Module homomorphism is an isomorphism. module In particular, the endomorphism ring of a simple module is a division ring.

Very simple proof

Since and are submodules of simple modules, they must either be trivial or equal to and respectively. If then and , hence is epic and monic and thus an -module isomorphism.

is an algebraically closed field and is a module over a K-monoid over , there are a few cases where one can conclude consists of homotheties, which is sometimes known as Schur’s first lemma. Namely

• If , by All elements of a finite-dimensional unital associative algebra are algebraic
• If , by Dixmier’s lemma
• If has a filtration like the universal enveloping algebra of a finite-dimensional Lie algebra, by Quillen’s lemma

which also rely on the result from Division algebra with only algebraic elements over an algebraically closed field.

Schur’s lemma for unitary group representations

Schur’s lemma is a statement about linear maps which “commute” with an irrep.¹²

Schur’s lemma, first form • Let be a finite-dimensional (complex) Irrep and a linear endomorphism. If commutes with , i.e.

for all , then for some . rep

Proof

Let be an eigenvalue of , and . Then for all . Therefore , meaning is -Invariant subspace. Since is irreducible and , . Therefore .

Schur’s lemma, second form • Let and be finite-dimensional unitary irreps and a linear map.

for all , then or and are unitarily equivalent. rep is thence called an intertwiner, which is unique up to scalar multiplication.

Proof

Taking the Hermitian conjugate of both sides gives for all , i.e. . Hence

thus commutes with , and by the first lemma, for some . Then for all and therefore which is real and nonnegative. Therefore either whence or and is unitary with the equivalence .

Corollaries

• Irreps of abelian groups are 1-dimensional

Schur’s lemma in abelian categories

Let be an abelian category and be simple objects. Then every nonzero morphism is an isomorphism. cat In particular, is a division ring.

Proof

Essentially, apply the Freyd-Mitchell theorem to the above proof for modules.

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Footnotes

1. 2023, Groups and representations, p. 31 ↩

2. 1996, Representations of finite and compact groups, §II.4, pp. 27–28. The proof offered here is virtually identical, but insists on using ∗-representations for reasons beyond me. ↩