CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Schur’s lemma for irreducible Lie-algebra representations

Statement

A nonzero intertwiner between irreducible $g$ -representations is an isomorphism, and the endomorphism ring of an irreducible representation is a division ring. If $k$ is algebraically closed and the representation is finite dimensional, every intertwining endomorphism is scalar.

Facts & Assumptions

Given: Irreducible representations of one Lie algebra $g$ over $k$ .

[L1]

Lie representations and their intertwiners are respectively unital $U (g)$ -modules and module maps (Lie representations are U(g)-modules).

[L2]

Schur's lemma for modules makes a nonzero map between simple modules an isomorphism and the endomorphism ring of a simple module a division ring (Schur's lemma for simple modules).

[L3]

A linear operator on a positive finite-dimensional vector space over an algebraically closed field has an eigenvalue (Every endomorphism of a nonzero finite-dimensional vector space over an algebraically closed field has an eigenvalue).

Proof

technique · direct

1.1

A stable subspace is exactly a $U (g)$ -submodule under [L1], so an irreducible Lie representation is a simple nonzero $U (g)$ -module. Applying [L2] proves the first two assertions.

L1L2

2.1

Now assume $k$ is algebraically closed and $V$ is finite-dimensional and irreducible. For $T \in {End}_{g} (V)$ , [L3] gives an eigenvalue $λ \in k$ . Then $T - λ I$ is still an intertwiner but is not invertible; the division-ring conclusion of step 1.1 forces $T - λ I = 0$ .

step 1.1L3algebra

3.1

Thus every such $T$ equals $λ I$ . The finite-dimensional and algebraic-closure hypotheses are used only in step 2.1 and are not claimed in the division-ring statement.

step 1.1step 2.1∎

Depends on

Used by

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Sources

Etingof, MIT 18.745 notes, §11.2, printed pp. 64–65 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, Theorem 4.29, printed p. 54 (standard reference, not scraped)