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

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Representation kernels are ideals

Statement

For a representation $ρ : g \to g l (V)$ , the kernel is an ideal of $g$ , and the representation is faithful if and only if this kernel is zero.

Facts & Assumptions

Given: A representation $ρ : g \to g l (V)$ over $k$ .

[L1]

A representation map is a Lie-algebra homomorphism (Representations of Lie algebras).

[L2]

The kernel of a Lie-algebra homomorphism is an ideal (Kernels, images, and the first isomorphism theorem for Lie algebras).

[L3]

Faithful means that $ρ$ is injective (Irreducible, completely reducible, and faithful representations).

Proof

technique · direct

1.1

Applying [L2] to the homomorphism in [L1] shows that $\ker ρ$ is an ideal.

L1L2

1.2

A linear map is injective exactly when its kernel is zero; by [L3], this says that the representation is faithful exactly when $\ker ρ = 0$ .

givenL3algebra

2.1

Thus both assertions hold, including for $V = 0$ : in that case $ρ$ is faithful precisely when $g = 0$ .

step 1.1step 1.2∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Kirillov, An Introduction to Lie Groups and Lie Algebras, §4.1, printed pp. 49–50 (standard reference, not scraped)