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

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Irreducible representations of solvable complex Lie algebras are one-dimensional

Statement

Every nonzero finite-dimensional irreducible complex representation of a finite-dimensional solvable complex Lie algebra is one-dimensional.

Facts & Assumptions

Given: A finite-dimensional solvable complex Lie algebra $g$ and a nonzero finite-dimensional irreducible $g$ -module $V$ .

[L1]

Lie's theorem gives a common eigenvector under these hypotheses (Lie's theorem).

[L2]

An irreducible nonzero representation has no invariant subspaces other than $0$ and the whole space (Irreducible, completely reducible, and faithful representations).

Proof

technique · direct

1.1

By [L1], choose a common eigenvector $0 \neq v \in V$ . Its line $k v$ is a nonzero $g$ -invariant subspace.

givenL1algebra

2.1

Irreducibility [L2] forces $k v = V$ , so $\dim_{C} V = 1$ . The zero module is excluded by the definition of irreducibility used here; if $g = 0$ , the same argument says an irreducible nonzero module is one-dimensional.

L2step 1.1∎

Depends on

Used by

Every irreducible real representation of a solvable Lie algebra is one-dimensional False statement

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kirillov, An Introduction to Lie Groups and Lie Algebras, Theorem 5.24 (standard reference, not scraped)