False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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

Statement

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

Facts & Assumptions

Given: The one-dimensional abelian real Lie algebra $a = R t$ and $V = R^{2}$ .

[L1]

The one-dimensional conclusion is proved over $C$ , under the complex form of Lie's theorem (Irreducible representations of solvable complex Lie algebras are one-dimensional).

[L2]

Solvability is termination of the derived series (Derived series and solvable Lie algebras).

[L3]

A Lie-algebra representation is a bracket-preserving linear map into the endomorphism algebra (Representations of Lie algebras).

[L4]

A nonzero Lie-algebra representation is irreducible when it has no stable subspace other than zero and the whole module (Irreducible, completely reducible, and faithful representations).

Refutation

technique · direct

1.1

Define $ρ (t) = R = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ . Since $a$ is abelian and $[R, R] = 0$ , this is a representation by [L3]. Also $a^{(1)} = 0$ , so $a$ is solvable by [L2].

givenL2L3algebra

2.1

Any nonzero proper subspace of $V$ is a real line. If such a line were $R$ -stable, a nonzero vector on it would be a real eigenvector of $R$ . But the characteristic polynomial of $R$ is $T^{2} + 1$ , which has no real root. Thus no nonzero proper stable subspace exists, so $V$ is irreducible by [L4].

L4step 1.1algebra

3.1

The representation in steps 1.1–2.1 is irreducible and two-dimensional, contradicting the proposed one-dimensional conclusion. It does not contradict [L1], whose scalar field is $C$ ; after complexification, $R$ has the two eigenlines with eigenvalues $i$ and $- i$ . The witness and all calculations are explicit and choice-free.

L1step 1.1step 2.1algebra∎

Depends on

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Sources

Knapp, Lie Groups Beyond an Introduction, field hypothesis in Lie's theorem (standard reference, not scraped)