TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Dominant simple highest-weight modules are finite-dimensional

Statement

Assume the Axiom of Choice. Let $g$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $h$ and a chosen positive system, and let $λ$ be dominant integral. Then the simple module $L (λ)$ of Unique simple quotient of the dominant cyclic module is finite-dimensional and is a simple highest weight module of highest weight $λ$ .

Facts & Assumptions

Given: The Axiom of Choice, such $g, h$ , a chosen positive system and a dominant integral $λ$ .

[A1]

The Axiom of Choice is assumed; it enters through the suppliers of [L1] (The Axiom of Choice).

[L1]

$M_{i n t} (λ)$ is finite dimensional (Simple-root integrability bounds the dominant cyclic module).

[L2]

$L (λ) = M_{i n t} (λ) / N_{λ}$ is a nonzero simple quotient of $M_{i n t} (λ)$ ; its canonical generator, the image of $v_{λ}$ , is nonzero and is killed by $n^{+}$ and has weight $λ$ (Unique simple quotient of the dominant cyclic module, Dominant cyclic highest-weight presentation).

[L3]

The quotient of a finite-dimensional module by a submodule is finite dimensional, and a nonzero simple module generated by a highest weight vector of weight $λ$ is a highest weight module of highest weight $λ$ (Highest-weight vectors and modules, Subrepresentations, quotient representations, and intertwiners, Irreducible, completely reducible, and faithful representations).

Proof

technique · direct

1.1

By [L1] the module $M_{i n t} (λ)$ is finite dimensional, and $L (λ)$ is its quotient by the submodule $N_{λ}$ , so $L (λ)$ is finite dimensional by [L3].

L1L2L3A1

1.2

By [L2] the image $\overset{ˉ}{v}$ of $v_{λ}$ in $L (λ)$ is nonzero, is killed by $n^{+}$ , and has weight $λ$ ; since $v_{λ}$ generates $M_{i n t} (λ)$ , its image generates $L (λ)$ , so $L (λ)$ is a highest weight module of highest weight $λ$ .

L2L3

2.1

The module $L (λ)$ is simple by [L2], hence a simple highest weight module of highest weight $λ$ that is finite dimensional.

L2step 1.1step 1.2∎

Depends on

Used by

Highest-weight classification Theorem
Simple highest-weight modules are classified by highest weight Theorem

Dependency tree · two levels

33 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

Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)