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

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An irreducible module is generated by its highest-weight vector

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, let $V$ be a finite-dimensional irreducible representation of $g$ (Irreducible, completely reducible, and faithful representations), and let $v \in V$ be a highest weight vector (Highest-weight vectors and modules). Then the subrepresentation generated by $v$ is $V$ ; equivalently $U (g) v = V$ .

Facts & Assumptions

Given: The Axiom of Choice, such $g, h$ , a finite-dimensional irreducible module $V$ , and a highest weight vector $v \in V$ .

[A1]

The Axiom of Choice is assumed; it enters through the definition of a highest weight vector, whose contract carries the assumption (The Axiom of Choice).

[L1]

The action of $g$ extends uniquely to a unital action of $U (g)$ on $V$ , and the subrepresentation generated by $v$ is $U (g) v$ (Lie representations are U(g)-modules, Highest-weight vectors and modules).

[L2]

A nonzero subrepresentation of an irreducible representation is the whole representation; the zero subspace and $V$ are the only subrepresentations of an irreducible $V \neq 0$ (Irreducible, completely reducible, and faithful representations, Subrepresentations, quotient representations, and intertwiners).

Proof

technique · direct

1.1L1A1

The vector $v$ is nonzero and lies in $U (g) v$ (as the image of $v$ under the unit of $U (g)$ ), so $U (g) v$ is a nonzero subrepresentation of $V$ .

2.1L1L2step 1.1

Since $V$ is irreducible, [L2] applied to the nonzero subrepresentation $U (g) v$ gives $U (g) v = V$ ; by [L1] this is the subrepresentation generated by $v$ .

3.1step 2.1∎

The subrepresentation generated by any highest weight vector is therefore all of $V$ .

Depends on

Used by

Rational highest-weight modules from adjoint Plücker vectors Lemma
The lowest weight space is the nilradical-invariant line Lemma
Highest weight of the dual representation Proposition
The highest-weight space is one-dimensional Proposition
Highest-weight classification Theorem
Simple highest-weight modules are classified by highest weight Theorem

Dependency tree · two levels

15 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)