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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

1.1

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 $V$ .

L1A1

2.1

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 $v$ .

L1L2step 1.1

3.1

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

step 2.1∎

Depends on

Used by

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)