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

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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.

Highest-weight classification

Statement

Assume the Axiom of Choice. Let $g$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $h$ and a fixed positive system. Then the maps $V ⟼ λ (V), λ ⟼ L (λ)$ are mutually inverse bijections between the isomorphism classes of finite-dimensional irreducible representations of $g$ and the dominant integral weights (Integral, dominant, and strictly dominant weights); here $λ (V)$ is the highest weight of $V$ (Highest-weight vectors and modules) and $L (λ)$ is the simple quotient of $M_{i n t} (λ)$ .

Facts & Assumptions

Given: The Axiom of Choice, such $g, h$ and a fixed positive system.

[A1]

The Axiom of Choice is assumed; it enters through the root-space theory used by the cited suppliers (The Axiom of Choice).

[L1]

Every nonzero finite-dimensional irreducible module $V$ contains a highest weight vector and, when generated by one, satisfies $V = U (n^{-}) C v$ with all weights $\leq λ$ ; the highest weight is unique (Every finite-dimensional irreducible module has a highest-weight vector, An irreducible module is generated by its highest-weight vector, Highest weight modules lie below the top weight, Highest-weight vectors and modules).

[L2]

The highest weight of a finite-dimensional irreducible module is dominant integral (Finite-dimensional highest weights are dominant integral).

[L3]

For dominant integral $λ$ the module $L (λ)$ is a finite-dimensional simple highest weight module of highest weight $λ$ (Unique simple quotient of the dominant cyclic module, Dominant simple highest-weight modules are finite-dimensional).

[L4]

Two finite-dimensional simple highest weight modules are isomorphic if and only if their highest weights agree (Simple highest-weight modules are classified by highest weight).

Proof

technique · direct

1.1L1L2A1

The assignment $V \mapsto λ (V)$ is well defined on isomorphism classes of nonzero finite-dimensional irreducible modules by [L1] and takes values in the dominant integral weights by [L2].

1.2L3

The assignment $λ \mapsto L (λ)$ is defined on all dominant integral weights by [L3], and $L (λ)$ is a nonzero finite-dimensional irreducible module whose highest weight is $λ$ .

1.3L1L2L3L4

For every nonzero finite-dimensional irreducible $V$ we have $V ≅ L (λ (V))$ : both are finite-dimensional simple highest weight modules with the same highest weight $λ (V)$ , so [L4] applies.

2.1L3step 1.3

Conversely, if $λ$ is dominant integral then the highest weight of $L (λ)$ is $λ$ by [L3]; hence the two assignments are inverse to one another on isomorphism classes.

3.1L4step 1.3step 2.1∎

Every dominant integral weight therefore occurs (through $L (λ)$ ), and no two distinct dominant integral weights give isomorphic modules by [L4]; every finite-dimensional irreducible representation occurs as $L (λ (V))$ by step 1.3. This is the asserted bijection.

Depends on

Used by

Every finite-dimensional module is a direct sum of highest-weight modules Corollary
Freudenthal recursion terminates from the highest weight Corollary
The Borel-Weil-Bott Euler character is a signed dual Weyl character Corollary
Tensor-product multiplicities for finite-dimensional simple modules Definition
The formal character of a finite-dimensional weight module Definition
An sl3 weight with cohomology in degree one Example
Finite-dimensional irreducible sl2 modules have exact integral central characters Example
Kostant multiplicity in the sl3 adjoint module Example
Standard and dual representations of slₙ Example
The Clebsch--Gordan tensor decomposition for sl2 Example
The Weyl dimension formula for a fundamental sl3 module Example
Three times three for sl3 Example
Weyl character and dimension formulas for sl2 Example
Minuscule weights have exactly the Weyl orbit as their weights Lemma
Positive root strings sum the Freudenthal correction Lemma
Projective orbit constructions for G/B and G/Pₐlpha Lemma
Rational highest-weight modules from adjoint Plücker vectors Lemma
The Casimir comparison on a weight space Lemma
The Chevalley–Eilenberg Laplacian is scalar on weight components Lemma
The extremal weight cochain of a Weyl element is closed and unique Lemma
The lowest weight space is the nilradical-invariant line Lemma
The shifted norm of a weight is maximal only at the top weight Lemma
Weyl alternation extracts a dominant highest-weight coefficient Lemma
Borel-Weil-Bott is compatible with Serre duality Proposition
Extremal Weyl-orbit weights Proposition
Highest weight of the dual representation Proposition
Root and weight lattice sandwich Proposition
Tensor-product multiplicities are character structure constants Proposition
Top summand in a tensor product Proposition
Highest weights for compact connected groups Theorem
Kostant's nilradical cohomology theorem Theorem
Kostant's weight multiplicity formula Theorem
Outside-alcove indecomposable tiltings are negligible Theorem
The Borel-Weil theorem Theorem
The Casselman–Osborne constraint on weights of nilradical cohomology Theorem
The Weyl denominator identity Theorem

Dependency tree · two levels

37 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

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