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

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Highest weight modules lie below the top weight

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 $V$ be a representation generated by a highest weight vector $v$ of weight $λ$ (Highest-weight vectors and modules). Then $V = U (n^{-}) v = U (n^{-}) C v,$ every weight $μ$ of $V$ is of the form $μ = λ - \sum_{i} n_{i} α_{i}$ with $n_{i} \in Z_{\geq 0}$ , so that $μ \leq λ$ in the root order (Root order on weights), and $V_{λ} = C v .$

Facts & Assumptions

Given: The Axiom of Choice, such $g, h$ , a chosen positive system, and a module $V$ generated by a highest weight vector $v$ of weight $λ$ .

[A1]

The Axiom of Choice is assumed; it enters through the root-space theory supplying [L1], [L2] and [L4] (The Axiom of Choice).

[L1]

The chosen positive system gives the direct sum $g = n^{-} \oplus h \oplus n^{+}$ with $n^{\pm}$ the span of the root spaces $g_{\pm α}$ , $α \in Φ^{+}$ (Triangular decomposition, Positive and negative nilpotent subalgebras and the Borel).

[L2]

For an ordered basis $x_{1}, \dots, x_{N}$ of a finite-dimensional complex Lie algebra, the monomials $x_{1}^{a_{1}} \dots x_{N}^{a_{N}}$ form a basis of its universal enveloping algebra (Poincaré–Birkhoff–Witt theorem); applied to $g$ with an ordered basis beginning with a basis of $n^{-}$ , continuing with a basis of $h$ and ending with a basis of $n^{+}$ , this gives $U (g) = U (n^{-}) U (h) U (n^{+})$ as a linear span, and applied to $n^{-}$ it gives that the monomials in a basis of $n^{-}$ span $U (n^{-})$ .

[L3]

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

[L4]

If $x \in g_{α}$ and $w \in V_{μ}$ , then $x \cdot w \in V_{μ + α}$ (Root vectors shift weights).

[L5]

Every positive root is a nonzero nonnegative integral combination of the simple roots, each root space is one-dimensional, and the simple roots are linearly independent (Simple roots form a signed integral basis, Root spaces of a complex semisimple Lie algebra are one-dimensional).

Proof

technique · direct

1.1A1L3

$V = U (g) v$ by [L3], because $V$ is the subrepresentation generated by $v$ .

2.1A1L1L2L3step 1.1

Applying [L2] and using that $n^{+} \cdot v = 0$ and $H \cdot v = λ (H) v$ for $H \in h$ gives $U (g) v = U (n^{-}) U (h) U (n^{+}) v = U (n^{-}) C v$ .

3.1A1L2L4L5step 2.1

Fix a basis $f_{1}, \dots, f_{N}$ of $n^{-}$ consisting of root vectors $f_{j} \in g_{- α^{(j)}}$ with $α^{(j)} \in Φ^{+}$ ; by [L2] the monomials in the $f_{j}$ span $U (n^{-})$ , so every element of $V$ is a linear combination of vectors $f_{j_{1}}^{a_{1}} \dots f_{j_{k}}^{a_{k}} \cdot v$ , and by [L4] and [L5] the weight of such a vector is $λ - \sum_{r} a_{r} α^{(j_{r})}$ , a functional of the form $λ - \sum_{i} n_{i} α_{i}$ with $n_{i} \geq 0$ .

4.1step 3.1

By step 3.1 every weight of $V$ lies in $λ - Q_{+}$ and satisfies $μ \leq λ$ in the root order (Root order on weights).

4.2L5step 3.1

For the top weight, a monomial $f_{j_{1}}^{a_{1}} \dots f_{j_{k}}^{a_{k}}$ has weight $λ$ exactly when $\sum_{r} a_{r} α^{(j_{r})} = 0$ ; since the $α^{(j_{r})}$ are nonzero elements of $Q_{+}$ and the simple roots are linearly independent by [L5], this forces $a_{r} = 0$ for every $r$ , so the only monomial of weight $λ$ is the empty one and $V_{λ} = C v$ .

5.1step 2.1step 4.1step 4.2∎

Steps 2.1, 4.1 and 4.2 prove the three assertions.

Depends on

Used by

Freudenthal recursion terminates from the highest weight Corollary
Tensor-product multiplicities for finite-dimensional simple modules Definition
Minuscule weights have exactly the Weyl orbit as their weights Lemma
Orthogonality identifies the Weyl numerator Lemma
Positive root strings sum the Freudenthal correction Lemma
Rational highest-weight modules from adjoint Plücker vectors Lemma
Simple-root integrability bounds the dominant cyclic module Lemma
The admissible-tableau count equals the Littlewood--Richardson coefficient Lemma
The augmentation kernel is the sum of the simple-reflection Verma submodules Lemma
The dominant cyclic generator survives 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
Unique simple quotient of the dominant cyclic module Lemma
Extremal Weyl-orbit weights Proposition
Highest weight of the dual representation Proposition
Littlewood--Richardson coefficients stabilise with rank Proposition
Tensor-product multiplicities are character structure constants Proposition
The highest-weight space is one-dimensional Proposition
Top summand in a tensor product Proposition
Highest weights for compact connected groups Theorem
Highest-weight classification Theorem
Outside-alcove indecomposable tiltings are negligible Theorem
Simple highest-weight modules are classified by highest weight Theorem
The Littlewood--Richardson tensor-product rule Theorem
Weak BGG resolution Theorem

Dependency tree · two levels

41 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)