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

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Unique simple quotient of the dominant cyclic module

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 $λ$ be dominant integral, and let $M = M_{i n t} (λ)$ with canonical generator $v_{λ}$ (Dominant cyclic highest-weight presentation). Then $M$ has a unique maximal proper submodule $N_{λ}$ , and the quotient $L (λ) : = M / N_{λ}$ is a nonzero simple $g$ -module; in particular $M$ has, up to isomorphism, a unique simple quotient.

Facts & Assumptions

Given: The Axiom of Choice, such $g, h$ , a chosen positive system, a dominant integral $λ$ , and $M = M_{i n t} (λ)$ with generator $v_{λ}$ .

[A1]

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

[L1]

$v_{λ} \neq 0$ has weight $λ$ , is killed by $n^{+}$ , and generates $M$ ; moreover $M = U (n^{-}) C v_{λ}$ , all weights of $M$ satisfy $μ \leq λ$ , and $M_{λ} = C v_{λ}$ (The dominant cyclic generator survives, Highest weight modules lie below the top weight).

[L2]

The action of $g$ extends to a unital action of $U (g)$ ; submodules are the $U (g)$ -submodules, and a submodule generated by one vector $v$ is $U (g) v$ (Lie representations are U(g)-modules, Highest-weight vectors and modules, Subrepresentations, quotient representations, and intertwiners).

[L3]

For an ordered basis of the abelian Lie algebra $h$ the PBW monomials form a basis of $U (h)$ , so $U (h)$ is the commutative polynomial algebra in these variables and acts on a weight vector of weight $μ$ through evaluation at $μ$ (Poincaré–Birkhoff–Witt theorem, Universal enveloping algebra).

[L4]

Every element of $M$ is a finite sum of weight vectors: [L1] gives the spanning set $U (n^{-}) C v_{λ}$ , PBW expresses its elements as finite linear combinations of monomials in negative-root vectors, and each such monomial sends a weight vector to a weight vector (or zero) by repeated application of Root vectors shift weights. [L1, L3]

Proof

technique · direct

1.1

Every proper submodule $N$ of $M$ misses $M_{λ} = C v_{λ}$ : if $v_{λ} \in N$ , then $N$ contains $U (g) v_{λ} = M$ by [L1] and [L2], so $N = M$ , contrary to being proper; hence $N \cap M_{λ} = 0$ .

L1L2A1

1.2

For a submodule $N$ , every element $n \in N$ has all its weight components in $N$ : write $n$ as a finite sum of weight vectors by [L4], with finite support $S$ ; for $μ \in S$ , Lagrange interpolation on the finitely many distinct functionals of $S$ gives a polynomial $p_{μ}$ with $p_{μ} (μ) = 1$ and $p_{μ} (ν) = 0$ for $ν \in S ∖ {μ}$ , and the corresponding element of $U (h)$ acts on each weight vector by these values by [L3]; since $N$ is $h$ -stable and $h \subseteq U (g)$ , the vector $p_{μ} \cdot n$ is the $μ$ -weight component of $n$ and lies in $N$ .

L3L4

2.1

Hence $N = (N \cap M_{μ})$ -direct sum over weights, and by step 1.1 every proper submodule $N$ satisfies $N \cap M_{λ} = 0$ .

step 1.1step 1.2

3.1

Let $N_{λ} = \sum {N : N is a proper submodule of M}$ ; this is a submodule of $M$ , and every element of it is a finite sum of elements lying in finitely many proper submodules, so by step 2.1 its $λ$ -component is zero; since $M_{λ} = C v_{λ} \neq 0$ , the submodule $N_{λ}$ is proper, and by construction it contains every proper submodule of $M$ . Thus $N_{λ}$ is the unique maximal proper submodule.

L1step 2.1

4.1

The quotient $L (λ) = M / N_{λ}$ is nonzero because $N_{λ}$ is proper, and it is simple: a nonzero proper submodule of $L (λ)$ would have as preimage a proper submodule of $M$ strictly containing $N_{λ}$ , contradicting the maximality of $N_{λ}$ ; hence the simple quotient is unique, since any simple quotient of $M$ has kernel a maximal proper submodule, which equals $N_{λ}$ .

L2step 3.1

5.1

Therefore $M$ has a unique maximal proper submodule and a unique simple quotient $L (λ)$ , as asserted.

step 4.1∎

Depends on

Used by

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

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)