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

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Simple-root integrability relations

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 with simple roots $α_{1}, \dots, α_{r}$ , let $λ$ be dominant integral with $m_{i} = ⟨ λ, α_{i}^{\lor} ⟩$ (Integral, dominant, and strictly dominant weights), and let $V$ be a finite-dimensional highest weight module of highest weight $λ$ with highest weight vector $v_{λ}$ (Highest-weight vectors and modules). For every simple root $α_{i}$ and a lowering vector $f_{i} \in g_{- α_{i}}$ with $[e_{i}, f_{i}] = h_{α_{i}}$ for a suitable $e_{i}$ (The root ${s l}_{2}$ triple), $f_{i}^{m_{i} + 1} v_{λ} = 0.$

Facts & Assumptions

Given: The Axiom of Choice, such $g, h$ , a dominant integral $λ$ with $m_{i} = ⟨ λ, α_{i}^{\lor} ⟩$ , a finite-dimensional highest weight module $V$ of highest weight $λ$ , and for each $i$ a pair $e_{i} \in g_{α_{i}}$ , $f_{i} \in g_{- α_{i}}$ forming, together with $h_{α_{i}}$ , a copy of ${s l}_{2}$ .

[A1]

The Axiom of Choice is assumed; it enters through the root-space theory supplying [L1] and through [L3] (The Axiom of Choice).

[L1]

For every $i$ , $[e_{i}, f_{i}] = h_{α_{i}}$ , $[h_{α_{i}}, e_{i}] = 2 e_{i}$ and $[h_{α_{i}}, f_{i}] = - 2 f_{i}$ (The root ${s l}_{2}$ triple).

[L2]

$v_{λ} \neq 0$ is killed by $n^{+}$ and satisfies $h_{α_{i}} \cdot v_{λ} = λ (h_{α_{i}}) v_{λ} = m_{i} v_{λ}$ (Highest-weight vectors and modules, Integral, dominant, and strictly dominant weights).

[L3]

A finite-dimensional ${s l}_{2}$ -module is a direct sum of irreducibles, and an irreducible submodule with highest weight $m \geq 0$ has dimension $m + 1$ and weights $m, m - 2, \dots, - m$ (Finite-dimensional representations of ${s l}_{2}$ ).

Proof

technique · direct

1.1A1L1L2

Fix $i$ and consider $W = U (span {e_{i}, f_{i}, h_{α_{i}}}) v_{λ}$ , the ${s l}_{2}$ -submodule of $V$ generated by $v_{λ}$ ; it is finite dimensional because $V$ is, and $e_{i} \cdot v_{λ} = 0$ while $h_{α_{i}} \cdot v_{λ} = m_{i} v_{λ}$ by [L1] and [L2].

2.1L3step 1.1

By [L3] write $W = ⨁_{a = 1}^{s} W_{a}$ as a direct sum of irreducible ${s l}_{2}$ -submodules, and write $v_{λ} = \sum_{a} v_{a}$ with $v_{a} \in W_{a}$ . Because every $W_{a}$ is stable under $e_{i}$ and $h_{α_{i}}$ , uniqueness of the direct sum and step 1.1 give $e_{i} v_{a} = 0$ and $h_{α_{i}} v_{a} = m_{i} v_{a}$ for every $a$ .

3.1L3step 2.1

For every $a$ with $v_{a} \neq 0$ , step 2.1 makes $v_{a}$ a highest weight vector of the irreducible module $W_{a}$ with highest weight $m_{i}$ . By [L3], $f_{i}^{m_{i} + 1} v_{a} = 0$ ; the same equality is trivial when $v_{a} = 0$ . Summing over $a$ gives $f_{i}^{m_{i} + 1} v_{λ} = 0$ in $W$ .

4.1step 3.1∎

Since $W \subseteq V$ , the vanishing of step 3.1 holds in $V$ ; as $i$ was arbitrary, $f_{i}^{m_{i} + 1} v_{λ} = 0$ for every simple root, which is the assertion.

Depends on

Used by

Simple highest-weight modules are classified by highest weight Theorem

Dependency tree · two levels

31 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)