Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22
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Dominance depends on a positive system

Statement

Assume the Axiom of Choice. Dominance of weights is defined canonically, without choosing positive roots.

Facts & Assumptions

Given: The Axiom of Choice, the root system Φ={α,−α} of sl2(C) with respect to h=Ch, where α(h)=2 (The special linear Lie algebra sl_2, Reduced crystallographic Euclidean root system), the two opposite positive systems Φ1+={α} and Φ2+={−α} (Positive systems and simple roots), the coroots hα=h and h−α=−h of Coroot of a Lie-algebra root, and the functional λ∈h∗ with λ(h)=1.

[A1]

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

[L1]

λ is dominant integral with respect to a positive system with base {β} exactly when ⟨λ,β∨⟩=λ(hβ)∈Z≥0 (Integral, dominant, and strictly dominant weights).

[L2]

For the root −α the coroot is h−α=−hα=−h, since the coroot is the normalised Killing dual and the dual vector changes sign with the functional (Coroot of a Lie-algebra root).

Refutation

technique · direct
1.1L1A1

With respect to the positive system {α} the functional λ is dominant integral: its only simple root is α and ⟨λ,α∨⟩=λ(h)=1∈Z≥0 by [L1].

1.2L1L2

With respect to the positive system {−α} the functional λ is not dominant: its simple root is −α and ⟨λ,(−α)∨⟩=λ(−h)=−1∉Z≥0 by [L1] and [L2].

2.1step 1.1step 1.2∎

The same functional is thus dominant for one choice of positive roots and non-dominant for the opposite choice, so there is no choice-free notion of dominance; the failed conclusion is that dominance could be decided without fixing positive roots.

Depends on

Used by

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Sources