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The dual action needs a minus sign

Statement

The dual action is $(x λ) (v) = λ (x v)$ , with no minus sign.

Facts & Assumptions

Given: The proposed plus-sign formula on the algebraic dual of a representation.

[L1]

The valid dual formula has a minus sign (Direct-sum, dual, Hom, and tensor representations).

Refutation

technique · direct computation in the standard sl₂-module over a characteristic-zero field

1.1givenalgebra

Write $P_{x} (λ) = λ \circ ρ (x)$ for the proposed plus-sign operator. Then $[P_{x}, P_{y}] (λ) = λ \circ (ρ (y) ρ (x) - ρ (x) ρ (y)) = - P_{[x, y]} (λ)$ , so the proposed operators reverse rather than preserve the Lie bracket.

2.1step 1.1L1algebra∎

In the standard two-dimensional ${s l}_{2}$ -module, $[e, f] = h$ and $P_{h} \neq 0$ . Hence $[P_{e}, P_{f}] = - P_{h} \neq P_{h}$ , since the field has characteristic zero. The plus formula is therefore not a representation; the two minus signs in the commutator of the formula in [L1] correct this reversal.

Depends on

Direct-sum, dual, Hom, and tensor representations

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Etingof, MIT 18.745 notes, dual representations in §11.2, printed pp. 62–63 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, §4.2, printed pp. 50–52 (standard reference, not scraped)