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Right-invariant fields identify $T_{e} G$ with the same bracket as left-invariant fields

Statement refuted

Ordinary right-invariant fields identify $T_{e} G$ with the same Lie bracket as ordinary left-invariant fields.

Facts & Assumptions

Given: ${A C}_{ω}$ and the upper-unitriangular $3$ -by- $3$ group.

[F1]

Right-invariant extensions carry the negative of the tangent bracket. Right-invariant fields carry the opposite Lie bracket.

[F2]

Matrix units obey $E_{i j} E_{k l} = δ_{j k} E_{i l}$ . Matrix units $E_{i j}$ and the Kronecker delta.

[F3]

Countable choice is the exact assumption inherited from [F1]. The Axiom of Countable Choice ( ${A C}_{ω}$ ).

Refutation

technique · counterexample

1.1F2algebra

Put $X = E_{12}$ and $Y = E_{23}$ . By [F2], their left-invariant tangent bracket is $[X, Y] = X Y - Y X = E_{13} \neq 0$ .

2.1F1step 1.1

By [F1], the corresponding right-invariant fields satisfy $[X^{R}, Y^{R}] = - [X, Y]^{R} = - E_{13}^{R}$ , not $E_{13}^{R}$ . This disproves the same-bracket assertion.

3.1F1F2F3step 1.1step 2.1∎

The witness is nonempty and three-dimensional; dimensions zero and one have zero bracket and cannot witness the sign error. There is no metric, degeneracy, interval, endpoint, or biconditional. ${A C}_{ω}$ is propagated exactly through [F1], with no additional choice.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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)
Robert L. Bryant, An Introduction to Lie Groups and Symplectic Geometry (standard reference, not scraped)

Right-invariant fields identify TeGTe​G with the same bracket as left-invariant fields