DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The left-translated distribution associated to a Lie subalgebra

Definition

Assume ${A C}_{ω}$ in the sense of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $G$ be a Lie group with Lie algebra $g = T_{e} G$ , and let $h \subseteq g$ be a Lie subalgebra. Its left-translated distribution is

$D_{g}^{h} = d (L_{g})_{e} (h) \subseteq T_{g} G .$

Under the smooth left trivialization $G \times g \to T G$ , $(g, X) \mapsto d (L_{g})_{e} X$ , this is the product subbundle $G \times h$ . Thus it is a smooth constant-rank distribution of rank $\dim h$ , including the cases $h = 0$ and $h = g$ . It is left invariant because $d (L_{a})_{g} D_{g}^{h} = D_{a g}^{h}$ .

The countable-choice assumption is inherited from the supplied smooth tangent-bundle trivialization; no additional choice is made in forming the displayed image of the supplied subspace.

Depends on

Used by

Dependency tree · two levels

17 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)