DefinitionDefinition: AI-adaptedProof: 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.

Left and right translations on a Lie group

Definition

Let $G$ be a Lie group and fix $g \in G$ . The left translation by $g$ and right translation by $g$ are respectively

$L_{g} : G ⟶ G, L_{g} (h) = g h,$

and

$R_{g} : G ⟶ G, R_{g} (h) = h g .$

Both maps are smooth: each is obtained from the smooth multiplication $m : G \times G \to G$ in Lie group by holding one argument fixed. The notation here is the ordinary right-translation convention. Kirillov writes a right action as $h \mapsto h g^{- 1}$ ; therefore this page's $R_{g}$ is that source's right action by $g^{- 1}$ .

The definition applies in dimensions zero and one and uses no metric or nondegeneracy condition. A Lie group is nonempty, its manifold is boundaryless by the page convention, and the one supplied element $g$ involves no choice from a family.

Depends on

Lie group

Used by

Right-trivialized differential of the Lie-group exponential Lemma
Maurer--Cartan form is a pointwise isomorphism and left invariant Proposition
Translations are diffeomorphisms and their differentials trivialize the tangent bundle Proposition
Baker–Campbell–Hausdorff theorem Theorem
Lie-group homomorphisms have constant rank Theorem
The differential of Ad is ad Theorem

Dependency tree · two levels

3 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)