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.

Lie group

Definition

A finite-dimensional real Lie group is a group $G$ , in the sense of Group and abelian group, together with the structure of a finite-dimensional real smooth manifold such that the maps

$m : G \times G ⟶ G, m (g, h) = g h,$

and

$inv : G ⟶ G, inv (g) = g^{- 1},$

are smooth in the sense of Smooth maps between manifolds with boundary. Its identity element is denoted by $e$ (or occasionally $1$ ).

Unless an item explicitly says otherwise, every Lie group on this page is real, finite-dimensional, and has no manifold boundary. Dimension zero is allowed; for example, any countable discrete group with its discrete zero-dimensional smooth structure is a Lie group. A Lie group cannot be empty because a group contains its identity. The definition applies unchanged in dimension one, uses no metric or nondegeneracy hypothesis, and makes no choice from a family of sets.

Depends on

Used by

Conjugation and the adjoint representation of a Lie group Definition
Homogeneous spaces of Lie groups Definition
Immersed, embedded, and closed Lie subgroups Definition
Left and right translations on a Lie group Definition
Left Maurer--Cartan form Definition
Lie-group homomorphism, isomorphism, and automorphism Definition
One-parameter subgroup of a Lie group Definition
Real and complex Lie groups Definition
Smooth left actions of Lie groups Definition
General and special linear Lie groups Example
Integer translations on the line Example
SL(n) as a closed Lie subgroup of GL(n) Example
The additive and multiplicative real Lie groups Example
The n-torus and its exponential lattice Example
Unitary and special unitary Lie groups Example
Differential at the identity determines a homomorphism from a disconnected Lie group False statement
The exponential map is globally injective on every connected Lie group False statement
The plus exponential convention is not a homomorphism for left actions False statement
Adjoint is a smooth Lie-group representation Proposition
Commuting Lie-algebra elements have multiplicative exponentials Proposition
Right-invariant fields carry the opposite Lie bracket Proposition
The tangent space at the identity is a Lie algebra Theorem

Dependency tree · two levels

9 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)