Lie pseudogroup
E140810
A Lie pseudogroup is a collection of local diffeomorphisms on a manifold that is closed under composition, inversion, and restriction, generalizing the concept of a Lie group to transformations defined only locally.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Lie pseudogroup canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1234898 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Lie pseudogroup Context triple: [Sophus Lie, hasConceptNamedAfter, Lie pseudogroup]
-
A.
Lie theory
Lie theory is a branch of mathematics that studies continuous symmetry through Lie groups and Lie algebras, with deep applications in geometry, analysis, and theoretical physics.
-
B.
Cartan connections
Cartan connections are a geometric framework generalizing affine and Riemannian connections that model curved spaces on homogeneous spaces, developed by Élie Cartan.
-
C.
Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
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D.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
-
E.
Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Lie pseudogroup Target entity description: A Lie pseudogroup is a collection of local diffeomorphisms on a manifold that is closed under composition, inversion, and restriction, generalizing the concept of a Lie group to transformations defined only locally.
-
A.
Lie theory
Lie theory is a branch of mathematics that studies continuous symmetry through Lie groups and Lie algebras, with deep applications in geometry, analysis, and theoretical physics.
-
B.
Cartan connections
Cartan connections are a geometric framework generalizing affine and Riemannian connections that model curved spaces on homogeneous spaces, developed by Élie Cartan.
-
C.
Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
-
D.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
-
E.
Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
pseudogroup ⓘ structure in differential geometry ⓘ |
| appearsIn |
equivalence of differential equations
ⓘ
theory of differential invariants ⓘ |
| characterizedBy |
infinitesimal symmetries
ⓘ
systems of partial differential equations ⓘ |
| closedUnder |
composition
ⓘ
gluing of local transformations ⓘ inversion ⓘ restriction of domain ⓘ |
| consistsOf | local diffeomorphisms ⓘ |
| definedAs | pseudogroup of local diffeomorphisms satisfying regularity conditions ⓘ |
| definedOn | smooth manifold ⓘ |
| developedBy | Élie Cartan ⓘ |
| example |
local conformal transformations
ⓘ
local contact transformations ⓘ local isometries of a Riemannian manifold ⓘ local symplectomorphisms ⓘ |
| field |
Lie theory
ⓘ
differential geometry ⓘ geometric analysis ⓘ |
| formalizedBy |
Charles Ehresmann
ⓘ
Shoshichi Kobayashi ⓘ |
| generalizes |
Lie group
ⓘ
local transformation group ⓘ |
| hasConcept | prolongation to jet spaces ⓘ |
| hasConstraint | transformations satisfy analytic or smooth regularity conditions ⓘ |
| hasInfinitesimalObject |
Lie algebra of vector fields
ⓘ
Lie algebroid ⓘ |
| hasOrigin | work of Sophus Lie ⓘ |
| hasProperty |
locality
ⓘ
sheaf-like behavior ⓘ |
| isSubsetOf | pseudogroup of all local diffeomorphisms on a manifold ⓘ |
| mayBe |
finite type
ⓘ
infinite type ⓘ |
| modeledBy | systems of Lie-type differential equations ⓘ |
| relatedTo |
Cartan connections
ⓘ
surface form:
Cartan geometry
theory of G-structures ⓘ
surface form:
G-structures
Lie groupoids ⓘ groupoids of local diffeomorphisms ⓘ jet bundles ⓘ transformation groups ⓘ |
| requires | smoothness conditions on transformations ⓘ |
| studiedIn | theory of symmetries of differential equations ⓘ |
| usedFor |
theory of G-structures
ⓘ
surface form:
Cartan’s method of moving frames
describing local symmetries of geometric structures ⓘ equivalence problems in differential geometry ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Lie pseudogroup Description of subject: A Lie pseudogroup is a collection of local diffeomorphisms on a manifold that is closed under composition, inversion, and restriction, generalizing the concept of a Lie group to transformations defined only locally.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.