Lie algebroid
E581260
A Lie algebroid is a geometric structure that generalizes Lie algebras and tangent bundles, encoding infinitesimal symmetries on manifolds via a vector bundle with a Lie bracket and an anchor map.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Lie algebroid canonical | 2 |
| Lie-Rinehart algebra | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6282397 — 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 algebroid Context triple: [Lie pseudogroup, hasInfinitesimalObject, Lie algebroid]
-
A.
Lie algebras
Lie algebras are algebraic structures used to study continuous symmetries, especially those arising from Lie groups, via a linearized, infinitesimal perspective.
-
B.
Lie pseudogroup
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.
-
C.
Lie bracket
The Lie bracket is a bilinear, antisymmetric operation on a Lie algebra that measures the noncommutativity of its elements and encodes its infinitesimal structure.
-
D.
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.
-
E.
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.
- 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 algebroid Target entity description: A Lie algebroid is a geometric structure that generalizes Lie algebras and tangent bundles, encoding infinitesimal symmetries on manifolds via a vector bundle with a Lie bracket and an anchor map.
-
A.
Lie algebras
Lie algebras are algebraic structures used to study continuous symmetries, especially those arising from Lie groups, via a linearized, infinitesimal perspective.
-
B.
Lie pseudogroup
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.
-
C.
Lie bracket
The Lie bracket is a bilinear, antisymmetric operation on a Lie algebra that measures the noncommutativity of its elements and encodes its infinitesimal structure.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
geometric structure
ⓘ
mathematical structure ⓘ |
| admits | cohomology theory ⓘ |
| anchorMapTarget | tangent bundle of the base manifold ⓘ |
| anchorMapType | vector bundle morphism ⓘ |
| appearsIn |
generalized geometry
ⓘ
integrability problems of Lie groupoids ⓘ |
| associatedWith | Lie groupoid ⓘ |
| bracketProperty |
Leibniz rule with respect to anchor
ⓘ
antisymmetric ⓘ satisfies Jacobi identity ⓘ |
| bracketType | R-bilinear map on sections ⓘ |
| cohomologyUsedFor |
characteristic classes
ⓘ
deformation theory ⓘ |
| compatibilityCondition | anchor map is Lie algebra homomorphism on sections ⓘ |
| definedOn | smooth manifold ⓘ |
| encodes |
infinitesimal symmetries
ⓘ
infinitesimal transformations on manifolds ⓘ |
| field |
Lie theory
ⓘ
Poisson geometry ⓘ differential geometry ⓘ |
| generalizes |
Lie algebra
ⓘ
tangent bundle ⓘ |
| hasAnchorMap | bundle map from the vector bundle to the tangent bundle ⓘ |
| hasBase | smooth manifold ⓘ |
| hasBracket | Lie bracket on space of sections ⓘ |
| hasComponent |
Lie bracket
ⓘ
anchor map ⓘ vector bundle ⓘ |
| hasDualObject | Lie groupoid ⓘ |
| hasGeneralization |
VB-algebroid
ⓘ
higher Lie algebroid ⓘ |
| hasMorphisms | Lie algebroid morphisms ⓘ |
| hasTotalSpace | vector bundle over a manifold ⓘ |
| introducedBy | Jean Pradines ⓘ |
| isInfinitesimalVersionOf | Lie groupoid ⓘ |
| morphismType | vector bundle maps compatible with brackets and anchors ⓘ |
| relatedConcept |
Courant algebroid
ⓘ
Lie-Rinehart algebra ⓘ
linked to:
Lie algebroid
Poisson manifold ⓘ |
| specialCase |
Lie algebra as Lie algebroid over a point
ⓘ
tangent bundle with usual Lie bracket of vector fields ⓘ |
| studiedIn | global analysis on manifolds ⓘ |
| usedIn |
Poisson geometry
ⓘ
deformation quantization ⓘ foliation theory ⓘ gauge theory ⓘ nonlinear connection theory ⓘ |
| yearIntroduced | 1967 ⓘ |
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 algebroid Description of subject: A Lie algebroid is a geometric structure that generalizes Lie algebras and tangent bundles, encoding infinitesimal symmetries on manifolds via a vector bundle with a Lie bracket and an anchor map.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Lie algebroid