Lie group
E142004
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Lie groups | 17 |
| Lie group canonical | 4 |
| Heisenberg group | 1 |
| Lie group actions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1234892 — 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 group Context triple: [Sophus Lie, hasConceptNamedAfter, Lie group]
-
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.
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 subgroup
A Lie subgroup is a subgroup of a Lie group that is itself a Lie group and an embedded submanifold, inheriting compatible smooth and group structures from the ambient Lie group.
-
D.
Lorentz group
The Lorentz group is the mathematical group of spacetime symmetries in special relativity, consisting of all rotations and boosts that preserve the Minkowski spacetime interval.
-
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 group Target entity description: A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
-
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.
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 subgroup
A Lie subgroup is a subgroup of a Lie group that is itself a Lie group and an embedded submanifold, inheriting compatible smooth and group structures from the ambient Lie group.
-
D.
Lorentz group
The Lorentz group is the mathematical group of spacetime symmetries in special relativity, consisting of all rotations and boosts that preserve the Minkowski spacetime interval.
-
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 (68)
| Predicate | Object |
|---|---|
| instanceOf |
differentiable manifold
ⓘ
group ⓘ mathematical structure ⓘ smooth manifold ⓘ topological group ⓘ |
| appearsIn |
gauge theory
ⓘ
general relativity ⓘ particle physics ⓘ quantum mechanics ⓘ string theory ⓘ |
| dimension | finite-dimensional (for finite-dimensional Lie groups) ⓘ |
| fieldOfStudy |
Lie theory
ⓘ
differential geometry ⓘ mathematics ⓘ representation theory ⓘ theoretical physics ⓘ |
| generalizationOf |
continuous groups of transformations
ⓘ
matrix groups ⓘ |
| hasAssociatedObject |
Cartan subgroup
ⓘ
Weyl group ⓘ
surface form:
Weyl group (for semisimple Lie groups)
maximal compact subgroup ⓘ root system (for semisimple Lie groups) ⓘ universal covering group ⓘ |
| hasExample |
Heisenberg Lie algebra
ⓘ
surface form:
Heisenberg group
circle group U(1) ⓘ complex numbers under addition ⓘ general linear group GL(n,C) ⓘ general linear group GL(n,R) ⓘ nonzero real numbers under multiplication ⓘ real numbers under addition ⓘ special linear group SL(n,C) ⓘ special linear group SL(n,R) ⓘ special orthogonal group SO(n) ⓘ special unitary group SU(n) ⓘ |
| hasPart | Lie algebra ⓘ |
| hasProperty |
Hausdorff
ⓘ
continuous symmetries ⓘ group operation is smooth ⓘ inversion map is smooth ⓘ inversion map is smooth diffeomorphism ⓘ locally Euclidean ⓘ multiplication map is smooth ⓘ second countable ⓘ |
| hasType |
abelian Lie group
ⓘ
compact Lie group ⓘ connected Lie group ⓘ finite-dimensional Lie group ⓘ infinite-dimensional Lie group ⓘ nilpotent Lie group ⓘ non-compact Lie group ⓘ reductive Lie group ⓘ semisimple Lie group ⓘ simply connected Lie group ⓘ solvable Lie group ⓘ |
| namedAfter | Sophus Lie ⓘ |
| relatedConcept |
Lie algebra
ⓘ
Lie group action ⓘ Lie group representation ⓘ Lie homomorphism ⓘ Lie ring ⓘ Lie semigroup ⓘ Lie subgroup ⓘ |
| studiedBy | Sophus Lie ⓘ |
| studiedIn | 19th century ⓘ |
| usedFor |
classification of symmetries in physics
ⓘ
representation theory of groups ⓘ study of continuous symmetries ⓘ study of differential equations ⓘ |
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 group Description of subject: A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
Referenced by (23)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Lie groups
this entity surface form:
Lie groups
this entity surface form:
Lie groups
this entity surface form:
Lie groups
this entity surface form:
Lie groups
this entity surface form:
Lie groups
this entity surface form:
Lie groups
this entity surface form:
Heisenberg group
subject surface form:
Élie Cartan
this entity surface form:
Lie groups
this entity surface form:
Lie groups
this entity surface form:
Lie groups
this entity surface form:
Lie groups
this entity surface form:
Lie groups
this entity surface form:
Lie group actions
this entity surface form:
Lie groups
subject surface form:
Marius Sophus Lie
this entity surface form:
Lie groups
subject surface form:
Sophus Lie
subject surface form:
Harish-Chandra
this entity surface form:
Lie groups
this entity surface form:
Lie groups
this entity surface form:
Lie groups