Lie subgroup
E140811
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Lie subgroup canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1234899 — 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 subgroup Context triple: [Sophus Lie, hasConceptNamedAfter, Lie subgroup]
-
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.
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B.
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.
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C.
Weyl group
A Weyl group is a finite reflection group associated with a root system that encodes the symmetries of Lie algebras and Lie groups in representation theory and geometry.
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D.
Cartan subalgebras
Cartan subalgebras are maximal abelian subalgebras of a Lie algebra consisting of semisimple elements, fundamental for classifying and understanding the structure of Lie algebras.
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E.
Cartan decomposition
Cartan decomposition is a fundamental structural result in Lie theory that expresses a Lie algebra or Lie group as a direct sum or product of subspaces or subgroups with specific symmetry properties, widely used in differential geometry and representation theory.
- 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 subgroup Target entity description: 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.
-
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.
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.
-
C.
Weyl group
A Weyl group is a finite reflection group associated with a root system that encodes the symmetries of Lie algebras and Lie groups in representation theory and geometry.
-
D.
Cartan subalgebras
Cartan subalgebras are maximal abelian subalgebras of a Lie algebra consisting of semisimple elements, fundamental for classifying and understanding the structure of Lie algebras.
-
E.
Cartan decomposition
Cartan decomposition is a fundamental structural result in Lie theory that expresses a Lie algebra or Lie group as a direct sum or product of subspaces or subgroups with specific symmetry properties, widely used in differential geometry and representation theory.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Lie group
ⓘ
embedded submanifold ⓘ mathematical concept ⓘ subgroup ⓘ |
| characterizedBy |
being a subgroup that is also an embedded submanifold
ⓘ
having inclusion map that is a smooth group homomorphism and an immersion ⓘ having smooth group operations induced from ambient Lie group ⓘ |
| correspondsTo | Lie subalgebra of the Lie algebra of the ambient Lie group under suitable conditions ⓘ |
| definedOn | Lie group ⓘ |
| example |
O(n) as a Lie subgroup of GL(n,ℝ)
ⓘ
SO(n) as a Lie subgroup of GL(n,ℝ) ⓘ U(n) as a Lie subgroup of GL(n,ℂ) ⓘ closed one-parameter subgroup generated by an element of the Lie algebra ⓘ maximal torus in a compact Lie group ⓘ torus T^n as a Lie subgroup of ℝ^n modulo ℤ^n ⓘ |
| field |
Lie theory
ⓘ
differential geometry ⓘ group theory ⓘ representation theory ⓘ |
| hasCondition |
must be a subgroup of the underlying abstract group of the Lie group
ⓘ
must be a submanifold with the subspace topology from the ambient Lie group ⓘ |
| hasMorphisms | smooth group homomorphisms between Lie subgroups ⓘ |
| hasProperty |
closed under group multiplication
ⓘ
closed under taking inverses ⓘ contains identity element of ambient Lie group ⓘ has finite-dimensional tangent spaces ⓘ inherits group structure from ambient Lie group ⓘ inherits smooth structure from ambient Lie group ⓘ is Hausdorff as a manifold ⓘ is a Lie group with the induced smooth structure ⓘ is a closed subset of the ambient Lie group when it is an embedded Lie subgroup ⓘ is a regular submanifold when embedded ⓘ is a subgroup stable under smooth conjugation maps of the ambient Lie group ⓘ is an embedded submanifold under standard definition ⓘ is an immersed submanifold of the ambient Lie group ⓘ is connected if it is a connected submanifold ⓘ is locally Euclidean ⓘ is second countable as a manifold ⓘ is smooth with respect to the ambient smooth structure ⓘ may be non-closed as a subset if only immersed ⓘ |
| hasTangentSpaceAtIdentity | Lie subalgebra of the ambient Lie algebra ⓘ |
| is | subgroup of a Lie group that is itself a Lie group ⓘ |
| relatedTo |
Lie algebra
ⓘ
Lie subalgebra ⓘ |
| usedIn |
classification of Lie groups
ⓘ
gauge theory in mathematical physics ⓘ representation theory of Lie groups ⓘ study of homogeneous spaces ⓘ symmetry analysis in 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 subgroup Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.