orthogonal group O(n+1,2)
E581259
The orthogonal group O(n+1,2) is the Lie group of linear transformations preserving a nondegenerate quadratic form of signature (n+1,2), playing a central role in conformal and Lie sphere geometry.
All labels observed (1)
| Label | Occurrences |
|---|---|
| orthogonal group O(n+1,2) canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6282374 — 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: orthogonal group O(n+1,2) Context triple: [Lie sphere geometry, usesGroup, orthogonal group O(n+1,2)]
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A.
orthogonal group O(n)
The orthogonal group O(n) is the group of all n×n real matrices that preserve the standard Euclidean inner product, representing rotations and reflections in n-dimensional space.
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B.
special orthogonal group SO(n)
The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
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C.
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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D.
Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
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E.
The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their 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: orthogonal group O(n+1,2) Target entity description: The orthogonal group O(n+1,2) is the Lie group of linear transformations preserving a nondegenerate quadratic form of signature (n+1,2), playing a central role in conformal and Lie sphere geometry.
-
A.
orthogonal group O(n)
The orthogonal group O(n) is the group of all n×n real matrices that preserve the standard Euclidean inner product, representing rotations and reflections in n-dimensional space.
-
B.
special orthogonal group SO(n)
The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
-
C.
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.
-
D.
Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
-
E.
The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Lie group
ⓘ
matrix group ⓘ orthogonal group ⓘ real algebraic group ⓘ |
| actsConformallyOn | n-dimensional sphere Sⁿ ⓘ |
| actsOn |
Möbius space of Sⁿ
ⓘ
real vector space of dimension n+3 ⓘ |
| actsTransitivelyOn | space of null lines in ℝ^{n+3} ⓘ |
| contains | reflections in nonisotropic vectors ⓘ |
| hasCartanDecomposition | 𝔰𝔬(n+1,2)=𝔨⊕𝔭 ⓘ |
| hasCartanInvolution | conjugation by diag(I_{n+1},−I₂) ⓘ |
| hasCenter | {±I} for n+3 ≠ 2 ⓘ |
| hasConnectedComponent | special orthogonal group SO(n+1,2) NERFINISHED ⓘ |
| hasDimension | (n+3)(n+2)/2 ⓘ |
| hasDoubleCover | Spin(n+1,2)→SO(n+1,2) ⓘ |
| hasFundamentalGroup | ℤ for n+3 ≥ 3 (via SO⁰(n+1,2)) ⓘ |
| hasIdentityComponent | SO⁰(n+1,2) ⓘ |
| hasIwasawaDecomposition | KAN with K ≅ O(n+1)×O(2) ⓘ |
| hasLieAlgebra | 𝔰𝔬(n+1,2) ⓘ |
| hasMaximalCompactSubgroup | O(n+1)×O(2) NERFINISHED ⓘ |
| hasMaximalTorusDimension | ⌊(n+3)/2⌋ ⓘ |
| hasParabolicSubgroupsCorrespondingTo | conformal stabilizers of points on Sⁿ ⓘ |
| hasRankOverℂ | ⌊(n+3)/2⌋ ⓘ |
| hasRealRank | 2 ⓘ |
| hasRootSystem | type B₂ or D₂ as real rank 2 form ⓘ |
| hasSignature | (n+1,2) ⓘ |
| hasSpecialSubgroup | SO(n+1,2) NERFINISHED ⓘ |
| hasSpinCover | Spin(n+1,2) NERFINISHED ⓘ |
| hasTwoComponents | true ⓘ |
| hasType | indefinite orthogonal group NERFINISHED ⓘ |
| hasWeylGroup | finite reflection group of type B₂ or D₂ depending on n ⓘ |
| identifiesWith | group of Möbius transformations of Sⁿ up to finite kernel ⓘ |
| isConformalGroupOf | round conformal structure on Sⁿ ⓘ |
| isDefinedOver | ℝ ⓘ |
| isGeneratedBy | orthogonal reflections ⓘ |
| isIsogenousTo | PO(n+1,2) NERFINISHED ⓘ |
| isIsometryGroupOf | quadratic space of signature (n+1,2) ⓘ |
| isNoncompact | true ⓘ |
| isRealFormOf | complex Lie group SO(n+3,ℂ) ⓘ |
| isReductive | true ⓘ |
| isSemisimple | true ⓘ |
| isSimpleModCenter | true for n+3 ≥ 5 ⓘ |
| isSubgroupOf | general linear group GL(n+3,ℝ) NERFINISHED ⓘ |
| isUsedIn |
AdS/CFT-related models via conformal symmetry
ⓘ
Lie sphere geometry NERFINISHED ⓘ conformal geometry ⓘ representation theory of real reductive groups ⓘ theory of automorphic forms ⓘ |
| preserves |
bilinear form of signature (n+1,2)
ⓘ
nondegenerate quadratic form of signature (n+1,2) ⓘ |
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: orthogonal group O(n+1,2) Description of subject: The orthogonal group O(n+1,2) is the Lie group of linear transformations preserving a nondegenerate quadratic form of signature (n+1,2), playing a central role in conformal and Lie sphere geometry.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.