Spin(2,d)
E590885
Spin(2,d) is the double-covering spin group of SO(2,d), serving as the relevant symmetry group for spinor fields in (d+1)-dimensional anti-de Sitter space.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Spin(2)\times Spin(d) | 1 |
| Spin(2,2) (AdS_{3} spin group) | 1 |
| Spin(2,3) (AdS_{4} spin group) | 1 |
| Spin(2,d) canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6397028 — 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: Spin(2,d) Context triple: [AdS isometry group SO(2,d), hasUniversalCover, Spin(2,d)]
-
A.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
-
B.
rotation group SU(2)
The rotation group SU(2) is the Lie group of 2×2 unitary matrices with determinant 1 that serves as the double cover of the three-dimensional rotation group SO(3) and underlies the quantum theory of angular momentum and spin.
-
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.
Poincaré group
The Poincaré group is the fundamental symmetry group of special relativity, combining spacetime translations with Lorentz transformations in four-dimensional Minkowski space.
-
E.
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.
- 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: Spin(2,d) Target entity description: Spin(2,d) is the double-covering spin group of SO(2,d), serving as the relevant symmetry group for spinor fields in (d+1)-dimensional anti-de Sitter space.
-
A.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
-
B.
rotation group SU(2)
The rotation group SU(2) is the Lie group of 2×2 unitary matrices with determinant 1 that serves as the double cover of the three-dimensional rotation group SO(3) and underlies the quantum theory of angular momentum and spin.
-
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.
Poincaré group
The Poincaré group is the fundamental symmetry group of special relativity, combining spacetime translations with Lorentz transformations in four-dimensional Minkowski space.
-
E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Lie group
ⓘ
double cover ⓘ spin group ⓘ universal covering group ⓘ |
| actsOn | spinor fields on AdS_{d+1} ⓘ |
| containsSubgroup |
Spin(2)
ⓘ
Spin(d) ⓘ |
| covers |
SO(2,d)
ⓘ
linked to:
SO(2,d)_{0}
|
| generalizes |
Spin(2,2) (AdS_{3} spin group)
ⓘ
linked to:
Spin(2,d)
Spin(2,3) (AdS_{4} spin group) ⓘ
linked to:
Spin(2,d)
Spin(2,4) (AdS_{5} spin group) ⓘ |
| hasCenter | \mathbb{Z}_{2} (for generic d) ⓘ |
| hasDimension | (d+2)(d+1)/2 ⓘ |
| hasFundamentalRepresentation | spinor representation ⓘ |
| hasLieAlgebra | \mathfrak{so}(2,d) ⓘ |
| hasMaximalCompactSubgroup |
Spin(2)\times Spin(d)
ⓘ
linked to:
Spin(2,d)
|
| hasProjectiveRepresentation | conformal group of d-dimensional Minkowski space ⓘ |
| hasRank | \min(2,d) ⓘ |
| hasSignature | (2,d) ⓘ |
| hasTopology | same as universal cover of SO(2,d)_{0} ⓘ |
| isCompact | false ⓘ |
| isConformalGroupOf | d-dimensional Minkowski space at the spin level ⓘ |
| isConnected | true ⓘ |
| isCoveringGroupOf | SO(2,d)_{0} ⓘ |
| isDefinedAs | \{x\in Cl^{0}(2,d)\mid x V x^{-1}=V,\ \forall V\subset \mathbb{R}^{2,d}\} ⓘ |
| isDefinedOver | \mathbb{R} ⓘ |
| isDoubleCoverOf |
SO(2,d)
ⓘ
linked to:
SO(2,d)_{0}
|
| isGlobalSymmetryOf | free spinor fields on AdS_{d+1} ⓘ |
| isIdentityComponentOf | Pin(2,d) ⓘ |
| isIsometryGroupOf |
a space of signature (2,d) at the spin level
ⓘ
spin structure on AdS_{d+1} ⓘ |
| isLocallyIsomorphicTo |
SO(2,d)
ⓘ
linked to:
SO(2,d)_{0}
|
| isNonAbelian | true ⓘ |
| isPartOf | AdS_{d+1} isometry supergroup in supersymmetric theories ⓘ |
| isRealFormOf | Spin(d+2,\mathbb{C}) ⓘ |
| isReductive | true ⓘ |
| isSemisimple | true ⓘ |
| isSimplyConnected | true ⓘ |
| isSubgroupOf |
Cl(2,d)^{\times}
ⓘ
Pin(2,d) ⓘ |
| isSymmetryGroupOf | spinor fields in (d+1)-dimensional anti-de Sitter space ⓘ |
| isUniversalCoverOf | SO(2,d)^{ ext{connected}} ⓘ |
| isUsedIn |
AdS/CFT correspondence
ⓘ
higher-spin gauge theories on AdS_{d+1} ⓘ supergravity on AdS_{d+1} ⓘ |
| isUsedToDefine | spin structures on AdS_{d+1} ⓘ |
| preserves | quadratic form of signature (2,d) ⓘ |
| relatesTo |
Clifford algebra Cl(2,d)
ⓘ
linked to:
Clifford algebra
|
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: Spin(2,d) Description of subject: Spin(2,d) is the double-covering spin group of SO(2,d), serving as the relevant symmetry group for spinor fields in (d+1)-dimensional anti-de Sitter space.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Spin(2,d)
linked to: Spin(2,d)
linked to: Spin(2,d)