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.
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) ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.