nLab anti de Sitter group



Riemannian geometry

Group Theory



The anti de Sitter group is the isometry group O(d1,2)O(d-1,2) of dd-dimensional anti de Sitter spacetime. (orthogonal group for signature (d1,2)(d-1,2))

This is analogous to how the Poincare group is the isometry group of Minkowski spacetime.

The connected component SO(d1,2)SO(d-1,2) of the anti de Sitter group is isomorphic to the connected component of the conformal group of d2,1\mathbb{R}^{d-2,1}. This is the basis of the AdS-CFT correspondence.


Exceptional isomorphisms

  • SO(6,2)SO(4,)SO(6,2) \simeq SO(4,\mathbb{H}) (where \mathbb{H} is the quaternions)
groupsymboluniversal coversymbolhigher coversymbol
orthogonal groupO(n)\mathrm{O}(n)Pin groupPin(n)Pin(n)Tring groupTring(n)Tring(n)
special orthogonal groupSO(n)SO(n)Spin groupSpin(n)Spin(n)String groupString(n)String(n)
Lorentz groupO(n,1)\mathrm{O}(n,1)\,Spin(n,1)Spin(n,1)\,\,
anti de Sitter groupO(n,2)\mathrm{O}(n,2)\,Spin(n,2)Spin(n,2)\,\,
conformal groupO(n+1,t+1)\mathrm{O}(n+1,t+1)\,
Narain groupO(n,n)O(n,n)
Poincaré groupISO(n,1)ISO(n,1)Poincaré spin groupISO^(n,1)\widehat {ISO}(n,1)\,\,
super Poincaré groupsISO(n,1)sISO(n,1)\,\,\,\,
superconformal group
geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d


The anti de Sitter Lie algebra is discussed for instance in

The representation theory and its Inönü-Wigner contraction to that of the Poincaré group is discussed in

  • Jouko Mickelsson, J. Niederle, Contractions of Representations of de Sitter Groups, Comm. Math. Phys. Volume 27, Number 3 (1972), 167-180. (Euclid)

  • Mauricio Ayala, Richard Haase, Group contractions and its consequences upon representations of different spatial symmetry groups (arXiv:hep-th/0206037)

  • Francisco J. Herranz, Mariano Santander, section 4 of (Anti)de Sitter/Poincare symmetries and representations from Poincare/Galilei through a classical deformation approach, J.Phys.A41:015204,2008 (arXiv:math-ph/0612059)

Last revised on July 10, 2018 at 09:24:54. See the history of this page for a list of all contributions to it.