# nLab anti de Sitter group

Contents

### Context

#### Riemannian geometry

Riemannian geometry

group theory

# Contents

## Idea

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

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

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

## Properties

### Exceptional isomorphisms

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

## References

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 05:24:54. See the history of this page for a list of all contributions to it.