nLab anti de Sitter group

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)

Related concepts

• anti de Sitter spacetime

• singleton representation

group	symbol	universal cover	symbol	higher cover	symbol
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 context	gauge group	stabilizer subgroup	local model space	local geometry	global geometry	differential cohomology	first order formulation of gravity
differential geometry	Lie group/algebraic group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group $Iso(d)$	rotation group $O(d)$	Cartesian space $\mathbb{R}^d$	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group $Iso(d-1,1)$	Lorentz group $O(d-1,1)$	Minkowski spacetime $\mathbb{R}^{d-1,1}$	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein 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 group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group $O(d,t+1)$	conformal parabolic subgroup	Möbius space $S^{d,t}$		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$	Lorentzian supergeometry	supergeometry	superconnection	supergravity
	super anti de Sitter group		super anti de Sitter spacetime
higher differential geometry	smooth 2-group $G$	2-monomorphism $H \to G$	homotopy quotient $G//H$	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) $H \to G$	homotopy quotient $G//H$ of ∞-action	higher Klein geometry	higher Cartan geometry	higher Cartan connection
examples			extended super Minkowski spacetime		extended supergeometry		higher supergravity: type II, heterotic, 11d

References

The anti de Sitter Lie algebra is discussed for instance in

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, volume 1, chapter I.3.8 of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

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)