# nLab de Sitter group

Contents

### Context

#### Riemannian geometry

Riemannian geometry

group theory

# Contents

## Idea

The de Sitter group is the isometry group of de Sitter spacetime. In spacetime dimension $d$ its connected component is isomorphic to $SO(d,1)$.

This is analgous to the Poincare group, which is the isometry group of Minkowski spacetime.

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

The 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)

• Thomas Basile, Xavier Bekaert, Nicolas Boulanger, Mixed-symmetry fields in de Sitter space: a group theoretical glance (arXiv:1612.08166)