nLab de Sitter group



Riemannian geometry

Group Theory



The de Sitter group is the isometry group of de Sitter spacetime. In spacetime dimension D=d+1D = d+1 its connected component is isomorphic to the orthogonal group SO(D,1)SO(D,1) in Lorentzian signature in one dimension higher.

This is analgous to the Poincaré group, which instead 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 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 de Sitter Lie algebra is discussed for instance in

On the representation theory of the de Sitter group and its Inönü-Wigner contraction to that of the Poincaré group:

  • 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)

  • Zimo Sun, A note on the representations of SO(1,d+1)SO(1,d+1) (arXiv:2111.04591)

  • Vladimir Schaub, A Walk Through Spin(1,d+1)Spin(1,d+1) [arXiv:2405.01659]

Last revised on May 6, 2024 at 08:39:41. See the history of this page for a list of all contributions to it.