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A classification (up to conjugacy) of all closed subgroups of the (identity component of the) Lorentz group $\mathrm{SO}(1,3)_0$ in terms of the subalgebras of its Lie algebra was given in

  • R. Shaw. The subgroup structure of the homogeneous Lorentz group. The Quaterly Journal of Mathematics, Oxford 21 (1970) 101-124

(see also the book of Hall: Symmetry and Curvature Structure in General Relativity).

My question is whether there exists a similar classification for $\mathrm{SO}(1,n)_0$, $n\ge 4$.

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up vote 3 down vote accepted

I would expect this to become impractical (although in practice there is an algorithm) as $n$ increases. There are results for $n=4$ in this paper:

Quantum numbers for particles in de Sitter space by J Patera, P Winternitz, H Zassenhaus. Published in the Journal of Mathematical Physics (1976) vol. 17 (5) pp. 717-728. (MathSciNet)

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Great. I'm mostly interested in the de Sitter case n=4 and I guess this paper will do the job. Thanks! – Tampopo Mar 15 '11 at 13:15
I'm glad this was useful. Can I ask why it is that you are interested in this? My interest in this lies in the classification of lorentzian homogeneous spaces with prescribed isometry, which is part of a long-term project on homogeneous supergravity backgrounds. – José Figueroa-O'Farrill Mar 15 '11 at 13:32
My interest in this comes from a novel constructive approach to quantum field theory on curved spacetimes. In the authors consider QFTs on Minkowski spacetime and use the action of the translation subgroup of the Poincare group to "deform" the theory. The whole procedure is similar to Rieffel deformations of $C^*$-algebras. So you can start with a free theory and end up with something interacting (but with a weakened form of locality). A natural question is whether other Abelian subgroups of the Lorentz group can be used to introduce interaction. – Tampopo Mar 15 '11 at 14:59

I'm not certain of the complete answer, but I think there's a rough classification as follows. If $H < SO_0(n,1)$, then either

  • $H$ is compact and is conjugate into $O(n)$
  • $H$ is solvable and preserves a 1-dimensional subspace on the light cone (i.e. an isotropic subspace). In this case, $H$ acts by conformal affine transformations of $\mathbb{R}^n$.
  • $H$ preserves a 2-dimensional subspace of signature $(1,1)$ (this could actually be subsumed in the previous case). Here, the action restricted to this subspace is $SO_0(1,1)\cong \mathbb{R}$. Then $H$ acts on the orthogonal space by a subgroup of $O(n-1)$. The action is a direct product of a closed subgroup of $O(n-1)$ and $\mathbb{R}$. However, this product is not canonical, since any generator of the $\mathbb{R}$ factor may be modified by a 1-parameter subgroup of the $O(n-1)$ factor.
  • $H$ fixes a higher dimensional subspace of signature $(k,1)$, $k>1$. Then $H$ splits as a product of $SO_0(k,1)$ and a compact subgroup of $O(n-k)$.

Classifying the compact subgroups of $O(n)$ is complicated, so I think this might the best one can say in general. I don't know a reference, and I hope I haven't overlooked any possibilities. The way I think about this is the action on hyperbolic $n$-space and its compactification by the sphere at infinity. Either the action fixes a point in $\mathbb{H}^n$, or it fixes a point at infinity, or it preserves a totally geodesic subspace of some dimension.

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