The Poincare group is not a simple group. Thus, it is an attractive idea to embed the Poincare group in a simple group (e.g. the conformal group). (Simple groups are the “atoms” all Lie groups are made out of. Thus it seems plausible that the Poincare group isn't the most fundamental spacetime group.)
In addition, we already know that classical Galilei group is a good approximation of the Poincare group as long as we only consider small velocities $ v \ll c$. Or expressed differently, the Poincare group becomes the Galilei group in the limit $c\rightarrow \infty$. Thus, we can ask if there is any group that yields the Poincare group in some limit. This question has already been answered in 1967 by Monique Levy‐Nahas in this paper. The answer is that the only possibility are $SO(4,1)$ and $SU(3,2)$, the deSitter and the anti deSitter group. Both are simple groups and preserve in addition to a maximal velocity, a maximal length scale $L$. In the limit of short length scales ($L\rightarrow \infty$), the deSitter and anti deSitter group becomes the Poincare group.
“Much of modern physics including quantum field theory and the Standard Model (SM) has been formulated through the spacetime symmetries of special relativity, encoded within the Poincare algebra, together with the principle of local gauge invariance. The Poincare algebra is sensitive to small perturbations in its structure constants however and therefore not stable. […] At the same time, given the instability of the Poincare algebra, its validity as underlying the SM symmetries is questionable. We argue that a natural path toward a more widely applicable and robust theory of elementary particles and their interactions is to replace the underlying unstable algebra by a larger and stable deformed one.” https://arxiv.org/pdf/1512.04339.pdf
“The hope that quantum theories with improved ultraviolet properties can be constructed has motivated a considerable research interest in deformations of space-time symmetries.” from Deformed Poincare containing the exact Lorentz algebra by Alexandros A. Kehagias et. al.
|The deSitter Group||Anti DeSitter Group|
|The Poincare Group|
|The Galilei Group|
Euclidean geometry can be understood in the sense of Klein as the study of invariants of subgroups of the Euclidean symmetry group. Thus, if we want to study generalized geometries, we can simply allow different symmetry groups instead of the Euclidean symmetry group. This is the basic idea of Klein geometry. However, one can generalize Euclidean geometry in a different direction by allowing instead of the flat Euclidean space curved spaces, which are commonly called Riemannian manifolds. The defining feature of a manifold is that it looks in the neighborhood of each point like the usual Euclidean space. A possible generalization is then that we also allow more general tangent spaces instead of the Euclidean space. This is idea behind Cartan Geometry. An important example is that the tangent space is the curved deSitter space, which is the idea behind MacDowell-Mansouri gravity.