Somehow along this other “X” I guess you’ve still got to break the tangent space of SO(4,1) into some particular copy of so(3,1) and a complement, so that the Cartan connection consists of an SO(3,1) connection plus a coframe field. What’s the “symmetry breaking” field in this case?

Anyway, to really understand this, what I would still like is a geometric description of how, after choosing a point to serve as the identity, any point in de Sitter space gives me an isometry of de Sitter space, and a geometric understanding of why these particular isometries form a subgroup of all of the isometries. I suspect that a really good geometric description of this should not involve talking about the Lie algebras.

But anyway, if you are writing stuff up about this de Sitter subgroup, and if you don’t already have such a geometric description, maybe we can figure it out after I read what you’ve got.

http://theoreticalatlas.wordpress.com/2012/10/08/observer-space-cartan-gr/

If I were trying to guess how this “X” works, I might start by breaking the Lie algebra into representations of SO(3,1):

and then exponentiating in the directions. This will locally give a 4d submanifold of SO(4,1) that looks like de Sitter space, but it sure won’t give me a subgroup, since isn’t a subalgebra. And also, it’ll only work locally. So, I guess you must mean something different.

Ah, nice paper. Hmm, I’ll have to write up this de Sitter subgroup stuff.

(By the way, sorry for dropping this thread before — just busy with other things about Cartan geometry and I didn’t have much to contribute immediately in the direction you were going.)

I really like this idea of “observer space”, this is close to what I have in mind for years, and I was also charmed by the way Cartan geometry explains MM theory, the synergy of the two is an exciting area!