## homogeneous spaces of semidirect product groups

Hello!

I could need some help on the following question:

What is the structure of the homogeneous space SL(2,C)(x)R^4/SU(2)xR^4
? (x) is a semidirect product (Poincare-Group), but x is direct. Is the
h.s. isomorphic to the hyperboloid in three dimensions, as it would be
in the case SL(2,C)/SU(2)? Are there any theorems on the question of
the cosets of affine groups with respect to direct product groups
containing the same inhomogenity? Thanks for your help.

P.S.: It might help to say that this problem arrises in induced reps of
the Poincare group.

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 On Thu, 23 Jun 2005 florian.beissner@web.de asked: > What is the structure of the homogeneous space SL(2,C)(x)R^4/SU(2)xR^4 where > (x) is a semidirect product (Poincare-Group), but x is direct. Did you inadvertently write (x) when you meant x and vice versa, or am I confused? I.e. one connected component of the Poincare group should be isomorphic to the semidirect product PSL(2,C) (x) R^4 Lorentz tnfms translations right? If so, presumably you have a left action by some Lie group G, with a closed Lie subgroup H, the isotropy subgroup of the action, and you form the homogeneous space G/H from the left cosets. Can you specify the left action you have in mind? Are you working over R or over C? A nice paper on the Lorentz group which should be helpful re questions of homogeneous spaces associated with the various Lie subalgebras of sl(2,C) is http://www.arxiv.org/abs/math-ph/0301014 BTW, someone should write an expository paper explaining the lovely connections between point symmetries of PDEs and ODEs and popular Kleinian geometries (homogeneous spaces). This involves learning a fairly large collection of not very difficult and only apparently unrelated ideas. The lattice of subalgebras of the Lorentz algebra nicely illustrates these connections. Some time ago, we started an exposition in this NG for Cartan's technique of computing crude de Rham cohomology using exterior algebra, starting from the Maurer-Cartan form, in order to topologically distinguish various Lie groups. At the very least, with a computer tool like Maple to help with multiplying out the matrix valued exterior forms, you should be able to compute the crude cohomology sequence for your group. This fails to distinguish many groups (and is unlikely to detect discrete quotients), but the folk at sci.math.research can probably help you compute more powerful sequences if you find initial results from the crude sequence encouraging. "T. Essel"