homogeneous spaces of semidirect product groups

florian.beissner@web.de

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.

tessel@tum.bot

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"