Reverting to the full groups generated by reflectors, and embedding both Mob(n) = PO(n, 1) and Lag(n+1) in Lie(n+1) = PO(n, 2) , via geometric algebra it turns out to be almost trivial that adjoining (the right sort of) translations to Möbius yields Laguerre in one higher dimension; conversely that Laguerre restricted to the unit sphere yields Moebius. So it does after all look as though overloaded nomenclature is the simple explanation for the temporarily alarming incompatibility earlier remarked. Current usage (pace Adam and company) seems to be favouring Lorentz == Möbius in 3-space; Poincaré == Laguerre in 4-space. It's a pity in particular that flatly contradictory Wikipedia definitions that started this hare do not carry some warning to this effect: how about ℗ standing for "physicists' nomenclature / theorem / whatever ...", along the lines of L-plates for learner drivers? https://en.wikipedia.org/wiki/Lorentz_group https://en.wikipedia.org/wiki/Spherical_wave_transformation Anyway, false alarm --- apologies again, everybody! WFL On 9/3/14, Warren D Smith <warren.wds@gmail.com> wrote:

The "Lorentz" (sub)group is normally not including translations and the "Poincare" group does include them.

In informal speech where it does not terribly matter, one often speaks of "Lorentz invariance" or some such, really meaning Poincare, or really meaning the manifold (GR) version of same, figuring the listener understands what you meant. If you really wanted to be careful & precise, you should actually define it each time you use it.

-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun