Re: conjugacy classes in Lorentz/deSitter groups


John Baez wrote:
> Does someone know the conjugacy classes in the groups
> SO(4,1) and SO(3,2)? I might be even happier if someone
> had worked out a classification of conjugacy classes for
> every real simple Lie group, but I need to know these cases
> for work on quantum gravity.

re SO(3,2),

there are 4 more parameters in the pseudorotation group so the task is
to understand those. Obviously one is the phase in the 4,5 plane. The
other three make up a unit four-vector or null four-vector, the former
being either spacelike or timelike. In all three cases the
infinitesimal transformation looks like a spacetime translation in the
direction of this unit-or-null four vector. If you contract out 5 with
a dimensional parameter, then this goes over directly into the Poincare
group.

For SO(4,1) I imagine the phase must go over to a scaling, but the
other results will likely be the same.

The Clifford algebra is easy,

[0, iy_m, -iy_m, 0], [0, iy_5, -iy_5, 0] m=1-4

>>From this you can actually work out the details of the CCs. In addition
to the 4 for the Lorentz group (boosts, rotations, screws, and null
rotations) you'll have the three types of pseudotranslation (spacelike,
timelike, lightlike) plus the phase.

-drl

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