Let S be a spin representation of the Euclidean
spin group Spin(d) and let Rd
be Euclidean d-space with Spin(d) action on it
in the canonical way, via the 2:1 cover to SO(d).
(I am being careful here: A spin rep.'', not
the spin rep.'')
Is there, for all d, an onto quadratic Spin(d)-equivariant map S→Rd?
If so, is there a `universal' (d-independent) construction of this map?
MOTIVATION: For d=2,3 I know these maps.
They are famous in celestial mechanics and yield the
standard regularizations of the Kepler problem,
or, what is the same, of binary collisions in the classical N-body problem.
They turn Kepler for negative energies into a harmonic oscillator.
Case d=2. I take Spin(2) to also be S1, but wrapped `twice' around
S1=SO(2). S=C. The quadratic map is w→w2.
This is the Levi-Civita regularization.
Case d=3. This is the standard Hopf map C2→R3,
or if you prefer, from the quaternions H to R3, sending q to qkˉq.
The astronomers call this Kuustanheimo-Steifel regularization.
WHERE I'VE LOOKED SO FAR: I tried to make sense out
of Deligne's discussion on spinors in the AMS two-volume set
from some Princeton year on string theory from a decade or so ago.
I understand that over the complexes, there is either exactly one or exactly two
spin representations, depending on the parity of d.
So even there , we don't get a `universal' d-dependent map.
Over the reals things decompose in a rather complicated
dimension dependent way (mod 8 probably) and there is no clear choice.
I also looked in Reese Harvey's book which I find too baroque and
signature depend to penetrate.
Case d=4. Here I am not sure. But I know Spin(4)=SU(2)×SU(2)
which I can think of as two copies of the unit quaternions, each acting on
``its own'' H.
I guess in this case I better take S=H×H
Then I get the desired quadratic map H×H→H=R4
as (q1,q2)↦q1ˉq2.
This post imported from StackExchange MathOverflow at 2015-02-20 16:59 (UTC), posted by SE-user Richard Montgomery