Subject: categories: Geometry of 2x2 real matrices

From: Vaughan Pratt <[email-masked]>
Date: Fri, 16 Apr 2004 18:40:02 -0700

I'm writing about applications to commonplace physics phenomena of "the other
4D Clifford algebra", namely R(2), the 2x2 real matrices.  (The quaternions
H of course being the prototype to which this is "the other".  The only
other Clifford algebras this size or smaller are smaller, namely R, C,
and the hyperbolic plane, call it J since we've already deducted H on line 3.)

(If anyone actually knows of such applications I'm all ears.)

Since a good many categorists are (were?) primarily motivated by geometry,
and particularly algebraic topology, I'm hoping this is a good place to
ask the following.

The geometry of the quaternions, certainly for its vector part (zero real
part), seems clear enough, namely that of ordinary Euclidean 3-space.
If x,y are unit vectors with 1,x,y,xy linearly independent then x,y,xy is a
right-handed coordinate system.  (But is there a coherent notion of handedness
for a triple like i,1,j or j,k,1 and if so what's it good for?  Should 3D
subspaces that include the real axis be considered simply connected with a
flat Euclidean metric like the physical 3D vector space spanned by i,j,k?
Does 360 degree rotation work the same there as in the vector subspace?)

Whereas the geometry of the quaternions feels like 3-space plus a perceptual
axis (or something like that), the geometry of R(2) feels more like the
four walls of a room.  The hands of clocks on the north and south walls
rotate through circular angles while those on the east and west walls
rotate through hyperbolic angles.  Clocks not parallel to some wall seem
to rotate through both kinds of angles simultaneously, at relative rates
that depend on the angle to the walls.  (This is the general idea but when
I try to flesh out the details too intuitively I find myself injecting too
much Euclidean geometry at first.)

In the Erlangen spirit I'm trying to come to grips with the geometrical
invariants for R(2).  The metrical invariants are more or less clear enough.
(That said, computing the geodesics seems painful---is there a simple
distillation that isn't equivalent to being dragged through the calculus of
variations?  An elementary guide to the Lie algebra for R(2) would be great.
I actually need to calculate those circular-to-hyperbolic angle ratios as
smoothly varying quantities along geodesics so any software that does this
could be very insightful.)

Where I'm running into serious trouble is with the topological invariants.
For example what's supposed to happen near the light cone?

For quaternions the only choices for the light cone (depending on what
you're using quaternions for) are x^2 = y^2+v^2+w^2 and x^2+y^2+v^2+w^2=0.
The former is a real cone, the vector part (x=0) of which is just the origin,
while the latter (the roots of the Euclidean squared norm for the quaternions)
is the origin regardless of the real part.

R(2) moves y^2 (y being a measure of asymmetry, 0 for symmetric matrices) over
to the other side, making the choices x^2+y^2=v^2+w^2 and x^2+y^2+v^2+w^2=0.
While the second hasn't changed, the first becomes y^2=v^2+w^2 when x=0.
Now what?  For example what are the pros and cons of compactly topologizing
the unit sphere y^2+v^2+w^2 <= 1 (not <) when you want to treat the cone
y^2=v^2+w^2 as unreachable from points on either side of it within that
sphere other than through the origin?  What is the physical significance
if any of a path through the origin?

One can always improvise answers but this is a time-consuming and unreliable
process that is greatly inferior to working out of a well-debugged cookbook.

An account of the topology of R(2) addressed to us Bears of Little Brain
would be very helpful.  Everything I've found to date about getting from
quaternions to other Clifford algebras goes straight to arbitrary Clifford
algebras over arbitrary fields, which is about as useful as telling kids
that ice cream comes from dairy farms -- academically interesting but
operationally useless when you just want one ice cream.

Vaughan Pratt