Subject: categories: Re: Modeling infinitesimals with 2x2 matrices

From: Vaughan Pratt <[email-masked]>
Date: Sat, 24 Apr 2004 15:46:40 -0700

Correction to my suggestion id+di = 0.  Don't do it.  id = di is fine
as it stands for refined complex numbers, which should be represented in
C(2) = 2x2 complex matrices (embeddable in R(4) - 4x4 real matrices)
as the obvious extension of the refined reals x+yd.

I shouldn't have been so smug about 4D Clifford algebras, this algebra of
refined complex numbers doesn't satisfy d^4 = 1, needed if d is to be a
Clifford generator.

And in fact although di =
1 0
0 0
we have id =
0 0
0 1
(I should have checked that more carefully.)

I thought about trying to make the infinitesimals points on the "light cone"
of R(2) (the singular matrices) but couldn't get that to work.  So 2x2
complex matrices with id = di is the best I could think of.  This works
for modeling the refined complex numbers (barring any other errors), but
with nothing left to motivate  id+di = 0.

The representation x+iy+dv+idw is fine, with idw = diw = wid etc., all is
commutative.  (I was hoping too hard for the excitement of noncommutativity,
this is boringly noninteractive as it stands.)

Vaughan