But if Q^3 is a scalar q, you have QRQQ = w RQQQ = w q R QRQQ = ww QQQR = w^2 q R so w must equal w^2 -> w can only be 0,1 I was looking at the same thing you described while trying to get an extension of geometric algebra (the even grades of 3-d geometric algebra are the quaternions) where instead of the product rule e_i^2 = (e_i . e_i) + (e_i /\ e_i) = 1+0 = 1 we have e_i^3 = 1 but I spotted the problem above and couldn't think of a solution for it (the commutative case being boring...) It reminded me of anyons, which only work in 2-D, so I think it might be possible to get an arbitrary phase on the exchange operator if you restrict yourself to powers of e_1 and e_2. Rich wrote:

I've been wondering about doing a cubic analog of the quaternions.

The idea is to have two generators Q and R with Q = cbrt(q) and R = cbrt(r), and the non-commutative multiplication rule R*Q = w Q*R (where w = cbrt(1) = (-1 + i sqrt3)/2 = e^(2 pi i/3)). -- Mike Stay staym@clear.net.nz http://www.xaim.com/staym