- #1

mnb96

- 715

- 5

it is known that

*pure-quaternions*(scalar part equal to zero) identify the [itex]\mathcal{S}^2[/itex] sphere. Similarly

*unit-quaternions*identify points on the [itex]\mathcal{S}^3[/itex] sphere.

Now let's consider quaternions as elements of the Clifford algebra [itex]\mathcal{C}\ell_{0,2}[/itex] and let's consider a quaternion [itex]\mathit{q} = a+b\mathbf{e}_1+c\mathbf{e}_2+d\mathbf{e}_{12}[/itex].

We now re-write

*q*in the following form:

[tex]\mathit{q} = (a+d\mathbf{e}_{12}) + \mathbf{e}_1(b - c\mathbf{e}_{12}) = \mathit{z_1} + \mathbf{e}_1 \mathit{z_2}[/tex]

We have esentially expressed a quaternion as an element of [itex]\mathbb{C}^2[/itex].

*** My question is:

if we

*assume*that [itex]z_1[/itex] and [itex]z_2[/itex] are

*unit complex-numbers*of the form [tex]e^{\mathbf{I} \theta}[/tex], can we find a manifold associated with this subset of quaternions?