How Do Quaternion and Spinor Structures Impact the Geometry of Vector Spaces?

[GcSanchez05]

Given a real vector space, I understand the significance of defining a complex structure. Now, if J is a complex structure on a real vector space, and we find an anti-commuting complex structure I, so that we have I, J, and K=IJ, what are some interesting properties that we have now on our vector space? Geometric properties?

 

[chiro]

Hey GcSanchez05.

Have you look at the extensions of the complex like the Quaternions and the Octonions? Quaternions can be thought in terms of rotations in 3-space and rotations are basically non-commutative since doing a rotation around Z then X is not the same in general as doing X then Z (as well as for other rotations).

You also have an object that's called a spinor (pronounced "spinnor") that acts very similar to a mobius loop: basically it takes 4pi to get back to where you started and after 2pi it goes "somewhere else" so to speak.

The big thing in all of this is to note where the periodicity is and also how many layers of periodicity exist and how they relate to each other. In the complex variables you have one main point which is the 2pi component of the argument: in a spinnor you have two levels of periodicity like you do when you run across the surface of a mobius loop.

Rotations in 3 dimensions are even crazier since you can have periodicity with respect infinitely many unit axis and because of the non-commutativity aspect, you can get all kinds of complex relationships when you consider applying multiple rotations one after the other (i.e multiplying quaternions and then considering how the string of multiplication screws up or adds more periodicity).

That's really the big thing about the complex numbers: complex numbers pretty much give periodicity and this is related to the angle measure in geometry which is really what all geometry is about (other than length).