Multiplication in projective space

Let #F# be a field and consider the projective space of dimension #n# over it with added the point #0#. It seems to me that there is a valid definition of multiplication by just entrywise multiplicating the elements. Of course both can be multiplied by #x \in F# but that goes for the product as well!

My question is whether the multiplication is well defined, and whether it is usual to consider this space a 'group' under that multiplication.

What for? You can also multiplicate vectors elementwise, but nobody does it because it is useless.

What you can do in higher dimensions is use the isomorphisms listed here:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/#2-Spheres

It's well defined, but this does not define a group. Any element with an entry of 0 is not invertible.

fresh_42 said:

What for?

Just a pet project. I'm interested in the coordinatewise multiplication simply because it's the easiest one to analyze. It's nice to get this information, because as you say it's regarded as a useless operation (and not a group operation) which is why I couldn't find my answer with google.

Structure seeker said:

Just a pet project. I'm interested in the coordinatewise multiplication simply because it's the easiest one to analyze. It's nice to get this information, because as you say it's regarded as a useless operation (and not a group operation) which is why I couldn't find my answer with google.

Maybe you can find a geometric meaning. Geometry is where projective originally came from. My first thought was the formula ##e^{i \varphi }\cdot e^{i \psi }=e^{i(\varphi +\psi)}## in one dimension, aka the projective plane. But that led to the isomorphisms I linked to: multiplication in orthogonal groups.

Why projective space?

Any projective space with an amount of entries divisible by 3 is OK, but indeed projective planes are of first interest. As to the geometric meaning, it kinda looks like it adds the two tangents of twice the angle with identity (the all one element) and applying the inverse tangent of the result to get half the angle of the newfound product.

Happy to say my pet project succeeded! In the end I used the nonzero field elements and an added zero point in the projective plane. I also define a strange 'addition' over which this multiplication is distributive. It's been fun

I'm not gonna give the results here. The 'addition' is noncommutative and nonassociative, I'll need to see first what the implications are. Just wanted to tell it succeeded.