Francis

@hallasurvivor This is a nice observation. Here is how I think about Puzzle 1: The unit quaternions are isomorphic to \( \mathbb{R}^3 \), and their product is given in terms of the dot product and cross product. Writing a general quaternion as \( (a, \vec{v}) \), with \( a \in \mathbb{R}, \vec{v} \in \mathbb{R}^3 \), the product can be written as (0,𝑣⃗)(0,𝑢⃗)=(−𝑣⃗⋅𝑢⃗,𝑣⃗×𝑢⃗). So taking \( \vec{v} \in S^2 \) to be any unit imaginary quaternion, we find that its square is \( -1 \).

@hallasurvivor This is related to the fact that \( Q[x] \) does not have unique factorization. When dealing with polynomial rings over commutative fields, we can prove unique factorization as a consequence of the Euclidean algorithm. And this doesn't work for skew-fields. The key point seems to be that you need to multiply things like \( ax^m bx^n = ab x^{m+n} \). This requires you to commute the indeterminate \( x \) with \( b \). Perhaps there's something more conceptual...

@hallasurvivor @highergeometer btw I think this can also be a good idea when submitting papers. You are way more likely to get the message of your project across to a referee if you explain it to them through a talk.

@highergeometer if you gave YouTube lectures explaining your papers that would be great! I wish everyone would do that.

@highergeometer why do you need the orbit space to be compact?

@highergeometer I see. I was wondering how that would matter if you were doing a sheafy thing. I guess proper is much more important since you can apply a linearization theorem in order to assume that locally your groupoid is the action groupoid of a compact group over an open set in a representation.