This question is related to my previous post:
Is this generalization of the Hopf map for quadratic field extensions surjective?
I still would like to know more and, while that post got several votes, no one other than me wrote anything unfortunately. Here I will ask related but different questions.
The "usual" Hopf map can be defined as:
$h: SU(2) \to S^2$
mapping $g \in SU(2)$ to $g \sigma_3 g^{-1}$, where
$\sigma_3 = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)$.
Note that the target space $S^2$ is a subset of the $3$-dimensional space of hermitian $2 \times 2$ matrices. One could have defined the Hopf map so that the target space $S^2$ is a subset of the Lie algebra $\mathfrak{su}(2)$ of $SU(2)$.
It makes sense to consider an algebraic generalization of the Hopf map over other fields than $\mathbb{R}$ and its quadratic extension $\mathbb{C}$. Does anyone know of a reference (or references) where such maps are considered, defined carefully, and where some of their most important properties are stated and proved? Most of my life, I have focused on $\mathbb{R}$ and $\mathbb{C}$, but there are many fields out there: finite fields, $\mathbb{Q}_p$, number fields, function fields and so on.