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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.

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  • $\begingroup$ I think that there is a "motivic Hopf map", coming from the map of algebraic varieties $\mathbf{A}^2-\{0\} \to \mathbf{P}^1$ (which is a map $S^{3,2} \to S^{2,1}$ in the motivic category) that exists in any base scheme. Over $\mathbf{C}$, this realizes to the usual Hopf fibration. $\endgroup$
    – skd
    Dec 2, 2018 at 0:39
  • $\begingroup$ @skd, thank you. I will look up what the notation means and get back to you. This is an opportunity for me to learn about the motivic category. $\endgroup$
    – Malkoun
    Dec 2, 2018 at 6:04
  • $\begingroup$ @skd, what is the notation $S^{k,m}$ please? Do you have any reference I can read to understand your statement? $\endgroup$
    – Malkoun
    Dec 2, 2018 at 6:24
  • $\begingroup$ Googling "motivic Hopf map" should give a few resources. $\endgroup$
    – skd
    Dec 2, 2018 at 15:17
  • $\begingroup$ yes, thank you. I am looking at some of the references now. Thank you. $\endgroup$
    – Malkoun
    Dec 2, 2018 at 15:24

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