Is there a definition of $\log(x)$ for quaternion/octonion $x$?

Question

I'm trying to implement $\log({\bf q})$ in python, where ${\bf q} = (q_0,\ldots,q_7) \in \mathbb{O}$ is an octonion. There is a well known definition of $\log({\bf q})$ for quaternions ${\bf q} = (s,v) \in \mathbb{H}$, which essentially boils down to $$\log({\bf q}) = \left( \log(\|{\bf q}\|),\left( \frac{1}{\|v\|} \arccos \frac{s}{\|{\bf q}\|}\right)v \right) \in \mathbb{H}$$ for ${\bf q} = (s,v)$ with $v \not= 0$ (see f.i. https://math.stackexchange.com/questions/2552/the-logarithm-of-quaternion). This definition can be extended to octonions ${\bf q}$ immediately with $v = (q_1,\ldots,q_7)$. With this definition we get $\exp(\log({\bf q})) = {\bf q}$ for a wide range of ${\bf q} \in \mathbb{O}$, but unfortunately $\log(\exp({\bf q})) \not= {\bf q}$ even for many ${\bf q} \in \mathbb{H}$. Is there /can there be any definition of $\log({\bf q})$ for ${\bf q} \in \mathbb{H}$ or ${\bf q} \in \mathbb{O}$ such that $\exp(\log({\bf q})) = {\bf q}$ and $\log(\exp({\bf q})) = {\bf q}$ for a wide range of ${\bf q}$?

Edit: If $|s| > \|v\|$ for ${\bf q} = (s,v)$ (the power series can be used in principle) we get $\log(\exp({\bf q})) = {\bf q}$. But what if $|s| < \|v\|$?

Answer 1

In characteristic zero, you can always just use the power series to define $\log(1+x)$ with a radius of convergence of $1$, since your algebras are at least power associative. It will satisfy $\exp(\log(1+x))=1+x$. But it won't have many other good properties such as taking products to sums, because of lack of commutativity. Two generator subalgebras of $\mathbb{O}$ are at least associative, but that isn't good enough.

Incidentally, it's better to write $\mathbb{H}$ instead of $\mathbb{Q}$ for Hamilton's quaternions, for reasons that I hope are obvious.

Edit: For your edited question, maybe what you're looking for is the paper of Gentili, Preselj and Vlacci, "Slice conformality and Riemann manifolds on quaternions and octonions." This paper constructs the analogue of the Riemann surface associated to $\mathbb{C}$.

Answer 2

The group $U_q$ of unit quaternions is a Lie group. As a manifold it is diffeomorhic to the unit sphere in $\newcommand{\bR}{\mathbb{R}}$ $\bR^4$. (This group is infact isomorphic to the matrix Lie group $SU(2)$=unitary matrices with determinant $1$.)

There is an exponential map, defined by the usual power series, from the vector space $V$ of purely imaginary quaternions to $U_q$. This map is onto.

Note that any $v\in V$ satisfies $v^2=-1$ so

$$\exp(tv)=\cos t +(\sin t)v. $$

This exponential defines a diffeomorphism

$$ \big\{\; v\in V;\;\;\Vert v\Vert <\pi\;\big\}\ni v\to \exp(v) \in U_q\setminus\{-1\}.$$

Its inverse can be described explicitly in terms of the spherical coordinates on the unit sphere.

Note that if $q$ is a unit quaternion, and $q=\cos t+(\sin t) v$, $t\in [0,\pi)$,$\DeclareMathOperator{\re}{\mathbf{Re}}$ then $t=\arccos\big(\re (q)\big)$. Then

$$\tag{1} \log (q)=tv=\frac{\arccos(\re q)}{\sin(\arccos(\re q))}\big(q-\re q\big). $$

For a general quaternion $q=s+tv$, $s\in\bR$, $v\in V$, we have

$$\exp(s+tv)= e^s\big(\; \cos t +(\sin t)v\;\big). $$

If $q\neq -\Vert q\Vert$ one can then define a logarithm of $q$ in an obvious way

$$\log q=\log \Vert q\Vert+\log\big(\Vert q\Vert^{-1} q\big). $$

The paths $t\mapsto \exp(tv)$ are the geodesic in the unit spfere that start at the Norh pole. Use this analogy for octonions.