## The quaternions $ℍ$

The quaternions is the $ℝ$-algebra

 $ℍ=ℝ\text{-span}\left\{1,i,j,k\right\}=\left\{{x}_{0}+{x}_{1}i+{x}_{2}j+{x}_{3}k\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{x}_{0},{x}_{1},{x}_{2},{x}_{3}\in ℝ\right\}$
with product determined by
 ${i}^{2}={j}^{2}={k}^{2}=-1,\phantom{\rule{2em}{0ex}}ij=-ji=k,\phantom{\rule{2em}{0ex}}jk=-kj=i,\phantom{\rule{2em}{0ex}}ki=-ik=j.$
The topology on $ℍ$ is given by
 $ℍ\simeq {ℝ}^{4},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}ℂ=\left\{{x}_{0}+{x}_{1}i\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{x}_{0},{x}_{1}\in ℝ\right\}$
is an $ℝ$-subalgebra of $ℍ$. Since $ij\ne ji$, $ℍ$ is not a $ℂ$-algebra.

Writing $ℍ$ as space-time,

 $ℍ=ℝ×{ℝ}^{3}=\left\{t+v\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}t\in ℝ,v\in {ℝ}^{3}\right\},$
the product in $ℍ$ is given by
 $\left({t}_{1}+{v}_{1}\right)\left({t}_{2}+{v}_{2}\right)={t}_{1}{t}_{2}-{v}_{1}\cdot {v}_{2}+\left({t}_{1}{v}_{2}+{t}_{2}{v}_{1}+{v}_{1}×{v}_{2}\right).$

The norm $N:ℍ\to {ℝ}_{\ge 0}$ is given by

 $N\left({x}_{0}+{x}_{1}i+{x}_{2}j+{x}_{3}k\right)={x}_{0}^{2}+{x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}={‖x‖}^{2}$,
where $‖\phantom{x}‖:{ℝ}^{4}\to {ℝ}_{\ge 0}$ is the usual Euclidean norm on ${ℝ}^{4}$. Then
 $N\left(xy\right)=N\left(x\right)N\left(y\right)$.
The conjugate $\stackrel{‾}{\phantom{P}}:ℍ\to ℍ$ is given by
 $\stackrel{‾}{{x}_{0}+{x}_{1}i+{x}_{2}j+{x}_{3}k}={x}_{0}-{x}_{1}i-{x}_{2}j-{x}_{3}k\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\stackrel{‾}{xy}=\stackrel{‾}{y}\phantom{\rule{.1em}{0ex}}\stackrel{‾}{x}$
so that $\stackrel{‾}{\phantom{P}}:ℍ\to ℍ$ is an antiautomorphism. Then
 $\begin{array}{cl}{\mathrm{GL}}_{1}\left(ℍ\right)& ={ℍ}^{×}=\left\{x\in ℍ\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}x\ne 0\right\}\\ \cup |& \\ {U}_{1}\left(ℍ\right)& =\left\{x\in ℍ\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}x{\stackrel{‾}{x}}^{t}=1\right\}=\left\{{x}_{0}+{x}_{1}i+{x}_{2}j+{x}_{3}k\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{x}_{0}^{2}+{x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}=1\right\}\end{array}$
and
 $\begin{array}{ccc}{ℍ}^{×}& \stackrel{\sim }{⟶}& {\left({ℝ}^{×}\right)}^{\circ }×{U}_{1}\left(ℍ\right)\\ x& ⟼& ‖x‖\cdot \frac{x}{‖x‖}\end{array}$
Here ${\left({ℝ}^{×}\right)}^{\circ }={ℝ}_{>0}$ is the connected component of the identity in the Lie group ${\mathrm{GL}}_{1}\left(ℝ\right)={ℝ}^{×}$. This polar decomposition is an example of the Cartan decomposition $G=K\cdot \left(\mathrm{exp}𝔭\right)$ (see Segal Theorem 4.1 and/or Knapp Prop. 1.2), where $K$ is a maximal compact subgroup of $G$, and $𝔤=𝔨\oplus 𝔭$ with $𝔤=\mathrm{Lie}\left(G\right)$ and $𝔭$ orthogonal to $𝔨=\mathrm{Lie}\left(K\right)$ with respect to the Killing form.

## Notes and References

The reference [Ch. VIII § 1.4, BouTop] provides a brief, but thorough, introduction to the quaternions.

## References

[BouTop] N. Bourbaki, General Topology, Chapter VI, Springer-Verlag, Berlin 1989. MR?????