The quaternions $ℍ$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 18 May 2011

The quaternions $ℍ$

The quaternions is the $ℝ$-algebra

$ℍ=ℝ\text{-span}\{1,i,j,k\}=\{x_0+x_{1}i+x_{2}j+x_{3}k|x_0,x_1,x_2,x_3\in ℝ\}$

with product determined by

$i^2=j^2=k^2=-1,ij=-ji=k,jk=-kj=i,ki=-ik=j.$

The topology on $ℍ$ is given by

$ℍ\simeq {ℝ}^4,\text{and}ℂ=\{x_0+x_{1}i|x_0,x_1\in ℝ\}$

is an $ℝ$-subalgebra of $ℍ$. Since $ij\ne ji$, $ℍ$ is not a $ℂ$-algebra.

Writing $ℍ$ as space-time,

$ℍ=ℝ\times {ℝ}^3=\{t+v|t\in ℝ,v\in {ℝ}^3\},$

the product in $ℍ$ is given by

$(t_1+v_1)(t_2+v_2)=t_1{t}_2-v_1\cdot v_2+(t_1{v}_2+t_2{v}_1+v_1\times v_2).$

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

$N(x_0+x_{1}i+x_{2}j+x_{3}k)=x_0^2+x_1^2+x_2^2+x_3^2={\Vert x\Vert }^2$,

where $\Vert \phantom{x}\Vert :{ℝ}^4\to {ℝ}_{\ge 0}$ is the usual Euclidean norm on ${ℝ}^4$. Then

$N\left(xy\right)=N\left(x\right)N\left(y\right)$.

The conjugate $\bar{\phantom{P}}:ℍ\to ℍ$ is given by

$\overline{x_0+x_{1}i+x_{2}j+x_{3}k}=x_0-x_{1}i-x_{2}j-x_{3}k\text{and}\overline{xy}=\bar{y}\bar{x}$

so that $\bar{\phantom{P}}:ℍ\to ℍ$ is an antiautomorphism. Then

$\begin{array}{cl}{\mathrm{GL}}_1\left(ℍ\right)& ={ℍ}^{\times }=\{x\in ℍ|x\ne 0\}\\ \cup |& \\ U_1\left(ℍ\right)& =\{x\in ℍ|x{\bar{x}}^t=1\}=\{x_0+x_{1}i+x_{2}j+x_{3}k|x_0^2+x_1^2+x_2^2+x_3^2=1\}\end{array}$

and

$\begin{array}{ccc}{ℍ}^{\times }& \stackrel{\sim }{⟶}& {\left({ℝ}^{\times }\right)}^{\circ }\times U_1\left(ℍ\right)\\ x& ⟼& \Vert x\Vert \cdot \frac{x}{\Vert x\Vert }\end{array}$

Here ${\left({ℝ}^{\times }\right)}^{\circ }={ℝ}_{>0}$ is the connected component of the identity in the Lie group ${\mathrm{GL}}_1\left(ℝ\right)={ℝ}^{\times }$. This polar decomposition is an example of the Cartan decomposition $G=K\cdot \left(\exp 𝔭\right)$ (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?????

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