Hamiltonian quaternions

Definition of $\mathbb{H}$

We define a unital associative algebra $\mathbb{H}$ over $\mathbb{R}$ , of dimension 4, by the basis $\{\mathbf{1},\mathbf{i},\mathbf{j},\mathbf{k}\}$ and the multiplication table

$1$	$i$	$j$	$k$
$i$	$-1$	$k$	$-j$
$j$	$-k$	$-1$	$i$
$k$	$j$	$-i$	$-1$

(where the element in row $x$ and column $y$ is $x y$ , not $y x$ ). Thus an arbitrary element of $\mathbb{H}$ is of the form

a\mathbf{1}+b\mathbf{i}+c\mathbf{j}+d\mathbf{k},\qquad a,b,c,d\in\mathbb{R}

(sometimes denoted by $\left<a,b,c,d\right>$ or by $a+\left<b,c,d\right>$ ) and the product of two elements $\left<a,b,c,d\right>$ and $\left<\alpha,\beta,\gamma,\delta\right>$ (order matters) is $\left<w,x,y,z\right>$ where

$\displaystyle w$	$\displaystyle=$	$\displaystyle a\alpha-b\beta-c\gamma-d\delta$
$\displaystyle x$	$\displaystyle=$	$\displaystyle a\beta+b\alpha+c\delta-d\gamma$
$\displaystyle y$	$\displaystyle=$	$\displaystyle a\gamma-b\delta+c\alpha+d\beta$
$\displaystyle z$	$\displaystyle=$	$\displaystyle a\delta+b\gamma-c\beta+d\alpha$

The elements of $\mathbb{H}$ are known as Hamiltonian quaternions.

Clearly the subspaces of $\mathbb{H}$ generated by $\{\mathbf{1}\}$ and by $\{\mathbf{1},\mathbf{i}\}$ are subalgebras isomorphic to $\mathbb{R}$ and $\mathbb{C}$ respectively. $\mathbb{R}$ is customarily identified with the corresponding subalgebra of $\mathbb{H}$ . (We shall see in a moment that there are other and less obvious embeddings of $\mathbb{C}$ in $\mathbb{H}$ .) The real numbers commute with all the elements of $\mathbb{H}$ , and we have

\lambda\cdot\left<a,b,c,d\right>=\left<\lambda a,\lambda b,\lambda c,\lambda d\right>

for $\lambda\in\mathbb{R}$ and $\left<a,b,c,d\right>\in\mathbb{H}$ .

Norm, conjugate, and inverse of a quaternion

Like the complex numbers ( $\mathbb{C}$ ), the quaternions have a natural involution called the quaternion conjugate. If $q=a\mathbf{1}+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$ , then the quaternion conjugate of $q$ , denoted $\overline{q}$ , is simply $\overline{q}=a\mathbf{1}-b\mathbf{i}-c\mathbf{j}-d\mathbf{k}$ .

One can readily verify that if $q=a\mathbf{1}+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$ , then $q\overline{q}=(a^{2}+b^{2}+c^{2}+d^{2})\mathbf{1}$ . (See Euler four-square identity.) This product is used to form a norm $\|\cdot\|$ on the algebra (or the ring) $\mathbb{H}$ : We define $\|q\|=\sqrt{s}$ where $q\overline{q}=s\mathbf{1}$ .

If $v,w\in\mathbb{H}$ and $\lambda\in\mathbb{R}$ , then

1.

$\|v\|\geq 0$ with equality only if $v=\left<0,0,0,0\right>=0$
2.

$\|\lambda v\|=|\lambda|\|v\|$
3.

$\|v+w\|\leq\|v\|+\|w\|$
4.

$\|v\cdot w\|=\|v\|\cdot\|w\|$

which means that $\mathbb{H}$ qualifies as a normed algebra when we give it the norm $\|\cdot\|$ .

Because the norm of any nonzero quaternion $q$ is real and nonzero, we have

\frac{q\overline{q}}{\|q\|^{2}}=\frac{\overline{q}q}{\|q\|^{2}}=\left<1,0,0,0\right>

which shows that any nonzero quaternion has an inverse:

q^{-1}=\frac{\overline{q}}{\|q\|^{2}}\;.

Other embeddings of $\mathbb{C}$ into $\mathbb{H}$

One can use any non-zero $q$ to define an embedding of $\mathbb{C}$ into $\mathbb{H}$ . If $\mathbf{n}(z)$ is a natural embedding of $z\in\mathbb{C}$ into $\mathbb{H}$ , then the embedding:

z\rightarrow q\mathbf{n}(z)q^{-1}

is also an embedding into $\mathbb{H}$ . Because $\mathbb{H}$ is an associative algebra, it is obvious that:

(q\mathbf{n}(a)q^{-1})(q\mathbf{n}(b)q^{-1})=q(\mathbf{n}(a)\mathbf{n}(b))q^{-1}

and with the distributive laws, it is easy to check that

(q\mathbf{n}(a)q^{-1})+(q\mathbf{n}(b)q^{-1})=q(\mathbf{n}(a)+\mathbf{n}(b))q^% {-1}

Rotations in 3-space

Let us write

U=\{q\in\mathbb{H}:||q||=1\}

With multiplication, $U$ is a group. Let us briefly sketch the relation between $U$ and the group $SO(3)$ of rotations (about the origin) in 3-space.

An arbitrary element $q$ of $U$ can be expressed $\cos\frac{\theta}{2}+\sin\frac{\theta}{2}(a\mathbf{i}+b\mathbf{j}+c\mathbf{k})$ , for some real numbers $\theta,a,b,c$ such that $a^{2}+b^{2}+c^{2}=1$ . The permutation $v\mapsto qv$ of $U$ thus gives rise to a permutation of the real sphere. It turns out that that permutation is a rotation. Its axis is the line through $(0,0,0)$ and $(a,b,c)$ , and the angle through which it rotates the sphere is $\theta$ . If rotations $F$ and $G$ correspond to quaternions $q$ and $r$ respectively, then clearly the permutation $v\mapsto qrv$ corresponds to the composite rotation $F\circ G$ . Thus this mapping of $U$ onto $SO(3)$ is a group homomorphism. Its kernel is the subset $\{1,-1\}$ of $U$ , and thus it comprises a double cover of $SO(3)$ . The kernel has a geometric interpretation as well: two unit vectors in opposite directions determine the same axis of rotation.

On the algebraic side, the quaternions provide an example of a division ring that is not a field.

Title	Hamiltonian quaternions
Canonical name	HamiltonianQuaternions
Date of creation	2013-03-22 12:35:42
Last modified on	2013-03-22 12:35:42
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	10
Author	mathcam (2727)
Entry type	Definition
Classification	msc 16W99
Synonym	quaternion
Related topic	EulerFourSquareIdentity
Related topic	QuaternionGroup
Related topic	HyperkahlerManifold
Related topic	MathematicalBiology
Defines	quaternion algebra