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Definition of $\Q$
We define a unital associative algebra $\Q$ over $\R$ , 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 $xy$ , not $yx$ ). Thus an arbitrary element of $\Q$ is of the form $$ a\mathbf{1} + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}, \qquad a,b,c,d \in \R$$ (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 \begin{eqnarray*} w&=& a\alpha - b\beta - c\gamma - d\delta \\ x&=& a\beta + b\alpha + c\delta - d\gamma \\ y&=& a\gamma - b\delta + c\alpha + d\beta \\ z&=& a\delta + b\gamma - c\beta + d\alpha \end{eqnarray*}The elements of $\Q$ are known as Hamiltonian quaternions.
Clearly the subspaces of $\Q$ generated by $\{\mathbf{1}\}$ and by $\{\mathbf{1},\mathbf{i}\}$ are subalgebras isomorphic to $\R$ and $\C$ respectively. $\R$ is customarily identified with the corresponding subalgebra of $\Q$ . (We shall see in a moment that there are other and less obvious embeddings of $\C$ in $\Q$ .) The real numbers commute with all the elements of $\Q$ , and we have $$\lambda \cdot \left<a,b,c,d\right> = \left<\lambda a, \lambda b, \lambda c, \lambda d\right>$$ for $\lambda \in \R$ and $\left<a,b,c,d\right> \in \Q$ .
Norm, conjugate, and inverse of a quaternion
Like the complex numbers ($\C$ ), 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) $\Q$ : We define $\|q\| = \sqrt{s}$ where $q\overline{q} = s\mathbf{1}$ .
If $v,w \in \Q$ and $\lambda \in \R$ , then
- $\|v\| \geq 0$ with equality only if $v = \left<0,0,0,0\right> = 0$
- $\|\lambda v\| = |\lambda| \|v\|$
- $\|v+w\| \leq \|v\| + \|w\|$
- $\|v \cdot w\| = \|v\| \cdot \|w\|$
which means that $\Q$ 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 $\C$ into $\Q$
One can use any non-zero $q$ to define an embedding of $\C$ into $\Q$ . If $\mathbf{n}(z)$ is a natural embedding of $z \in \C$ into $\Q$ , then the embedding: $$ z \rightarrow q \mathbf{n}(z) q^{-1} $$ is also an embedding into $\Q$ . Because $\Q$ 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 $$U=\{q\in\Q:||q||=1\}$$ 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.
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