Quaternions

A quaternion is often represented as $q = a + bi + cj + dk$, where $a, b, c,$ and $d$ are real numbers, and $i, j,$ and $k$ are the fundamental quaternion units. These units have the following multiplication rules:

$i^2 = j^2 = k^2 = ijk = -1$

$ij = k,$ but $ji = -k$

$jk = i,$ but $kj = -i$

$ki = j,$ but $ik = -j$

Quaternions are often written as $\mathbb{H}$. If we have an unitary quaternion $u$ we have:

The transformation:

\begin{array}{ccl} \mathbb{H}& \longrightarrow & \mathbb{H}\\ q & \longrightarrow & u \cdot q\\ \end{array} $$

is an isometry, since $|uq-up|=|u|\cdot|q-p|=|q-p|$.

The transformation:

$$ \begin{array}{ccl} P& \longrightarrow & P\\ q & \longrightarrow & u \cdot q\cdot u^{-1}\\ \end{array} $$

is a rotation in the pure imaginary quaternions $P=\mathbb{R} i+\mathbb{R} j+\mathbb{R} k \cong \mathbb{R}^3$

The transformation:

\begin{array}{ccl} \mathbb{H}& \longrightarrow & \mathbb{H}\\ q & \longrightarrow & -u \cdot \bar{q}\cdot u\\ \end{array} $$

is a reflection in $\mathbb{H}$ and viceversa.

Any rotation in $\mathbb{H}=\mathbb{R}^4$ has the form:

$$ \begin{array}{ccl} \mathbb{H}& \longrightarrow & \mathbb{H}\\ q & \longrightarrow & v \cdot q\cdot w\\ \end{array} $$

for $v, w$ unitary quaternions.

Unitary quaternions are equivalent to the special unitary group SU(2).

As Clifford algebras: Observe that $\text{Cl}_{0,2}(\mathbb{R})$ is a four-dimensional algebra spanned by $\{1, e_1, e_2, e_1e_2\}$ which behave like quaternions. Also quaternions can be understood as the even subalgebra of $Cl(3,0)$.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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