Quaternions

A quaternion is often represented as $q = a + b i + c j + d k$ , 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:

$i^{2} = j^{2} = k^{2} = i j k = - 1$

$i j = k,$ but $j i = - k$

$j k = i,$ but $k j = - i$

$k i = j,$ but $i k = - j$

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

The transformation:

\begin{array}{ccl} H & ⟶ & H \\ q & ⟶ & u \cdot q \end{array}

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

The transformation:

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

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

The transformation:

\begin{array}{ccl} H & ⟶ & H \\ q & ⟶ & - u \cdot \bar{q} \cdot u \end{array}

is a reflection in $H$ and viceversa.

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

\begin{array}{ccl} H & ⟶ & H \\ q & ⟶ & 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 ${Cl}_{0, 2} (R)$ is a four-dimensional algebra spanned by ${1, e_{1}, e_{2}, e_{1} e_{2}}$ which behave like quaternions. Also quaternions can be understood as the even subalgebra of $C l (3, 0)$ .

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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