# Abstract Algebra/Quaternions

## Contents

The set of Quaternions is an algebraic structure first studied by the Irish mathematician William Rowan Hamilton, in the hopes of constructing a generalization for complex numbers. When first discovered, quaternions generated a lot of excitement among mathematicians and physicists alike, for it was hoped that quaternions would provide a "unified theory" of mechanics and electromagnetism. Although these hopes proved to be unfounded, quaternions are still considered interesting as well as useful mathematical entities.

## Definition

A Quaternion is an ordered 4-tuple $q=(a,b,c,d)$, where $a,b,c,d\in\mathbb{R}$. A quaternion is often denoted as $q=a+bi+cj+dk$ (Observe the analogy with complex numbers). The set of all quaternions is denoted by $\mathbb{H}$.

It is straightforward to define component-wise addition and scalar multiplication on $\mathbb{H}$, making it a real vector space.

The rule for multiplication was a product of Hamilton's ingenuity. He discovered what are known as the Bridge-stone Equations:

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

From the above equations alone, it is possible to derive rules for the pairwise multiplication of $i$, $j$, and $k$:

$ij=k, jk=i, ki=j$ (positive cyclic permutations)

$ji=-k, kj=-i, ik=-j$ (negative cyclic permutations).

Using these, it is easy to define a general rule for multiplication of quaternions. Because quaternion multiplication is not commutative, $\mathbb{H}$ is not a field. However, every nonzero quaternion has a multiplicative inverse (see below), so the quaternions are an example of a non-commutative division ring. It is important to note that the non-commutative nature of quaternion multiplication makes it impossible to define the quotient $p/q$ of two quaternions $p$ and $q$ unambiguously, as the quantities $pq^{-1}$ and $q^{-1}p$ are generally different.

Like the more familiar complex numbers, the quaternions have a conjugation, often denoted by a superscript star: $q^*$. The conjugate of the quaternion $q=a+bi+cj+dk$ is $q^*=a-bi-cj-dk$. As is the case for the complex numbers, the product $qq^*$ is always a positive real number equal to the sum of the squares of the quaternion's components. Using this fact, it is fairly easy to show that the multiplicative inverse of a general quaternion $q$ is given by

$q^{-1}=\frac{q^*}{qq^*}$

where division is defined since $qq^*$ is a scalar. Note that, unlike in the complex case, the conjugate $q^*$ of a quaternion $q$ can be written as a polynomial in $q$:

$q^*=-\frac{1}{2}(q+iqi+jqj+kqk)$.

The quaternions are isomorphic to the Clifford algebra C2(R) and the even subalgebra of C3(R).

## Pauli Spin Matrices

Quaternions are closely related to the Pauli spin matrices of Quantum Mechanics. The Pauli matrices are often denoted as
$\sigma_1=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$ , $\sigma_2=\begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}$ , $\sigma_3=\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$

(Where $i$ is the well known quantity $\sqrt{-1}$ of complex numbers)

The 2×2 identity matrix is sometimes taken as $\sigma_0$. It can be shown that $S$, the real linear span of the matrices $\sigma_0$, $i\sigma_1$, $i\sigma_2$ and $i\sigma_3$, is isomorphic to the set of all quaternions, $\mathbb{H}$. For example, take the matrix product below:

$\begin{pmatrix}i & 0 \\ 0 & -i\end{pmatrix} \begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix} = \begin{pmatrix}0 & i \\ i & 0\end{pmatrix}$

Or, equivalently:

$i\sigma_3 i\sigma_2 = i\sigma_1$

All three of these matrices square to the negative of the identity matrix. If we take $1=\sigma_0$, $i=i\sigma_3$, $j=i\sigma_2$, and $k=i\sigma_1$, it is easy to see that the span of the these four matrices is "the same as" (that is, isomorphic to) the set of quaternions $\mathbb{H}$.

## Exercise

1. Using the Bridge-stone equations, explicitly state the rule of multiplication for general quaternions, that is, given $q_1=a_1+b_1i+c_1j+d_1k$ and $q_2=a_2+b_2i+c_2j+d_2k$, give the components of their product $q=q_1q_2$

## Reference

• E.T. Bell, Men of Mathematics, Simon & Schuster, Inc.
• The Wikipedia article on Pauli Spin Matrices