Abstract Algebra/Quaternions

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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 Cℓ₂(R) and the even subalgebra of Cℓ₃(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

Using the Bridge-stone equations, explicitly state the rule of multiplication for general quaternions, that is, given $q_{1}=a_{1}+b_{1}i+c_{1}j+d_{1}k$ and $q_{2}=a_{2}+b_{2}i+c_{2}j+d_{2}k$ , give the components of their product $q=q_{1}q_{2}$

Reference

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

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