Abstract Algebra/Quaternions

The algebra of Quaternions is a structure first studied by the Irish mathematician William Rowan Hamilton which extends the two-dimensional complex numbers to four dimensions. Multiplication is non-commutative in quaternions, a feature which enables its representation of three-dimensional rotation. Hamilton's provocative discovery of quaternions founded the field of hypercomplex numbers. Suggestive methods like dot products and cross products implicit in quaternion products enabled algebraic description of geometry now widely applied in science and engineering.

Definitions

Quaternion plaque on Broom Bridge, Dublin, which says:

Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication i² = j² = k² = ijk = −1 & cut it on a stone of this bridge

A Quaternion corresponds to an ordered 4-tuple $q=(a,b,c,d)$, where $a,b,c,d\in \mathbb {R}$. A quaternion is denoted $q=a+bi+cj+dk$. The sum $bi+cj+dk$ is called the vector part of q, and a is the real part. Hamiltion coined the term vector in this context. Subsequent developments have extended the usage of the term vector to any element of a linear space. The vectors in H form a 3-dimensional subspace V.

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.

Multiplication follows the rules of the "quaternion group" Q₈ = {1, -1, i, -i, j, -j, k, -k} that Hamilton carved into a stone of Broom Bridge, Dublin:

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

The rules for the pairwise multiplication of $i$, $j$, and $k$ are:

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

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

Using these, one can define a general rule for multiplication of quaternions. Because quaternion multiplication is not commutative, $\mathbb {H}$ is a not a field. However, every nonzero quaternion has a multiplicative inverse (see below), so the quaternions are an example of a 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. The norm of a quaternion is the square root of $qq^{*}$.

If pq is the product of two quaternions, then $(pq)(pq)^{*}=(pp^{*})(qq^{*}),$ implying that $\mathbb {H}$ forms a composition algebra.

The multiplicative inverse of a non-zero quaternion $q$ is given by

$q^{-1}={\frac {q^{*}}{qq^{*}}}$ where division is defined since $qq^{*}\neq 0.$

Unlike in the complex case, the conjugate $q^{*}$ of a quaternion $q$ can be computed algebraically:

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

Versors and elliptic space

William Kingdon Clifford used Hamilton’s quaternions to explicate rotation geometry as an elliptic space with its own variety of lines, parallels, and surfaces. The ideas were reviewed in 1948 by Lemaitre and Coxeter and that sketch has these definitions:

A versor is a quaternion of norm one, thus it lies on a 3-dimensional sphere found in the 4-space of quaternions. The versors are given by Euler's formula for complex numbers where the imaginary unit is taken from the unit sphere in the 3-space of vector quaternions:

$v=\cos c+s\sin c=e^{cs},\ \ s^{2}=-1.$

The distance between two versors u and v is $d(u,v)=\arccos(uv^{*}+vu^{*})/2.$

A right parataxy on elliptic space is effected by multiplying on the right by a versor $v=e^{cs}.$ Similarly a left parataxy arises from left multiplication. In recognition of his contribution to elliptic geometry, a parataxy is called a Clifford translation.

The general displacement of elliptic space is a combination of two parataxies, one left, one right:$x\mapsto uxv.$ Note that if $u=v^{*},$ then the real line in the quaternions is fixed and the displacement is a rotation of the 3-space of quaternion vectors.

The term line is appropriated for elliptic geometry. These lines are not straight, but they are parametrized by real numbers. Each line is associated with a right versor like s when c = π/2 in v. Then $L=\{e^{cs}:c\in R\}$ is a typical elliptic line. It corresponds to the axis of the rotation

$x\mapsto e^{cs}xe^{-cx}.$

Now for u not on L, there are two Clifford parallels to L through u:

$\{ue^{cs}:c\in R\},\quad \{e^{cs}u:c\in R\}.$

For fixed right versors r and s, a Clifford surface can be formed as a union of Clifford parallels or as

$\{e^{cs}e^{dr}:b,c\in R\}.$

To form elliptic space from versors, two versors u and v are equivalent if u + v = 0. Modulo this equivalence, the versors, their algebra and geometry, represent elliptic space.

Linear viewpoint

Quaternions may be represented by 2×2 matrices with complex number entries: the place of $i,j,k$ is taken by these arrays:

${\begin{pmatrix}i&0\\0&-i\end{pmatrix}},\quad {\begin{pmatrix}0&1\\-1&0\end{pmatrix}},\quad {\begin{pmatrix}0&i\\i&0\end{pmatrix}}.$

One uses matrix multiplication to verify that these expressions obey the rules of presentation of Q₈.

M(2,C) denotes the full algebra of 2×2 complex matrices, which has eight real dimensions, and sustains a representation of $\mathbb {H}$ as a four-dimensional subalgebra. The linear properties of $\mathbb {H}$ and M(2,C) assure the fidelity of the representation once the copy of Q₈ has been identified.

Quaternions, like other associative hypercomplex systems of the 19th century, eventually were viewed as matrix algebras in the 20th century. However, in 1853 Hamilton included biquaternions in his book of Lectures on Quaternions.

Biquaternions are quaternions with complex number coefficients, sometimes called complex quaternions. Biquaternions form an algebra isomorphic to M(2,C). If the rows or columns of a matrix are proportional, then the determinant is zero, and there is no inverse. Nevertheless, such matrices have been used in physical science to represent events on a light-path from the origin. Authors Silberstein and Lanczos refer to this algebra as the biquaternions, but other writers have abandoned the label: Elie Cartan used M(2,C) extensively in The Theory of Spinors (1938), and Wolfgang Pauli, in his matrix mechanics of the atom, caused himself to be associated with M(2,C).

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 $\sigma _{0}$.

Thus $S$, the real linear span of the matrices $\sigma _{0}$, $i\sigma _{1}$, $i\sigma _{2}$ and $i\sigma _{3}$, is isomorphic to $\mathbb {H}$. For example, take this matrix product:

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

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 $\mathbb {H}$.

Exercises

1. Using the presentation equations of Q₈, write out the full product of two quaternions. In other words, given $q_{1}=a_{1}+b_{1}i+c_{1}j+d_{1}k$ and $q_{2}=a_{2}+b_{2}i+c_{2}j+d_{2}k$, find the components of their product $q=q_{1}q_{2}.$
2. Show the composition algebra property $(pq)(pq)^{*}=(pp^{*})(qq^{*}).$ Hint: use w: Euler's four-square identity.

Axial pencils

The axis of a pencil is the real axis in the 4-algebra

Hamilton's quaternions provide a picture of a pencil of complex number planes that fill out his hyperspace. Another pair of pencils provide alternative descriptions of 4-space as made up of planar algebras: The hyperspace is $\mathbb {R} ^{4}$ with the first coordinate taken as the real line, and as the axis of the various pencils. The Hamilton case uses the sphere of imaginary units

$S^{2}=\{q\in H:q^{2}=-1\}=\{xi+yj+zk:x^{2}+y^{2}+z^{2}=1\}.$

Any pair of antipodal points on this sphere generates a plane isomorphic to the ordinary complex plane $\mathbb {C} .$

The second and third pencils derive from the findings of James Cockle and Arthur Cayley. Cayley set up an arithmetic of matrix multiplication which has expedited modern science. For instance, the so-called imaginary units are represented by ${\begin{pmatrix}0&-1\\1&0\end{pmatrix}}$ which has multiplicative square equal to the negative of the identity matrix. But then there is also by ${\begin{pmatrix}0&1\\1&0\end{pmatrix}}$ which generates an algebraic plane distinct from ordinary complex numbers. This algebra split-binarions, has inverse proportion included as a structural feature, such as found in economics or spacetime. In fact, the Lorentz boost is exhibited by a split-binarion multiplication. The inherent relation was recognized in the 19th century by J. Cockle, W.K. Clifford, and A. Macfarlane in the English world and by some Serbians.

The second pencil is an imaginary one without linear representation. As Hamilton had a sphere of imaginary units, Macfarlane would have a sphere of hyperbolic units u with u² = +1. The full algebra of split-binarions is $A=\{q=x+yu:x,y\in \mathbb {R} \}$ Any pair of elements that are polar opposite on this sphere generate a plane isomorphic to the split-binarions A. The 4-algebra containing this pencil is the hyperbolic quaternion algebra. As Oliver Heaviside and Willard Gibbs advocated a positive dot product for vectors, they have been associated with hyperbolic quaternions. When this algebra drew attention in the 1890s a "great vector debate" ensued in various publications including Nature. When the failure of the algebra to satisfy the associative law of multiplication was noted, it was realized that no matrix representation would be found.

Each plane of the pencil can represent a Lorentz boost. However, rotations of the vector subspace, an operation within the reach of Hamilton's structure, is beyond the means of hyperbolic quaternions, hence the Lorentz group cannot be represented with Macfarlane's algebra.

The third pencil arises from both imaginary units and hyperbolic units as found in the ring of 2x2 real matrices. In this figure there must be noted nilpotent matrices such as by ${\begin{pmatrix}0&1\\0&0\end{pmatrix}}$ which correspond to dual numbers in the matrix algebra. Such planes separate the complex and split-binarion planes, and are included in the pencil. The axis of the pencil is the line of matrices that are real multiples of the identity matrix. Over an alternate basis this ring is known as split-quaternions, and the pencil has three types of planar subrings.