Abstract Algebra/Quaternions

The algebra of Quaternions is an 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 ${\displaystyle q=(a,b,c,d)}$ , where ${\displaystyle a,b,c,d\in \mathbb {R} }$ . A quaternion is denoted ${\displaystyle q=a+bi+cj+dk}$ . The sum ${\displaystyle bi+cj+dk}$  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 ${\displaystyle \mathbb {H} }$ . It is straightforward to define component-wise addition and scalar multiplication on ${\displaystyle \mathbb {H} }$ , 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:

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

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

${\displaystyle ij=k,\ \ jk=i,\ \ ki=j}$  (positive cyclic permutations)
${\displaystyle ji=-k,\ \ kj=-i,\ \ ik=-j}$  (negative cyclic permutations).

Using these, it is easy to define a general rule for multiplication of quaternions. Because quaternion multiplication is not commutative, ${\displaystyle \mathbb {H} }$  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 ${\displaystyle p/q}$  of two quaternions p and q unambiguously, as the quantities ${\displaystyle pq^{-1}}$  and ${\displaystyle q^{-1}p}$  are generally different.

Like the more familiar complex numbers, the quaternions have a conjugation, often denoted by a superscript star: ${\displaystyle q^{*}}$ . The conjugate of the quaternion ${\displaystyle q=a+bi+cj+dk}$  is ${\displaystyle q^{*}=a-bi-cj-dk}$ . As is the case for the complex numbers, the product ${\displaystyle qq^{*}}$  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 ${\displaystyle qq^{*}}$ .

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

The multiplicative inverse of a non-zero quaternion ${\displaystyle q}$  is given by

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

Unlike in the complex case, the conjugate ${\displaystyle q^{*}}$  of a quaternion ${\displaystyle q}$  can be written as a polynomial in q:

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

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:

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

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

A right parataxy on elliptic space is effected by multiplying on the right by a versor ${\displaystyle v=e^{cs}.}$  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:${\displaystyle x\mapsto uxv.}$  Note that if ${\displaystyle u=v^{*},}$  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 ${\displaystyle L=\{e^{cs}:c\in R\}}$  is a typical elliptic line. It corresponds to the axis of the rotation

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

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

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

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

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

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 ${\displaystyle i,j,k}$  is taken by these arrays:

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

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 ${\displaystyle \mathbb {H} }$  as a four-dimensional subalgebra. The linear properties of ${\displaystyle \mathbb {H} }$  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

${\displaystyle \sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}}$  , ${\displaystyle \sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}}$  , ${\displaystyle \sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$

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

The 2×2 identity matrix is sometimes taken as ${\displaystyle \sigma _{0}}$ .

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

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

Or, equivalently,   ${\displaystyle i\sigma _{3}\ i\sigma _{2}=i\sigma _{1}.}$

All three of these matrices square to the negative of the identity matrix. If we take ${\displaystyle 1=\sigma _{0}}$ , ${\displaystyle i=i\sigma _{3}}$ , ${\displaystyle j=i\sigma _{2}}$ , and ${\displaystyle k=i\sigma _{1}}$ , 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 ${\displaystyle \mathbb {H} }$ .

Exercises

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