Inabstract algebra, thebiquaternions are the numbersw +xi +yj +zk, wherew,x,y, andz arecomplex numbers, or variants thereof, and the elements of{1,i,j,k} multiply as in thequaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:

• Biquaternions when the coefficients arecomplex numbers.
• Split-biquaternions when the coefficients aresplit-complex numbers.
• Dual quaternions when the coefficients aredual numbers.

This article is about theordinary biquaternions named byWilliam Rowan Hamilton in 1844.^[1] Some of the more prominent proponents of these biquaternions includeAlexander Macfarlane,Arthur W. Conway,Ludwik Silberstein, andCornelius Lanczos. As developed below, the unitquasi-sphere of the biquaternions provides a representation of theLorentz group, which is the foundation ofspecial relativity.

The algebra of biquaternions can be considered as atensor productC ⊗_RH, whereC is thefield of complex numbers andH is thedivision algebra of (real)quaternions. In other words, the biquaternions are just thecomplexification of the quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of2 × 2 complex matricesM₂(C). They are also isomorphic to severalClifford algebras includingC ⊗_RH = Cl^[0]
₃(C) = Cl₂(C) = Cl_1,2(R),^[2] thePauli algebraCl_3,0(R),^[3][4] and the even partCl^[0]
_1,3(R) = Cl^[0]
_3,1(R) of thespacetime algebra.^[5]

Let{1,i,j,k} be the basis for the (real)quaternionsH, and letu,v,w,x be complex numbers, then

${\displaystyle q=u\mathbf{1}+v\mathbf{i}+w\mathbf{j}+x\mathbf{k}}$

is abiquaternion.^[6] To distinguish square roots of minus one in the biquaternions, Hamilton^[7][8] andArthur W. Conway used the convention of representing the square root of minus one in the scalar fieldC byh to avoid confusion with thei in the quaternion group.Commutativity of the scalar field with the quaternion group is assumed:

${\displaystyle h\mathbf{i}=\mathbf{i}h,h\mathbf{j}=\mathbf{j}h,h\mathbf{k}=\mathbf{k}h.}$

Hamilton introduced the termsbivector,biconjugate,bitensor, andbiversor to extend notions used with real quaternionsH.

Hamilton's primary exposition on biquaternions came in 1853 in hisLectures on Quaternions. The editions ofElements of Quaternions, in 1866 byWilliam Edwin Hamilton (son of Rowan), and in 1899, 1901 byCharles Jasper Joly, reduced the biquaternion coverage in favour of the real quaternions.

Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a4-dimensionalalgebra over the complex numbersC. The algebra of biquaternions isassociative, but notcommutative. A biquaternion is either aunit or azero divisor. The algebra of biquaternions forms acomposition algebra and can be constructed frombicomplex numbers. See§ As a composition algebra below.

Note that thematrix product

.

Becauseh is theimaginary unit, each of these three arrays has a square equal to the negative of theidentity matrix.When this matrix product is interpreted asij =k, then one obtains asubgroup of matrices that isisomorphic to thequaternion group. Consequently,

represents biquaternionq =u1 +vi +wj +xk.Given any2 × 2 complex matrix, there are complex valuesu,v,w, andx to put it in this form so that thematrix ringM(2,C) is isomorphic^[9] to the biquaternionring. This representation also shows that the 16-element group

${\displaystyle \{\pm \mathbf{1},\pm h,\pm \mathbf{i},\pm h\mathbf{i},\pm \mathbf{j},\pm h\mathbf{j},\pm \mathbf{k},\pm h\mathbf{k}\}}$

is isomorphic to thePauli group, thecentral product of acyclic group of order 4 and thedihedral group of order 8. Concretely, thePauli matrices

correspond respectively to the elements－hk, －hj, and－hi.

Considering the biquaternion algebra over the scalar field of real numbersR, the set

${\displaystyle \{\mathbf{1},h,\mathbf{i},h\mathbf{i},\mathbf{j},h\mathbf{j},\mathbf{k},h\mathbf{k}\}}$

forms abasis so the algebra has eight realdimensions. The squares of the elementshi,hj, andhk are all positive one, for example,(hi)² =h²i² = (−1)(−1) = +1.

Thesubalgebra given by

${\displaystyle \{x+y(h\mathbf{i}):x,y\in \mathbb{R}\}}$

isring isomorphic to the plane ofsplit-complex numbers, which has an algebraic structure built upon theunit hyperbola. The elementshj andhk also determine such subalgebras.

Furthermore,${\displaystyle \{x+y\mathbf{j}:x,y\in \mathbb{C}\}}$ is a subalgebra isomorphic to thebicomplex numbers.

A third subalgebra calledcoquaternions is generated byhj andhk. It is seen that(hj)(hk) = (−1)i, and that the square of this element is−1. These elements generate thedihedral group of the square. Thelinear subspace with basis{1,i,hj,hk} thus is closed under multiplication, and forms the coquaternion algebra.

In the context ofquantum mechanics andspinor algebra, the biquaternionshi,hj, andhk (or their negatives), viewed in theM₂(C) representation, are calledPauli matrices.

The biquaternions have twoconjugations:

• thebiconjugate or biscalar minusbivector is${\displaystyle q^{\ast }=w-x\mathbf{i}-y\mathbf{j}-z\mathbf{k},}$ and
• thecomplex conjugation of biquaternion coefficients${\displaystyle \overline{q}=\overline{w}+\overline{x}\mathbf{i}+\overline{y}\mathbf{j}+\overline{z}\mathbf{k}}$

where${\displaystyle \overline{z}=a-bh}$ when${\displaystyle z=a+bh,a,b\in \mathbb{R},h^2=-\mathbf{1}.}$

Note that${\displaystyle (pq{)}^{\ast }=q^{\ast }{p}^{\ast },\overline{pq}=\overline{p}\overline{q},\overline{q^{\ast }}={\overline{q}}^{\ast }.}$

Clearly, if${\displaystyle q{q}^{\ast }=0}$ thenq is a zero divisor. Otherwise${\displaystyle \{q{q}^{\ast }{\}}^{-\mathbf{1}}}$ is a complex number. Further,${\displaystyle q{q}^{\ast }=q^{\ast }q}$ is easily verified. This allows the inverse to be defined by

• ${\displaystyle q^{-1}=q^{\ast }\{q{q}^{\ast }{\}}^{-1}}$, if${\displaystyle q{q}^{\ast }\ne 0.}$

Consider now the linear subspace^[10]

${\displaystyle M=\{q:q^{\ast }=\overline{q}\}=\{t+x(h\mathbf{i})+y(h\mathbf{j})+z(h\mathbf{k}):t,x,y,z\in \mathbb{R}\}.}$

M is not a subalgebra since it is notclosed under products; for example${\displaystyle (h\mathbf{i})(h\mathbf{j})=h^2\mathbf{i}\mathbf{j}=\mathbf{k}\notin M.}$ Indeed,M cannot form an algebra if it is not even amagma.

Proposition: Ifq is inM, then${\displaystyle q{q}^{\ast }=t^2-x^2-y^2-z^2.}$

Proof: From the definitions,

Definition: Let biquaterniong satisfy${\displaystyle g{g}^{\ast }=1.}$ Then theLorentz transformation associated withg is given by

${\displaystyle T(q)=g^{\ast }q\overline{g}.}$

Proposition: Ifq is inM, thenT(q) is also inM.

Proof:${\displaystyle (g^{\ast }q\overline{g}{)}^{\ast }={\overline{g}}^{\ast }{q}^{\ast }g=\overline{g^{\ast }}\overline{q}g=\overline{g^{\ast }q\overline{g})}.}$

Proposition:${\displaystyle T(q)(T(q){)}^{\ast }=q{q}^{\ast }}$

Proof: Note first thatgg* = 1 implies that the sum of the squares of its four complex components is one. Then the sum of the squares of thecomplex conjugates of these components is also one. Therefore,${\displaystyle \overline{g}(\overline{g}{)}^{\ast }=1.}$ Now

${\displaystyle (g^{\ast }q\overline{g})(g^{\ast }q\overline{g}{)}^{\ast }=g^{\ast }q(\overline{g}{\overline{g}}^{\ast })q^{\ast }g=g^{\ast }q{q}^{\ast }g=q{q}^{\ast }.}$

As the biquaternions have been a fixture oflinear algebra since the beginnings ofmathematical physics, there is an array of concepts that are illustrated or represented by biquaternion algebra. Thetransformation group${\displaystyle G=\{g:g{g}^{\ast }=1\}}$ has two parts,${\displaystyle G\cap H}$ and${\displaystyle G\cap M.}$ The first part is characterized by${\displaystyle g=\overline{g}}$ ; then the Lorentz transformation corresponding tog is given by${\displaystyle T(q)=g^{-1}qg}$ since${\displaystyle g^{\ast }=g^{-1}.}$ Such a transformation is arotation by quaternion multiplication, and the collection of them isSO(3)${\displaystyle \cong G\cap H.}$ But this subgroup ofG is not anormal subgroup, so noquotient group can be formed.

To view${\displaystyle G\cap M}$ it is necessary to show some subalgebra structure in the biquaternions. Letr represent an element of thesphere of square roots of minus one in the real quaternion subalgebraH. Then(hr)² = +1 and the plane of biquaternions given by${\displaystyle D_r=\{z=x+yhr:x,y\in \mathbb{R}\}}$ is a commutative subalgebra isomorphic to the plane of split-complex numbers. Just as the ordinary complex plane has a unit circle,${\displaystyle D_r}$ has aunit hyperbola given by

${\displaystyle \exp (ahr)=\cosh (a)+hr\sinh (a),a\in R.}$

Just as the unit circle turns by multiplication through one of its elements, so the hyperbola turns because${\displaystyle \exp (ahr)\exp (bhr)=\exp ((a+b)hr).}$ Hence these algebraic operators on the hyperbola are calledhyperbolic versors. The unit circle inC and unit hyperbola inD_r are examples ofone-parameter groups. For every square rootr of minus one inH, there is a one-parameter group in the biquaternions given by${\displaystyle G\cap D_r.}$

The space of biquaternions has a naturaltopology through theEuclidean metric on8-space. With respect to this topology,G is atopological group. Moreover, it has analytic structure making it a six-parameterLie group. Consider the subspace of bivectors${\displaystyle A=\{q:q^{\ast }=-q\}}$. Then theexponential map${\displaystyle \exp :A\to G}$ takes the real vectors to${\displaystyle G\cap H}$ and theh-vectors to${\displaystyle G\cap M.}$ When equipped with thecommutator,A forms theLie algebra ofG. Thus this study of asix-dimensional space serves to introduce the general concepts ofLie theory. When viewed in the matrix representation,G is called thespecial linear groupSL(2,C) inM(2,C).

Many of the concepts ofspecial relativity are illustrated through the biquaternion structures laid out. The subspaceM corresponds toMinkowski space, with the four coordinates giving the time and space locations of events in a restingframe of reference. Any hyperbolic versorexp(ahr) corresponds to avelocity in directionr of speedc tanha wherec is thevelocity of light. The inertial frame of reference of this velocity can be made the resting frame by applying theLorentz boostT given byg = exp(0.5ahr) since then${\displaystyle g^{\star }=\exp (-0.5ahr)=g^{\ast }}$ so that${\displaystyle T(\exp (ahr))=1.}$Naturally thehyperboloid${\displaystyle G\cap M,}$ which represents the range of velocities for sub-luminal motion, is of physical interest. There has been considerable work associating this "velocity space" with thehyperboloid model ofhyperbolic geometry. In special relativity, thehyperbolic angle parameter of a hyperbolic versor is calledrapidity. Thus we see the biquaternion groupG provides agroup representation for theLorentz group.^[11]

After the introduction ofspinor theory, particularly in the hands ofWolfgang Pauli andÉlie Cartan, the biquaternion representation of the Lorentz group was superseded. The new methods were founded onbasis vectors in the set

${\displaystyle \{q:q{q}^{\ast }=0\}=\left\{w+x\mathbf{i}+y\mathbf{j}+z\mathbf{k}:w^2+x^2+y^2+z^2=0\right\}}$

which is called thecomplex light cone. The aboverepresentation of the Lorentz group coincides with what physicists refer to asfour-vectors. Beyond four-vectors, theStandard Model of particle physics also includes other Lorentz representations, known asscalars, and the(1, 0) ⊕ (0, 1)-representation associated with e.g. theelectromagnetic field tensor. Furthermore, particle physics makes use of theSL(2,C) representations (orprojective representations of the Lorentz group) known as left- and right-handedWeyl spinors,Majorana spinors, andDirac spinors. It is known that each of these seven representations can be constructed as invariant subspaces within the biquaternions.^[12]

Although W. R. Hamilton introduced biquaternions in the 19th century, its delineation of itsmathematical structure as a special type ofalgebra over a field was accomplished in the 20th century: the biquaternions may be generated out of thebicomplex numbers in the same way thatAdrian Albert generated the real quaternions out of complex numbers in the so-calledCayley–Dickson construction. In this construction, a bicomplex number(w,z) has conjugate(w,z)* = (w, –z).

The biquaternion is then a pair of bicomplex numbers(a,b), where the product with a second biquaternion(c,d) is

${\displaystyle (a,b)(c,d)=(ac-d^{\ast }b,da+b{c}^{\ast }).}$

If${\displaystyle a=(u,v),b=(w,z),}$ then thebiconjugate${\displaystyle (a,b{)}^{\ast }=(a^{\ast },-b).}$

When(a,b)* is written as a 4-vector of ordinary complex numbers,

${\displaystyle (u,v,w,z{)}^{\ast }=(u,-v,-w,-z).}$

The biquaternions form an example of aquaternion algebra, and it has norm

${\displaystyle N(u,v,w,z)=u^2+v^2+w^2+z^2.}$

Two biquaternionsp andq satisfyN(pq) =N(p)N(q), indicating thatN is a quadratic form admitting composition, so that the biquaternions form acomposition algebra.

• Biquaternion algebra
• Hypercomplex number
• Hypercomplex analysis
• Joachim Lambek
• MacFarlane's use
• Quotient ring
• Quaternion Lorentz Transformations
• Biquaternion functions

The WikibookAssociative Composition Algebra has a page on the topic of:Biquaternions

• Composition algebras
• Quaternions
• Ring theory
• Special relativity
• William Rowan Hamilton

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