In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:

• Biquaternions when the coefficients are complex numbers.
• Split-biquaternions when the coefficients are split-complex numbers.
• Dual quaternions when the coefficients are dual numbers.

This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844.^[1] Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a representation of the Lorentz group, which is the foundation of special relativity.

The algebra of biquaternions can be considered as a tensor product C ⊗_R H, where C is the field of complex numbers and H is the division algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2 × 2 complex matrices M₂(C). They are also isomorphic to several Clifford algebras including C ⊗_R H = Cl^[0]
₃(C) = Cl₂(C) = Cl_1,2(R),^[2] the Pauli algebra Cl_3,0(R),^[3][4] and the even part Cl^[0]
_1,3(R) = Cl^[0]
_3,1(R) of the spacetime algebra.^[5]

Definition

Let {1, i, j, k} be the basis for the (real) quaternions H, and let u, v, w, x be complex numbers, then

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

is a biquaternion.^[6] To distinguish square roots of minus one in the biquaternions, Hamilton^[7][8] and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field C by h to avoid confusion with the i 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 terms bivector, biconjugate, bitensor, and biversor to extend notions used with real quaternions H.

Hamilton's primary exposition on biquaternions came in 1853 in his Lectures on Quaternions. The editions of Elements of Quaternions, in 1866 by William Edwin Hamilton (son of Rowan), and in 1899, 1901 by Charles 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 a 4-dimensional algebra over the complex numbers C. The algebra of biquaternions is associative, but not commutative. A biquaternion is either a unit or a zero divisor. The algebra of biquaternions forms a composition algebra and can be constructed from bicomplex numbers. See § As a composition algebra below.

Place in ring theory

Algebraic properties

The biquaternions have two conjugations:

• the biconjugate or biscalar minus bivector is ${\displaystyle q^{*}=w-x\mathbf {i} -y\mathbf {j} -z\mathbf {k} \!\ ,}$ and
• the complex conjugation of biquaternion coefficients ${\displaystyle {\bar {q}}={\bar {w}}+{\bar {x}}\mathbf {i} +{\bar {y}}\mathbf {j} +{\bar {z}}\mathbf {k} }$

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

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

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

• ${\displaystyle q^{-1}=q^{*}\lbrace qq^{*}\rbrace ^{-1}}$, if ${\displaystyle qq^{*}\neq 0.}$

Associated terminology

As the biquaternions have been a fixture of linear algebra since the beginnings of mathematical physics, there is an array of concepts that are illustrated or represented by biquaternion algebra. The transformation group ${\displaystyle G=\lbrace g:gg^{*}=1\rbrace }$ has two parts, ${\displaystyle G\cap H}$ and ${\displaystyle G\cap M.}$ The first part is characterized by ${\displaystyle g={\bar {g}}}$ ; then the Lorentz transformation corresponding to g is given by ${\displaystyle T(q)=g^{-1}qg}$ since ${\displaystyle g^{*}=g^{-1}.}$ Such a transformation is a rotation by quaternion multiplication, and the collection of them is SO(3) ${\displaystyle \cong G\cap H.}$ But this subgroup of G is not a normal subgroup, so no quotient group can be formed.

To view ${\displaystyle G\cap M}$ it is necessary to show some subalgebra structure in the biquaternions. Let r represent an element of the sphere of square roots of minus one in the real quaternion subalgebra H. Then (hr)² = +1 and the plane of biquaternions given by ${\displaystyle D_{r}=\lbrace z=x+yhr:x,y\in \mathbb {R} \rbrace }$ 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 a unit hyperbola given by

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

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 called hyperbolic versors. The unit circle in C and unit hyperbola in D_r are examples of one-parameter groups. For every square root r of minus one in H, there is a one-parameter group in the biquaternions given by ${\displaystyle G\cap D_{r}.}$

The space of biquaternions has a natural topology through the Euclidean metric on 8-space. With respect to this topology, G is a topological group. Moreover, it has analytic structure making it a six-parameter Lie group. Consider the subspace of bivectors ${\displaystyle A=\lbrace q:q^{*}=-q\rbrace }$. Then the exponential map ${\displaystyle \exp :A\to G}$ takes the real vectors to ${\displaystyle G\cap H}$ and the h-vectors to ${\displaystyle G\cap M.}$ When equipped with the commutator, A forms the Lie algebra of G. Thus this study of a six-dimensional space serves to introduce the general concepts of Lie theory. When viewed in the matrix representation, G is called the special linear group SL(2,C) in M(2, C).

Many of the concepts of special relativity are illustrated through the biquaternion structures laid out. The subspace M corresponds to Minkowski space, with the four coordinates giving the time and space locations of events in a resting frame of reference. Any hyperbolic versor exp(ahr) corresponds to a velocity in direction r of speed c tanh a where c is the velocity of light. The inertial frame of reference of this velocity can be made the resting frame by applying the Lorentz boost T given by g = exp(0.5ahr) since then ${\displaystyle g^{\star }=\exp(-0.5ahr)=g^{*}}$ so that ${\displaystyle T(\exp(ahr))=1.}$ Naturally the hyperboloid ${\displaystyle G\cap M,}$ which represents the range of velocities for sub-luminal motion, is of physical interest. There has been considerable work associating this "velocity space" with the hyperboloid model of hyperbolic geometry. In special relativity, the hyperbolic angle parameter of a hyperbolic versor is called rapidity. Thus we see the biquaternion group G provides a group representation for the Lorentz group.^[11]

After the introduction of spinor theory, particularly in the hands of Wolfgang Pauli and Élie Cartan, the biquaternion representation of the Lorentz group was superseded. The new methods were founded on basis vectors in the set

${\displaystyle \{q\$ :\ qq^{*}=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 the complex light cone. The above representation of the Lorentz group coincides with what physicists refer to as four-vectors. Beyond four-vectors, the standard model of particle physics also includes other Lorentz representations, known as scalars, and the (1, 0) ⊕ (0, 1)-representation associated with e.g. the electromagnetic field tensor. Furthermore, particle physics makes use of the SL(2, C) representations (or projective representations of the Lorentz group) known as left- and right-handed Weyl spinors, Majorana spinors, and Dirac spinors. It is known that each of these seven representations can be constructed as invariant subspaces within the biquaternions.^[12]

As a composition algebra

Although W. R. Hamilton introduced biquaternions in the 19th century, its delineation of its mathematical structure as a special type of algebra over a field was accomplished in the 20th century: the biquaternions may be generated out of the bicomplex numbers in the same way that Adrian Albert generated the real quaternions out of complex numbers in the so-called Cayley–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

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

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

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

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

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

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

Two biquaternions p and q satisfy N(pq) = N(p) N(q), indicating that N is a quadratic form admitting composition, so that the biquaternions form a composition algebra.

Functions of Complex Quaternions

The exponential function is well-defined by its power series, which converges over the entire domain, even for complex quaternions. Since the basic circular and hyperbolic functions cos, sin, cosh, sinh are linear combinations of exponential functions, they too are well-defined. Some preliminaries: In this section I J K are used for the basis quaternions so that i may be used as the square root of -1. Here I I = -1, I J = K, etc. Let x = a + b I + c J+ d K be a complex quaternion with complex coefficients a, b, c, d and with norm a² + b² + c² + d². The norm of a product is the product of the norms. The complex quaternions do not form a normed division algebra since there are non-zero elements with zero norm. These elements have no inverse. A simple expression for exp(x) is sought. The vector part v = b I + c J+ d K has the property that its square is the scalar -(b² + c² + d²), which is the negative of its norm. The vector quaternion v is called basis-like if its square is -1. Let the complex s be the square root of - v v. Any power series in the vector part v can be expressed as a sum of an even power series in s and an odd power series in s which multiplies the vector part v divided by s. If these two power series are those of common functions, then this gives a simple way to evaluate. Now exp(a + b I + c J + d K) = exp(a) exp(b I + c J + d K) since a is in the complex sub-algebra which commutes with the complex quaternion algebra. The second exponential factor can be expressed as a cos(s) term plus a sin(s)/s term multiplying the vector part v. This assumes s to be non-zero. For the case for which the vector part N of the quaternion has zero norm, exp(a + b N) = exp(a) ( 1 + b N).

The sqrt function can be handled even though it has a branch cut. There are more than two multiple values. For the complex numbers, any multiple by a vector quaternion with square +1 such as i I is also a possible value. The case for which the vector part has square zero is a special case. Indeed, a quaternion which is a non-zero vector quaternion with square zero has no square root. For a quaternion with a vector part having non-zero norm, such as a + b I with b non-zero, there are four possibilities for the multiplying factor, namely 1, -1, i I, and -i I. Consider the complex quaternion x = a + b I. The vector part, assumed to have a non-zero norm, can always be scaled so that it is basis-like with norm +1 with the complex scaling factor being absorbed into b. No generality is lost by representing this quaternion vector by I since only the property that its square is -1 is needed in this discussion. Its square root must have the form c + d I. This gives two complex equations for the two complex unknowns c and d. If a is non-zero, its square root can be factored out so that now the square root to be found is of 1 + A I. Within a common sign, c = (sqrt(1+ i A)+sqrt(1- iA))/2 and d= -i (sqrt(1+iA)-sqrt(1-iA))/2. Notice that it doesn’t matter which sign is picked for sqrt(1+iA) and which sign is picked for sqrt(1-iA) so long as the choices are consistent. If a is zero, then the square root of I needs to be found. It is (1+I)/sqrt(2 ) within a sign. As an example,^[13] sqrt(2 + 3*i*I + J) = 1.09868 + i * 0.45509 + (0.482696 + i * 1.16533) * I + (0.388443 + i * -0.160899) * J. The special case a + b N where a is non-zero and N is a null vector quaternion has a square root c + d N given by c = sqrt(a) and d = b / (2* sqrt(a)).

The logarithm function log(1+x) is not well defined and its power series only converges in a region near the origin. As a cautionary example, consider log(-1). Some possible values are pi*i, pi*I, pi*J, pi*K, and pi times any quaternion vector with norm +1. Admittedly this is a pathological case. If x has the form c + d I, then exp(x) always has the form a + b I. It is assumed that the vector part has a non-zero norm so that it can be scaled to be basis-like with norm +1 and be represented by I. It is only for complex numbers with b equal to zero that the pathology arises. Otherwise only the commutative subalgebra generated by 1, i, and I need be considered. For instance, log(I) has the values (pi/2)*I + 2 m pi i + 2 n pi I, where m and n are integers. There will always be a grid of solutions 2 m pi i + 2 n pi I centered on a particular solution. Let b/a = tan(theta) where theta is complex. Now exp(theta I) = cos(theta) + sin(theta) I even for complex theta. So 1 + (b/a) I = cos(theta) ( 1 + tan(theta) I). So c = log( a / cos(theta) ) and d = atan(b/a). As an example, log(2+(1+i)*I) = 0.748933 + i * 0.231824 + (0.553574 + i * 0.402359) * I. The log of a quaternion that is a null vector does not exist. For N a null vector quaternion, log(1 + N) = N.

The circular and hyperbolic functions are defined in terms of the exp function. Examination of the cosine and sine functions and their inverses demonstrates the important concepts for all. They are defined by cos(x) = (exp(i x) + exp(-i x))/2 and sin(x) = (exp(i x) - exp(-i x))/(2 i). Using DeMoivre's formula exp(i x) = cos(x) + i sin(x) and the identity cos²(x) + sin²(x) = 1 lets x be expressed in terms of either cos(x) or sin(x) using the sqrt and log functions, giving their respective inverses arccosine and arcsine. Finding those values of x for which cos(x) or sin(x) have the same values reveals the multi-valued behavior of their inverses. Except for special cases, x = a + b I, where I is a basis-like vector quatornion of norm +1 and b is non-zero. For integers m and n, adding m pi + n pi i I to x takes exp(i x) into itself for even m + n and into its negative for odd m + n. The even case leaves both cos(x) and sin(x) unchanged. The odd case together with negating x leaves sin(x) unchanged. A special case is when x = a + N, where N is a non-zero vector quaternion having zero norm. The 2D grid then becomes the 1D grid m pi. The other special case is when x = a, where a is complex. Then adding to x any term of the form n pi i I, where I is any basis-like vector quaternion of norm +1, takes exp(i x) into itself for even n and into its negative for odd n. The hyperbolic functions are just the circular functions with the complex plane rotated by 90 degrees.

The complex quaternion function exp can be used to perform Lorentz boosts. This cannot be done with the real quaternions. The following material adds briefly to the sections Relation to Lorentz transformations and As a composition algebra in this article. A 4-vector in special relativity has the form X = t + x i I + y i J + z i K, where t, x, y, and z are real. Its norm is the Minkowski invariant t² - x² - y² - z². Let a ^T superscript denote a "transpose" operation taking I into -I, J into -J, and K into -K. The reason for the "transpose" name is that I, J, and K can be represented as 4x4 real anti-symmetric matrices and that is what the matrix transpose does. The transpose of a product is the product of the transposes in reverse order, as in matrix algebra. Let q* denote complex conjugation of the quaternion q. This operation does not do anything to I, J, and K. This definition is again reasonable since I, J, K can be expressed as 4x4 anti-symmetric real matrices. Let an overbar denote the complex conjugate transpose. If q = a + b I + c J + d K, then q = a* - b* I - c* J - d* K. Note that X = X. Let q have norm 1. Consider X' = q X q. If q has norm 1, then since the complex quaternions are a composition algebra, X' and X have the same norm. Also X' has the same form as X with real scalar part and imaginary spatial part. The complex quaternion B = exp(i I α/2) = cosh(α/2) + i sinh(α/2) I does a boost in the x direction using X' = B X B. Here tanh(α) = v/c is the boost. Similarly, the quaternion R = exp(I θ/2) = cos(θ/2) + sin(θ/2) I does a spatial rotation about the x axis by angle θ using X' = R X R. R is also a real quaternion, unlike the boost quaternion. An aside on the 4x4 real anti-symmetric matrix representation: In terms of the Pauli spin matrices, I, J, and K multiply as i σ_x, -i σ_y, and i σ_z, respectively. The unit imaginary i can be represented as a 2x2 real antisymmetric matrix, zero can be represented as the 2x2 zero matrix, and 1 can be represented as the 2x2 identity matrix. This gives the 4x4 matrix representation. Any group of 2x2 complex matrices can be made 4x4 real with the same group multiplication properties. The group here are the matrices i σ_x, -i σ_y, and i σ_z, the 2x2 identity matrix, and their negatives. One question remains: Suppose q has norm one. What Lorentz transformation is it associated with? It always represents some Lorentz transformation since X' = q X q is a linear transformation of X, preserves the Minkowski invariant, and preserves the form X = X. Any proper Lorentz transformation can be represented uniquely either as a boost B₁ followed by a rotation R₂ or as a rotation R₁ followed by a boost B₂. Note that B = B and that R R = 1. To be definite, let q = B R. Then q q = B R R B = B B = B B. Its square root gives B. Then R = B⁻¹ q. As a check, let Q = q q. Note that Q = Q and that Q has a positive real scalar part and a pure imaginary spatial part. Its norm is one since q has norm one. So its scalar part is not only positive real but is also greater than or equal to one. So it has the form of a boost. As a further check, R R = q B⁻¹ B⁻¹ q = q (q q)⁻¹ q = 1. That R R = 1 means that R is a rotation except for a possible phase factor. That B R has norm 1 means that the phase factor is 0 or 180 degrees. That doesn't matter. Changing the sign of q still gives the same Lorentz transformation.

See also

• Biquaternion algebra
• Hypercomplex number
• Joachim Lambek
• MacFarlane's use
• Quotient ring

Citations

References

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