Split-biquaternion

In mathematics, a split-biquaternion is a hypercomplex number of the form

q = w + xi + yj + zk \!

where w, x, y, and z are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient w, x, y, z spans two real dimensions, the split-biquaternion is an element of an eight-dimensional vector space. Considering that it carries a multiplication, this vector space is an algebra over the real field, or an algebra over a ring where the split-complex numbers form the ring. This algebra was introduced by William Kingdon Clifford in an 1873 article for the London Mathematical Society. It has been repeatedly noted in mathematical literature since then, variously as a deviation in terminology, an illustration of the tensor product of algebras, and as an illustration of the direct sum of algebras. The split-biquaternions have been identified in various ways by algebraists; see the Synonyms section below.

Modern definition

A split-biquaternion is a member of the Clifford algebra Cℓ_0,3(R). This is the geometric algebra generated by three orthogonal imaginary unit basis directions, {e₁, e₂, e₃} under the combination rule

e_i e_j = \Bigg\{ \begin{matrix} -1 & i=j, \\ - e_j e_i & i \not = j \end{matrix}

giving an algebra spanned by the 8 basis elements {1, e₁, e₂, e₃, e₁e₂, e₂e₃, e₃e₁, e₁e₂e₃}, with (e₁e₂)² = (e₂e₃)² = (e₃e₁)² = −1 and (ω = e₁e₂e₃)² = +1. The sub-algebra spanned by the 4 elements {1, i = e₁, j = e₂, k = e₁e₂} is the division ring of Hamilton's quaternions, H = Cℓ_0,2(R). One can therefore see that

C\ell_{0,3}(\mathbb{R}) = \mathbb{H} \otimes \mathbb{D}

where D = Cℓ_1,0(R) is the algebra spanned by {1, ω}, the algebra of the split-complex numbers. Equivalently,

C\ell_{0,3}(\mathbb{R}) = \mathbb{H} \oplus \mathbb{H}.

Split-biquaternion group

The split-biquaternions form an associative ring as is clear from considering multiplications in its basis {1, ω, i, j, k, ωi, ωj, ωk}. When ω is adjoined to the quaternion group one obtains a 16 element group

( {1, i, j, k, −1, −i, −j, −k, ω, ωi, ωj, ωk, −ω, −ωi, −ωj, −ωk}, × ).

Direct sum of two quaternion rings

The direct sum of the division ring of quaternions with itself is denoted $\mathbb{H} \oplus \mathbb{H}$ . The product of two elements $(a \oplus b)$ and $(c \oplus d)$ is $a c \oplus b d$ in this direct sum algebra.

Proposition: The algebra of split-biquaternions is isomorphic to $\mathbb{H} \oplus \mathbb{H}.$

proof: Every split-biquaternion has an expression q = w + z ω where w and z are quaternions and ω² = +1. Now if p = u + v ω is another split-biquaternion, their product is

pq = uw + vz + (uz + vw) \omega .\!

The isomorphism mapping from split-biquaternions to $\mathbb{H} \oplus \mathbb{H}$ is given by

p \mapsto (u + v) \oplus (u - v) , \quad q \mapsto (w + z) \oplus (w - z).

In $\mathbb{H} \oplus \mathbb{H}$ , the product of these images, according to the algebra-product of $\mathbb{H} \oplus \mathbb{H}$ indicated above, is

(u + v)(w + z) \oplus (u - v)(w - z).

This element is also the image of pq under the mapping into $\mathbb{H} \oplus \mathbb{H}.$ Thus the products agree, the mapping is a homomorphism; and since it is bijective, it is an isomorphism.

Though split-biquaternions form an eight-dimensional space like Hamilton’s biquaternions, on the basis of the Proposition it is apparent that this algebra splits into the direct sum of two copies of the real quaternions.

Hamilton biquaternion

The split-biquaternions should not be confused with the (ordinary) biquaternions previously introduced by William Rowan Hamilton. Hamilton's biquaternions are elements of the algebra

C\ell_2(\mathbb{C}) = \mathbb{H} \otimes \mathbb{C}.

Synonyms

The following terms and compounds refer to the split-biquaternion algebra:

elliptic biquaternions – Clifford (1873), Rooney(2007)
Clifford biquaternion – Joly (1902), van der Waerden (1985)
dyquaternions – Rosenfeld (1997)
$\mathbb{D} \otimes \mathbb{H}$ where D = split-complex numbers – Bourbaki (1994), Rosenfeld (1997)
$\mathbb{H} \oplus \mathbb{H}$ , the direct sum of two quaternion algebras – van der Waerden (1985)

The octonions of Alexander MacAulay (1898) are sometimes confused with split-biquaternions, but they are different.

References

William Kingdon Clifford (1873), "Preliminary Sketch of Biquaternions", Paper XX, Mathematical Papers, p. 381.
Alexander MacAulay (1898) Octonions: A Development of Clifford's Biquaternions, Cambridge University Press.
P.R. Girard (1984), "The quaternion group and modern physics", European Journal of Physics, 5:25-32.
Joe Rooney (2007) "William Kingdon Clifford", in Marco Ceccarelli, Distinguished figures in mechanism and machine science, Springer.
Charles Jasper Joly (1905) Manual of Quaternions, page 21, MacMillan & Co.
Boris Rosenfeld (1997) Geometry of Lie Groups, page 48, Kluwer ISBN 0-7923-4390-5 .
Nicolas Bourbaki (1994) Elements of the History of Mathematics, J. Meldrum translator, Springer.
B. L. van der Waerden (1985) A History of Algebra, page 188, Springer, ISBN 0-387-13610-X .

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