Clifford biquaternion

A Clifford 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

<math>e_i e_j = \Bigg\{ \begin{matrix} -1 & i=j, \\

                                  - e_j e_i &   i \not = j \end{matrix} </math>

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

<math>Cl_{0,3}(\mathbb{R}) = \mathbb{H} \otimes \mathbb{D}</math>

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

<math>Cl_{0,3}(\mathbb{R}) = \mathbb{H} \oplus \mathbb{H}</math>

Clifford biquaternion group

The Clifford biquaternions form an associative ring as is clear from considering multiplications in its basis. 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},•).

Hamilton biquaternion

The Clifford biquaternions should not be confused with the biquaternions previously introduced by William Rowan Hamilton. Hamilton's biquaternions are elements of the algebra

<math>Cl_{0,2}(\mathbb{C}) = \mathbb{H} \otimes \mathbb{C}</math>

See also

• split-octonions

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.