Clifford biquaternion

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A Clifford biquaternion is a member of the Clifford algebra Cℓ0,3(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 = Cℓ0,2(R)

One can therefore see that

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

where D = Cℓ1,0(R) is the algebra spanned by {1, ω}, the algebra of the split-complex numbers.

Equivalently,

Cl_{0,3}(\mathbb{R}) = \mathbb{H} \oplus \mathbb{H}

Contents

[edit] 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},•).

[edit] 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

Cl_{0,2}(\mathbb{C}) = \mathbb{H} \otimes \mathbb{C}

[edit] See also

[edit] References

  • William Kingdon Clifford (1873), "Preliminary Sketch of Biquaternions", Paper XX, Mathematical Papers, p.181.
  • Alexander MacAuley (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.
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