Split-biquaternion
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In mathematics, a split-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
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
where D = Cℓ1,0(R) is the algebra spanned by {1, ω}, the algebra of the split-complex numbers.
Equivalently,
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[edit] Split-biquaternion group
The split-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 split-biquaternions should not be confused with the (ordinary) biquaternions previously introduced by William Rowan Hamilton. Hamilton's biquaternions are elements of the algebra
[edit] See also
[edit] 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.
