Split-biquaternion - Citizendia

In mathematics, a split-biquaternion is a member of the Clifford algebra Cℓ_0,3(R). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, Clifford algebras are a type of Associative algebra. 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. In Mathematical physics, a geometric algebra is a Multilinear algebra described technically as a Clifford algebra over a real vector space equipped

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

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

where D = Cℓ_1,0(R) is the algebra spanned by {1, ω}, the algebra of the split-complex numbers. In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real

Equivalently,

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

1 Split-biquaternion group
2 Hamilton biquaternion
3 See also
4 References

Split-biquaternion group

The split-biquaternions form an associative ring as is clear from considering multiplications in its basis. In Mathematics, associativity is a property that a Binary operation can have In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those 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},•). In Group theory, the quaternion group is a non-abelian group of order 8

Hamilton biquaternion

The split-biquaternions should not be confused with the (ordinary) biquaternions previously introduced by William Rowan Hamilton. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Hamilton's biquaternions are elements of the algebra

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

References

William Kingdon Clifford (1873), "Preliminary Sketch of Biquaternions", Paper XX, Mathematical Papers, p. The biquaternions are the numbers w + xi + yj + zk \ \! where w x y and z are complex numbers and the elements of {1 i j k} multiply as in the Quaternion group In Mathematics, the split-octonions are a Nonassociative extension of the Quaternions (or the Split-quaternions. William Kingdon Clifford FRS ( May 4, 1845 &ndash March 3, 1879) was an English Mathematician and 381.
Alexander MacAulay (1898) Octonions: A Development of Clifford's Biquaternions, Cambridge University Press. Alexander MacAulay (1863 - 1931 was an explorer of Clifford biquaternion theory and was the first professor of Mathematics and Physics at the
P. R. Girard (1984), "The quaternion group and modern physics", European Journal of Physics, 5:25-32.

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Contents

Split-biquaternion group

Hamilton biquaternion

See also

References