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Concept
Split-biquaternion
Summary
In mathematics, a split-biquaternion is a hypercomplex number of the form
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 below.
A split-biquaternion is ring isomorphic to 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 ω2 = (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,
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}, × ).
Since elements {1, i, j, k} of the quaternion group can be taken as a basis of the space of split-biquaternions, it may be compared to a vector space. But split-complex numbers form a ring, not a field, so vector space is not appropriate.
Official source
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