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In [[mathematics]], a '''split-biquaternion''' is a [[hypercomplex number]] of the form
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:<math>q = w + xi + yj + zk \!</math>
where ''w'', ''x'', ''y'', and ''z'' are [[split-complex number]]s and i, j, and k multiply as in the [[quaternion group]].  Since each [[coefficient]] ''w'', ''x'', ''y'', ''z'' spans two [[real number|real]] [[dimension]]s, 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 a field|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 modules#Direct sum of algebras|direct sum of algebras]].
The split-biquaternions have been identified in various ways by algebraists; see the ''Synonyms'' section below.
 
==Modern denomination==
A split-biquaternion is a member of the [[Clifford algebra]] ''C''&#x2113;<sub>0,3</sub>('''R''').  This is the [[geometric algebra]] generated by three orthogonal imaginary unit basis directions,  {''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>} 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, ''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>, ''e''<sub>1</sub>''e''<sub>2</sub>,  ''e''<sub>2</sub>''e''<sub>3</sub>, ''e''<sub>3</sub>''e''<sub>1</sub>, ''e''<sub>1</sub>''e''<sub>2</sub>''e''<sub>3</sub>}, with (''e''<sub>1</sub>''e''<sub>2</sub>)<sup>2</sup> = (''e''<sub>2</sub>''e''<sub>3</sub>)<sup>2</sup> = (''e''<sub>3</sub>''e''<sub>1</sub>)<sup>2</sup> = &minus;1 and (&omega; = ''e''<sub>1</sub>''e''<sub>2</sub>''e''<sub>3</sub>)<sup>2</sup> = +1.
 
The sub-algebra spanned by the 4 elements {1, ''i'' = ''e''<sub>1</sub>, ''j'' = ''e''<sub>2</sub>,  ''k'' = ''e''<sub>1</sub>''e''<sub>2</sub>} is  the [[division ring]] of Hamilton's [[quaternions]], '''H''' = ''C''&#x2113;<sub>0,2</sub>('''R''')
 
One can therefore see that
:<math>C\ell_{0,3}(\mathbb{R}) = \mathbb{H} \otimes \mathbb{D}</math>
 
where '''D''' = ''C''&#x2113;<sub>1,0</sub>('''R''') is the algebra spanned by {1, &omega;}, the algebra of the [[split-complex number]]s.
 
Equivalently,
:<math>C\ell_{0,3}(\mathbb{R}) = \mathbb{H} \oplus \mathbb{H}.</math>
 
<!--
The idea is to replace the [[complex number]]s used in an ordinary (Hamilton) [[biquaternion]] with [[split-complex number]]s.Thus  q = w + x i + y j + z k , with w, x, y, z &isin; '''D''' is a Clifford biquaternion. Such a number can also be written q = r + s &omega; , r, s &isin; '''H''', &omega;<sup>2</sup> = + 1 , '''H''' the [[division ring]] of Hamilton's [[quaternions]]. -->
 
==Split-biquaternion group==
The split-biquaternions form an [[associativity|associative]] [[ring theory|ring]] as is clear from considering multiplications in its [[basis (linear algebra)|basis]] {1, &omega;, i, j, k, &omega;i, &omega;j, &omega;k,}. When &omega; is adjoined to the [[quaternion group]] one obtains a 16 element group
:( {1, i, j, k, &minus;1, &minus;i, &minus;j, &minus;k, &omega;, &omega;i, &omega;j, &omega;k, &minus;&omega;, &minus;&omega;i, &minus;&omega;j, &minus;&omega;k}, × ).
 
==Direct sum of two quaternion rings==
The direct sum of the division ring of quaternions with itself is denoted <math>\mathbb{H} \oplus \mathbb{H}</math>. The product of two elements <math>(a \oplus b)</math> and <math> (c \oplus d)</math> is <math> a c \oplus b d </math> in this [[direct sum of modules#Direct sum of algebras|direct sum algebra]].
 
'''Proposition:''' The algebra of split-biquaternions is isomorphic to <math>\mathbb{H} \oplus \mathbb{H}.</math>
 
proof: Every split-biquaternion has an expression ''q'' = ''w'' + ''z'' ω  where ''w'' and ''z''  are quaternions and ω<sup>2</sup> = +1. Now if ''p'' = ''u'' + ''v'' ω is another split-biquaternion, their product is
:<math> pq = uw + vz + (uz + vw) \omega .\!</math>
 
The isomorphism mapping from split-biquaternions to <math>\mathbb{H} \oplus \mathbb{H}</math> is given by
:<math>p \mapsto (u + v) \oplus (u - v) , \quad q \mapsto (w + z) \oplus (w - z).</math>
In <math>\mathbb{H} \oplus \mathbb{H}</math>, the product of these images, according to the algebra-product of <math>\mathbb{H} \oplus \mathbb{H}</math> indicated above, is
:<math>(u + v)(w + z) \oplus (u - v)(w - z).</math>
This element is also the image of pq under the mapping into <math>\mathbb{H} \oplus \mathbb{H}.</math>
Thus the products agree, the mapping is a homomorphism; and since it is [[bijective]], it is an isomorphism.
 
Though split-biquaternions form an [[eight-dimensional space]] like Hamilton’s biquaternions, on the basis of the Proposition it is apparent that this algebra splits into the direct sum of two copies of the real quaternions.
 
== Hamilton biquaternion ==
The split-biquaternions should not be confused with the (ordinary) biquaternions previously introduced by [[William Rowan Hamilton]].  Hamilton's [[biquaternion]]s are elements of the algebra
:<math>C\ell_2(\mathbb{C}) = \mathbb{H} \otimes \mathbb{C}.</math>
 
==Synonyms==
The following terms and compounds refer to the split-biquaternion algebra:
* elliptic biquaternions – Clifford (1873), Rooney(2007)
* octonions – [[Alexander MacAulay]] (1898)
* Clifford biquaternion – Joly (1902), van der Waerden (1985)
* dyquaternions – Rosenfeld (1997)
* <math>\mathbb{D} \otimes \mathbb{H}</math> where '''D''' = [[split-complex number]]s – Bourbaki (1994), Rosenfeld (1997)
* <math>\mathbb{H} \oplus \mathbb{H}</math>, the [[direct sum of modules#Direct sum of algebras|direct sum]] of two quaternion algebras – van der Waerden (1985)
 
==See also==
* [[split-octonion]]s
 
==References==
*[[William Kingdon Clifford]] (1873), "Preliminary Sketch of Biquaternions", Paper XX, ''Mathematical Papers'', p.&nbsp;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.
*Joe Rooney (2007) "William Kingdon Clifford", in Marco Ceccarelli, ''Distinguished figures in mechanism and machine science'', Springer.
*[[Charles Jasper Joly]] (1905) ''Manual of Quaternions'', page 21, MacMillan & Co.
*Boris Rosenfeld (1997) ''Geometry of Lie Groups'', page 48, [[Kluwer]] ISBN 0-7923-4390-5 .
*[[Nicolas Bourbaki]] (1994) ''Elements of the History of Mathematics'', J. Meldrum translator, Springer.
*[[B. L. van der Waerden]] (1985) ''A History of Algebra'', page 188, Springer, ISBN 0-387-13610-X .
 
[[Category:Quaternions]]
[[Category:Clifford algebras]]
 
[[de:Biquaternion#Clifford Biquaternion]]

Latest revision as of 00:19, 26 September 2014

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