Stein factorization

2013-04-06T22:08:36Z

84.133.101.35: /* Proof */

In [[abstract algebra]], the '''biquaternions''' are the numbers {{nowrap|''w'' + ''x'' i + ''y'' j + ''z'' k}}, where ''w'', ''x'', ''y'', and ''z'' are complex numbers and the elements of {{nowrap|{1, i, j, k}<nowiki/>}} multiply as in the [[quaternion group]]. As there are three types of complex number, so there are three types of biquaternion:
* (Ordinary) biquaternions when the coefficients are (ordinary) [[complex number]]s
* [[Split-biquaternion]]s when ''w'', ''x'', ''y'', and ''z'' are [[split-complex number]]s
* [[Dual quaternion]]s when ''w'', ''x'', ''y'', and ''z'' are [[dual numbers]].

This article is about the ordinary biquaternions named by [[William Rowan Hamilton]] in 1844 (see Proceedings of Royal Irish Academy 1844 & 1850 page 388). Some of the more prominent proponents of these biquaternions include [[Alexander Macfarlane]], [[Arthur W. Conway]], [[Ludwik Silberstein]], and [[Cornelius Lanczos]]. As developed below, the [[Hyperboloid#Relation to the sphere|unit quasi-sphere]] of the biquaternions provides a presentation of the [[Lorentz group]], which is the foundation of [[special relativity]].

The algebra of biquaternions can be considered as a [[tensor product of algebras|tensor product]] {{nowrap|'''C''' ⊗ '''H'''}} (taken over the reals) where '''C''' is the [[field (mathematics)|field]] of complex numbers and '''H''' is the algebra of (real) [[quaternions]]. In other words, the biquaternions are just the [[complexification]] of the (real) quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2×2 complex matrices ''M''2('''C'''). They can be classified as the [[Clifford algebra]] {{nowrap|1=Cℓ2('''C''') = Cℓ03('''C''')}}. This is also isomorphic to the [[Pauli algebra]] Cℓ3,0('''R'''), and the even part of the [[spacetime algebra]] Cℓ01,3('''R''').

==Definition==
Let {1, ''i'', ''j'', ''k''} be the basis for the (real) [[quaternion]]s, and let ''u'', ''v'', ''w'', ''x'' be [[complex number]]s, then
:''q'' = ''u'' 1 + ''v'' i + ''w'' j + ''x'' k
is a ''biquaternion''.<ref>Hamilton (1853) page 639</ref>
To distinguish square roots of minus one in the biquaternions, Hamilton<ref>Hamilton (1853) page 730</ref><ref>Hamilton (1899) ''Elements of Quaternions'', 2nd edition, page 289</ref> and [[Arthur W. Conway]] used the convention of representing the square root of minus one in the scalar field '''C''' by h since there is an i in the [[quaternion group]]. Then
: h i = i h, h j = j h, and h k = k h since h is a scalar.
Hamilton's primary exposition on biquaternions came in 1853 in his ''Lectures on Quaternions'', now available in the ''Historical Mathematical Monographs'' of Cornell University. The two editions of ''Elements of Quaternions'' (1866 & 1899) reduced the biquaternion coverage in favor of the real quaternions. He introduced the terms [[bivector (complex)|bivector]], ''biconjugate, bitensor'', and ''biversor''.

Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a 4-dimensional [[algebra over a field|algebra]] over the complex numbers. The algebra of biquaternions is [[associative]], but not [[commutative]]. A biquaternion is either a [[unit (ring theory)|unit]] or a [[zero divisor]].

==Place in ring theory==

===Linear representation===
Note the matrix product
:<math>\begin{pmatrix}h & 0\\0 & -h\end{pmatrix}\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}</math> = <math>\begin{pmatrix}0 & h\\h & 0\end{pmatrix}</math>
where each of these three arrays has a square equal to the negative of the [[identity matrix]].
When this matrix product is interpreted as i j = k, then one obtains a [[subgroup]] of the matrix group that is [[isomorphism|isomorphic]] to the [[quaternion group]]. Consequently
:<math>\begin{pmatrix}u+hv & w+hx\\-w+hx & u-hv\end{pmatrix}</math> represents biquaternion ''q'' = ''u'' 1 + ''v'' i + ''w'' j + ''x'' k.
Given any 2 × 2 complex matrix, there are complex values ''u'', ''v'', ''w'', and ''x'' to put it in this form so that the [[matrix ring]] is isomorphic<ref>[[Leonard Dickson]] (1914) ''Linear Algebras'', §13 "Equivalence of the complex quaternion and matric algebras", page 13</ref> to the biquaternion [[ring (mathematics)|ring]].

===Subalgebras===
Considering the biquaternion algebra over the scalar field of real numbers '''R''', the set
{1, h, i, hi, j, hj, k, hk } forms a [[basis (linear algebra)|basis]] so the algebra has eight real [[dimension]]s.
Note the squares of the elements hi, hj, and hk are all plus one, for example,
:<math>(hi)^2 \ = \ h^2 i^2 \ = \ (-1) (-1) \ =\ +1 .</math>
Then the [[subalgebra]] given by
<math>\lbrace x + y(hi) : x, y \in R \rbrace </math> is [[ring isomorphism|ring isomorphic]] to the plane of [[split-complex number]]s, which has an algebraic structure built upon the [[unit hyperbola]]. The elements ''hj'' and ''hk'' also determine such subalgebras. Furthermore,
<math>\lbrace x + yj : x,y \in C \rbrace </math> is a subalgebra isomorphic to the [[tessarine]]s.

A third subalgebra called [[coquaternion]]s is generated by hj and hk. First note that
(hj)(hk) = (−1) i, and that the square of this element is −1. These elements generate the [[dihedral group]] of the square. The [[linear subspace]] with basis {1, i, hj, hk} thus is closed under multiplication, and forms the coquaternion algebra.

In the context of [[quantum mechanics]] and [[spinor]] algebra, the biquaternions hi, hj, and hk (or their negatives), viewed in the M(2,C) representation, are called [[Pauli matrices]].

==Algebraic properties==
The biquaternions have two ''conjugations'':
* the '''biconjugate''' or biscalar minus [[bivector (complex)|bivector]] is <math>q^* = w - xi - yj - zk \!\ ,</math> and
* the [[complex conjugation]] of biquaternion coefficients <math>q^{\star} = w^{\star} + x^{\star} i + y^{\star} j + z^{\star} k \!</math>
where <math>z^{\star} = a - bh</math> when <math>z = a + bh,\quad a,b \in R,\quad h^2 = -1.</math>

Note that <math>(pq)^* = q^* p^*, \quad (pq)^{\star} = p^{\star} q^{\star} , \quad (q^*)^{\star} = (q^{\star})^*.</math>

Clearly, if <math>q q^* = 0 \!</math> then ''q'' is a zero divisor. Otherwise <math>\lbrace q q^* \rbrace^{-1} \!</math> is defined over the complex numbers. Further, <math>q q^* = q^* q \!</math> is easily verified. This allows an inverse to be defined as follows:
* <math>q^{-1} = q^* \lbrace q q^* \rbrace^{-1}\!</math>, iff <math>qq^* \neq 0.</math>

===Relation to Lorentz transformations===
Consider now the linear subspace
<ref>Lanczos (1949) Equation 94.16 page 305. The following algebra compares to Lanczos, except he uses ~ to signify quaternion conjugation and * for complex conjugation</ref>
:<math>M = \lbrace q : q^* = q^{\star} \rbrace = \lbrace t + x(hi) + y(hj) + z(hk) : t, x, y, z \in R \rbrace .</math>
''M'' is not a subalgebra since it is not [[closure (mathematics)|closed under products]]; for example <math>(hi)(hj) = h^2 ij = -k \notin M.</math>. Indeed, ''M'' cannot form an algebra if it is not even a [[magma (algebra)|magma]].

'''Proposition:''' If ''q'' is in ''M'', then <math>q q^* = t^2 - x^2 - y^2 - z^2 \!</math>.

proof: <math>q q^* = (t+xhi+yhj+zhk)(t-xhi-yhj-zhk) \!</math>
:<math> = t^2 - x^2(hi)^2 - y^2(hj)^2 - z^2(hk)^2 = t^2 - x^2 - y^2 - z^2. \!</math>

'''Definition:''' Let biquaternion ''g'' satisfy ''g g'' * = 1. Then the '''Lorentz transformation''' associated with ''g'' is given by
:<math>T(q) = g^* q g^{\star}.</math>

'''Proposition:''' If ''q'' is in ''M'', then ''T(q)'' is also in ''M''.

proof: <math>(g^* q g^{\star})^* = (g^{\star})^* q^* g = (g^*)^{\star} q^{\star} g = (g^* q g^{\star})^{\star}.</math>

'''Proposition:''' <math>\quad T(q) (T(q))^* = q q^* \!</math>

proof: Note first that ''g g'' * = 1 means that the sum of the squares of its four complex components is one. Then the sum of the squares of the ''complex conjugates'' of these components is also one. Therefore, <math>g^{\star} (g^{\star})^* = 1.</math> Now
:<math>(g^* q g^{\star})(g^* q g^{\star})^* = g^* q g^{\star} (g^{\star})^* q^* g = g^* q q^* g = q q^*.</math>

==Associated terminology==
As the biquaternions have been a fixture of [[linear algebra]] since the beginnings of [[mathematical physics]], there is an array of concepts that are illustrated or represented by biquaternion algebra. The [[transformation group]] <math>G = \lbrace g : g g^* = 1 \rbrace \!</math> has two parts, <math>G \cap H</math> and <math>G \cap M.</math> The first part is characterized by <math>g = g^{\star}</math> ; then the Lorentz transformation corresponding to ''g'' is given by <math>T(q) = g^{-1} q g \!</math> since <math>g^* = g^{-1}. \!</math> Such a transformation is a [[quaternions and spatial rotation|rotation by quaternion multiplication]], and the collection of them is [[O(3)]] <math>\cong G \cap H .</math> But this subgroup of ''G'' is not a [[normal subgroup]], so no [[quotient group]] can be formed.

To view <math>G \cap M</math> it is necessary to show some subalgebra structure in the biquaternions. Let ''r'' represent an element of the [[quaternion#Square roots of −1|sphere of square roots of minus one]] in the real quaternion subalgebra '''H'''. Then (''hr'')2 = +1 and the plane of biquaternions given by <math>D_r = \lbrace z = x + yhr : x, y \in R \rbrace</math> is a commutative subalgebra isomorphic to the plane of [[split-complex number]]s. Just as the ordinary complex plane has a unit circle, <math>D_r \!</math> has a [[unit hyperbola]] given by
:<math>exp(ahr) = \cosh(a) + hr\ \sinh(a),\quad a \in R. \!</math>
Just as the unit circle turns by multiplication through one of its elements, so the hyperbola turns because <math>\exp(ahr) \exp(bhr) = \exp((a+b)hr). \!</math> Hence these algebraic operators on the hyperbola are called [[versor#Hyperbolic versor|hyperbolic versors]]. The unit circle in '''C''' and unit hyperbola in ''D''''r'' are examples of [[one-parameter group]]s. For every square root ''r'' of minus one in '''H''', there is a one-parameter group in the biquaternions given by <math>G \cap D_r.</math>

The space of biquaternions has a natural [[topology]] through the [[Euclidean metric]] on 8-space. With respect to this topology, ''G'' is a [[topological group]]. Moreover, it has analytic structure making it a six-parameter [[Lie group]]. Consider the subspace of [[bivector (complex)|bivector]]s <math>A = \lbrace q : q^* = -q \rbrace \!</math>. Then the [[exponential map]]
<math>\exp:A \to G</math> takes the real vectors to <math>G \cap H</math> and the ''h''-vectors to <math>G \cap M.</math> When equipped with the [[commutator]], ''A'' forms the [[Lie algebra]] of ''G''. Thus this study of a [[six-dimensional space]] serves to introduce the general concepts of [[Lie theory]]. When viewed in the matrix representation, ''G'' is called the [[special linear group]] [[SL(2,C)]] in M(2,C).

Many of the concepts of [[special relativity]] are illustrated through the biquaternion structures laid out. The subspace ''M'' corresponds to [[Minkowski space]], with the four coordinates giving the time and space locations of events in a resting [[frame of reference]]. Any hyperbolic versor exp(''ahr'') corresponds to a [[velocity]] in direction ''r'' of speed ''c'' tanh ''a'' where ''c'' is the [[velocity of light]]. The inertial frame of reference of this velocity can be made the resting frame by applying the [[Lorentz boost]] ''T'' given by ''g'' = exp(0.5''ahr'') since then <math>g^{\star} = \exp(-0.5ahr) = g^*</math> so that <math>T(\exp(ahr)) = 1 .</math>
Naturally the [[hyperboloid]]
<math>G \cap M,</math> which represents the range of velocities for sub-luminal motion, is of physical interest. There has been considerable work associating this "velocity space" with the [[hyperboloid model]] of [[hyperbolic geometry]]. In special relativity, the [[hyperbolic angle]] parameter of a hyperbolic versor is called [[rapidity]]. Thus we see the biquaternion group ''G'' provides a [[group representation]] for the [[Lorentz group]].

After the introduction of [[spinor]] theory, particularly in the hands of [[Wolfgang Pauli]] and [[Élie Cartan]], the biquaternion representation of the Lorentz group was superseded. The new methods were founded on [[basis (linear algebra)|basis vectors]] in the set
:<math>\lbrace q \ :\ q q^* = 0 \rbrace = \lbrace w + xi + yj + zk \ :\ w^2 + x^2 + y^2 + z^2 = 0 \rbrace </math>
which is called the "complex light cone".

==See also==

*[[Conic octonion]]s (isomorphism)
*[[Hyperbolic quaternion#MacFarlane's hyperbolic quaternion paper of 1900|MacFarlane's use]]
*[[Quotient ring#Quaternions and alternatives|Quotient ring]]
*[[Hypercomplex number]]

==Notes==
{{reflist|2}}

==References==
* Proceedings of the Royal Irish academy November 1844 (NA) and 1850 page 388 from google books [http://books.google.com/books?id=ggoFAAAAQAAJ&pg=PA388&dq=proceedings+of+royal+irish+academy+1844+Hamilton&hl=en&ei=WysiTPLwMcKRnwepmoDBDw&sa=X&oi=book_result&ct=result&resnum=5&ved=0CD4Q6AEwBA]
* Arthur Buchheim (1885) [http://www.jstor.org/stable/2369176 "A Memoir on biquaternions"], [[American Journal of Mathematics]] 7(4):293 to 326 from [[Jstor]] early content.
*{{citation|first=Arthur W.|last=Conway|authorlink=Arthur W. Conway|year=1911|title=On the application of quaternions to some recent developments in electrical theory|journal=Proceedings of the Royal Irish Academy|volume=29A|pages=1–9}}.
* [[William Rowan Hamilton]] (1853) ''Lectures on Quaternions'', Article 669. This historical mathematical text is available on-line courtesy of [http://historical.library.cornell.edu/math/ Cornell University].
*Hamilton (1866) ''[http://books.google.com/books?id=fIRAAAAAIAAJ Elements of Quaternions]'' [[University of Dublin]] Press. Edited by William Edwin Hamilton, son of the deceased author.
*Hamilton (1899) ''Elements of Quaternions'' volume I, (1901) volume II. Edited by [[Charles Jasper Joly]]; published by [[Longmans, Green & Co.]].
*Kravchenko, Vladislav (2003), ''Applied Quaternionic Analysis'', Heldermann Verlag ISBN 3-88538-228-8.
*{{citation|first=Cornelius|last=Lanczos|authorlink=Cornelius Lanczos|year=1949|title=The Variational Principles of Mechanics|publisher=University of Toronto Press|pages=304–312}}.
*{{citation|first=Ludwik|last=Silberstein|authorlink=Ludwik Silberstein|year=1912|title=Quaternionic form of relativity|journal=Philosophy Magazine|series=Series 6|volume=23|pages=790–809}}.
*{{citation|first=Ludwik|last=Silberstein|authorlink=Ludwik Silberstein|title=The Theory of Relativity|year=1914}}.
*{{citation|last=Synge|first=J. L.|year=1972|title=Quaternions, Lorentz transformations, and the Conway-Dirac-Eddington matrices|journal=Communications of the Dublin Institute for Advanced Studies|series=Series A|volume=21}}.
*{{citation|first=P. R.|last=Girard|year=1984|title=The quaternion group and modern physics|journal=[[European Journal of Physics]]|volume=5|pages=25–32|bibcode = 1984EJPh....5...25G |doi = 10.1088/0143-0807/5/1/007 }}.
*{{citation|last=Kilmister|first=C. W.|year=1994|title=Eddington's search for a fundamental theory|publisher=Cambridge University Press|isbn=0-521-37165-1|pages=121, 122, 179, 180}}.
*{{citation|first1=Stephen J.|last1=Sangwine|first2=Todd A.|last2=Ell|first3=Nicolas|last3=Le Bihan|year=2010|arxiv=1001.0240|title=Fundamental representations and algebraic properties of biquaternions or complexified quaternions|journal=[[Advances in Applied Clifford Algebras]]|pages=1–30|doi=10.1007/s00006-010-0263-3}}.
*{{citation|first1=Stephen J.|last1=Sangwine|first2=Daniel|last2=Alfsmann|year=2010|arxiv=0812.1102|title=Determination of the biquaternion divisors of zero, including idempotents and nilpotents|journal=[[Advances in Applied Clifford Algebras]]|volume=20|issue=2|pages=401–410|bibcode = 2008arXiv0812.1102S }}.
*{{citation|first=M.|last=Tanişli|year=2006|year=Gauge transformation and electromagnetism with biquaternions|journal=[[Europhysics Letters]]|volume=74|issue=4|page=569}}.

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