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BiquaternionFrom Wikipedia, the free encyclopedia

In abstract algebra, the biquaternions are the numbers where w,x, y, and zare complex numbers and the elements of {1, i,j, k}

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 numbers split-biquaternions when w,x, y, and zare split-complex numbers Study biquaternions or dual quaternions when w,x, y, and zare dual numbers.

The following 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 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 CH (taken over the reals) where C is the 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 22 complex matrices M2(C). In terms of Clifford algebra they can be classified as C2(C) = C03(C). This

is also isomorphic to the Pauli algebra C3,0(R), and the even part of the space-time algebra C01,3(R).

Contents

1 Definition 2 Place in ring theory

2.1 Linear representation 2.2 Subalgebras

3 Algebraic properties 3.1 Relation to Lorentz Transformations

4 Associated terminology 5 See also 6 Notes 7 References

Definition

Let {1, i,j, k} be the basis for the (real) quaternions, and let u, v, w,xbe complex numbers, then

q = u 1 + v i+ w j+x k

is a biquaternion.[1] To distinguish square roots of minus one in the biquaternions, Hamilton[2][3] 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 iin the quaternion group. Then

hi = ih, hj = jh, and hk = kh 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 ofElements of Quaternions (1866 & 1899) reduced the biquaternion coverage in favor of the real quaternions. He

introduced the terms 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 the complex numbers. The algebra of biquaternions is associative, but not commutative. A biquaternion is either a unit or a zero divisor.

Place in ring theory

Linear representation

Note the matrix product

=

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 isomorphic to the quaternion group. Consequently

represents biquaternion q = = u 1 + v i+ w j+x k.

Given any 2 2 complex matrix, there are complex values u, v, w, andxto put it in this form so that the matrix ring is isomorphic[4] to the biquaternion

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 so the algebra has eight real

dimensions. Note the squares of the elements hi, hj, and hkare all plus one, for example,

Then the subalgebra given by is ring isomorphic to the plane of split-complex numbers, which has an algebraic structure built

upon the unit hyperbola. The elements hjand hkalso determine such subalgebras. Furthermore, is a subalgebra isomorphic to

the tessarines.

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A third subalgebra called coquaternions is generated by hjand 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 quaternion conjugation , and

the complex conjugation of quaternion coefficients

where when

Note that

Clearly, if then q is a zero divisor. Otherwise is defined over the complex numbers. Further, is easily verified. This allows

an inverse to be defined as follows:

, iff

Relation to Lorentz Transformations

Consider now the linear subspace [5]

Mis not a subalgebra since it is not closed under products; for example . Indeed, Mcannot form an algebra if it is not

even a magma.

Proposition: Ifq is in M, then .

proof:

Definition: Let biquaternion gsatisfy g g* = 1. Then the Lorentz transformation associated with gis given by

Proposition: Ifq is in M, then T(q) is also in M.

proof:

Proposition:

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, Now

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 t ransformation group has two parts, and The fi rst part is

characterized by ; then the Lorentz transformation corresponding to gis given by since Such a transformation is a

rotation by quaternion multiplication, and the collection of them is O(3) But this subgroup ofG is not a normal subgroup, so no quotient group

can be formed.

To view it is necessary to show some subalgebra structure in the biquaternions. Let rrepresent an element of the sphere of square roots of

minus one in the real quaternion subalgebra H. Then (hr)2 = +1 and the plane of biquaternions given by is a

commutative subalgebra isomorphic to the plane of split-complex numbers. Just as the ordinary complex plane has a unit circle, has a unit hyperbola

given by

Just as the unit circle turns by multiplication through one of its elements, so the hyperbola turns because

Hence these algebraic operators on the hyperbola are called hyperbolic versors. The unit circle in C and unit hyperbola in Drare examples of one-

parameter groups. For every square root rof minus one in H, there is a one-parameter group in the biquaternions given by

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 vector biquaternions . Then the

exponential map takes the real vectors to and the h-vectors to When equipped with the commutator,A forms the Lie

algebra ofG. 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 Mcorresponds 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 avelocity in direction rof speed ctanh a where cis the velocity of light. The inertial f rame of reference of this velocity can be made the resting frame by

applying the Lorentz boost Tgiven by g= exp(0.5ahr) since then so that Naturally the hyperboloid

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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 calledrapidity. 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 lie Cartan, the biquaternion representation of the Lorentz groupwas superseded. The new methods were founded on basis vectors in the set

which is called the "complex light cone".

See also

Conic octonions (isomorphism) MacFarlane's use Quotient ring Hypercomplex number

Notes

1. ^ Hamilton (1853) page 6392. ^ Hamilton (1853) page 7303. ^ Hamilton (1899) Elements of Quaternions, 2nd edition, page 2894. ^ Leonard Dickson (1914) Linear Algebras, 13 "Equivalence of the complex quaternion and matric algebras", page 135. ^ Lanczos (1949) Equation 94.16 page 305. The following algebra compares to Lanczos, except he uses ~ to signify quaternion conjugation and * for complex

conjugation

References Proceedings of the Royal Irish academy November 1844 (NA) and 1850 page 388 from google books [1] (http://books.google.com/books?

id=ggoFAAAAQAAJ&pg=PA388&dq=proceedings+of+royal+irish+academy+1844+Hamilton&hl=en&ei=WysiTPLwMcKRnwepmoDBDw&sa=X&oi= Conway, Arthur W. (1911), "On the application of quaternions to some recent developments in electrical theory", Proceedings of the Royal Irish

Academy29A: 19. William Rowan Hamilton (1853) Lectures on Quaternions, Article 669. This historical mathematical text is available on-line courtesy of Cornell

University (http://historical.library.cornell.edu/math/) . Hamilton (1866) Elements of Quaternions (http://books.google.com/books?id=fIRAAAAAIAAJ) 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. Lanczos, Cornelius (1949), The Variational Principles of Mechanics, University of Toronto Press, pp. 304312. Silberstein, Ludwik (1912), "Quaternionic form of relativity", Philosophy Magazine, Series 6 23: 790809. Silberstein, Ludwik (1914), The Theory of Relativity. Synge, J. L. (1972), "Quaternions, Lorentz transformations, and the Conway-Dirac-Eddington matrices", Communications of the Dublin Institute for

Advanced Studies, Series A 21. Girard, P. R. (1984), "The quaternion group and modern physics", European Journal of Physics 5: 2532, Bibcode 1984EJPh....5...25G

(http://adsabs.harvard.edu/abs/1984EJPh....5...25G) , doi:10.1088/0143-0807/5/1/007 (http://dx.doi.org/10.1088%2F0143-0807%2F5%2F1%2F007) . Kilmister, C. W. (1994), Eddington's search for a fundamental theory, Cambridge University Press, pp. 121, 122, 179, 180, ISBN 0-521-37165-1. Sangwine, Stephen J.; Ell, Todd A.; Le Bihan, Nicolas (2010), "Fundamental representations and algebraic properties of biquaternions or

complexified quaternions",Advances in Applied Clifford Algebras: 130, arXiv:1001.0240 (http://arxiv.org/abs/1001.0240) , doi:10.1007/s00006-010-0263-3 (http://dx.doi.org/10.1007%2Fs00006-010-0263-3) .

Sangwine, Stephen J.; Alfsmann, Daniel (2010), "Determination of the biquaternion divisors of zero, including idempotents and nilpotents",Advances in Applied Clifford Algebras 20 (2): 401410, arXiv:0812.1102 (http://arxiv.org/abs/0812.1102) , Bibcode 2008arXiv0812.1102S(http://adsabs.harvard.edu/abs/2008arXiv0812.1102S) .

Tanili, M. (Gauge transformation and electromagnetism with biquaternions), Europhysics Letters 74 (4): 569.

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