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In [[abstract algebra]], the [[algebra over a field|algebra]] of '''hyperbolic quaternions''' is a [[nonassociative algebra]] over the [[real numbers]] with elements of the form
:<math>q = a + bi + cj + dk, \quad a,b,c,d \in \Bbb{R} \!</math>
where multiplication is determined with rules that are similar to (but different from) multiplication in the [[quaternions]].

The four-dimensional algebra of hyperbolic quaternions incorporates some of the features of the older and larger algebra of [[biquaternion]]s. They both contain subalgebras isomorphic to the [[split-complex number]] plane. Furthermore, just as the quaternion algebra '''H''' can be viewed as a [[quaternion#H as a union of complex planes|union of complex planes]], so the hyperbolic quaternion algebra is a union of split-complex number planes sharing the same [[real line]].

It was [[Alexander Macfarlane]] who promoted this concept in the 1890s as his '''Algebra of Physics''', first through the [[American Association for the Advancement of Science]] in 1891, then through his 1894 book of five ''Papers in Space Analysis'', and in a series of lectures at [[Lehigh University]] in 1900 (see Historical Review below).

==Algebraic structure==
Like the [[quaternions]], the set of hyperbolic quaternions form a [[vector space]] over the [[real numbers]] of [[dimension]] 4.  A [[linear combination]] 

:<math>q = a+bi+cj+dk</math> 

is a '''hyperbolic quaternion''' when  <math>a, b, c,</math> and <math>d</math> are real numbers and the basis set <math>\{1,i,j,k\}</math> has these products: 

:<math>ij=k=-ji</math>
:<math>jk=i=-kj</math>
:<math>ki=j=-ik</math>
:<math>i^2=+1=j^2=k^2</math>         

Using the [[distributive property]], these relations can be used to multiply any two hyperbolic quaternions.

Unlike the ordinary quaternions, the hyperbolic quaternions are not [[associative]]. For example, <math>(ij)j = kj = -i</math>, while <math>i(jj) = i</math>. In fact, this example shows that the hyperbolic quaternions are not even an [[alternative algebra]].

The first three relations show that products of the (non-real) basis elements are ''anti-commutative''.  Although this basis set does not form a [[group (mathematics)|group]], the set 

:<math>\{1,i,j,k,-1,-i,-j,-k\}</math>

forms a [[quasigroup]]. One also notes that any subplane of the set ''M'' of hyperbolic quaternions that contains the real axis forms a plane of [[split-complex number]]s. If 

:<math>q^*=a-bi-cj-dk</math>  

is the conjugate of <math>q</math>, then the product 

:<math>q(q^*)=a^2-b^2-c^2-d^2</math>

is the [[quadratic form]] used in spacetime theory.
In fact, the [[bilinear form]] called the [[Minkowski space#The Minkowski inner product|Minkowski inner product]] arises as the negative of the real part of the hyperbolic quaternion product ''pq''* :
:<math> -p_0q_0 + p_1q_1 + p_2q_2 + p_3q_3</math>.
Note that the set of [[unit (ring theory)|units]] U = {''q'' : ''qq''* ≠ 0 } is not closed under multiplication. See the references (external link) for details.

==Discussion==

The hyperbolic quaternions form a [[nonassociative ring]]; the failure of [[associativity]] in this algebra curtails the facility of this algebra in transformation theory. Nevertheless, 
this algebra put a focus on analytical kinematics by suggesting a [[mathematical model]]:
When one selects a unit vector ''r'' in the hyperbolic quaternions, then ''r'' <sup>2</sup> = +1. The plane <math>D_r = \lbrace t + x r : t, x \in R \rbrace </math> with hyperbolic quaternion multiplication is a commutative and associative subalgebra isomorphic to the split-complex number plane.
The [[versor#Hyperbolic versor|hyperbolic versor]] <math>\exp(a r) = \cosh(a)  + r \ \sinh(a) </math> transforms D<sub>r</sub> by
:<math> t + x r \quad \mapsto \quad \exp(a r) (t + x r) = </math>
:<math> (\cosh(a) t + x \sinh(a)) + (\sinh(a) t + x \cosh(a)) r . \!</math>
Since the direction ''r'' in space is arbitrary, this hyperbolic quaternion multiplication can express any [[Lorentz boost]] using the parameter ''a'' called [[rapidity]]. However, the hyperbolic quaternion algebra is deficient for representing the full [[Lorentz group]] (see [[biquaternion]] instead).

Writing in 1967 about the dialogue on vector methods in the 1890s, a historian commented
:''The introduction of another system of vector analysis, even a sort of compromise system such as Macfarlane's, could scarcely be well received by the advocates of the already existing systems and moreover probably acted to broaden the question beyond the comprehension of the as-yet uninitiated reader.''<ref>Crowe (1967) 191</ref>

==Geometry==
Later, Macfarlane published an article in the ''Proceedings of the Royal Society at Edinburgh'' in 1900. In it he treats a model for [[hyperbolic space]] H<sup>3</sup> on the [[hyperboloid]]  

:<math>H^3 = \{ q \in M: q(q^*)=1 \} \!</math>.

This [[isotropic]] model is called the [[hyperboloid model]] and consists of all the [[versor#Hyperbolic_versor|hyperbolic versors]] in the ring of hyperbolic quaternions.

==Historical review==<!-- This section is linked from [[Quasigroup]] -->
The 1890s felt the influence of the posthumous publications of [[William Kingdon Clifford|W. K. Clifford]] and the ''continuous groups'' of [[Sophus Lie]]. An example of a [[one-parameter group]] is the [[versor#Hyperbolic versor|hyperbolic versor]] with the [[hyperbolic angle]] parameter. This parameter is part of the [[polar decomposition#Alternative planar decompositions|polar decomposition]] of a split-complex number. But it is a startling aspect of finite mathematics that makes the hyperbolic quaternion ring different:

The basis <math>\{1,\,i,\,j,\,k\}</math> of the vector space of hyperbolic quaternions is not [[closure (mathematics)|closed]] under multiplication: for example, <math>ji=-\!k</math>. Nevertheless, the set <math>\{1,\,i,\,j,\,k,\,-\!1,\,-\!i,\,-\!j,\,-\!k\}</math> is closed under multiplication. It satisfies all the properties of an abstract group except the associativity property; being finite, it is a [[latin square]] or [[quasigroup]], a peripheral [[mathematical structure]]. Loss of  the associativity property of multiplication as found in quasigroup theory is not consistent with [[linear algebra]] since all linear transformations compose in an associative manner. Yet physical scientists were calling in the 1890s for mutation of the squares of <math>i</math>,<math>j</math>, and <math>k</math> to be <math>+1</math> instead of <math>-1</math> :
The [[Yale University]] physicist [[Willard Gibbs]] had pamphlets with the plus one square in his three-dimensional vector system. [[Oliver Heaviside]] in England wrote columns in the ''Electrician'', a trade paper, advocating the positive square. In 1892 he brought his work together in ''Transactions of the Royal Society A''<ref>See pages 427 to 430 of volume 183</ref> where he says his vector system is
:simply the elements of Quaternions without quaternions, with the notation simplified to the uttermost, and with the very inconvenient ''minus'' sign before scalar product done away with.

So the appearance of Macfarlane's hyperbolic quaternions had some motivation, but the disagreeable non-associativity precipitated a reaction. [[Cargill Gilston Knott]] was moved to offer the following:

'''Theorem''' (Knott<ref>C.G. Knott (1892) "Recent Innovations in Vector Theory" ''Proceedings of the Royal Society in Edinburgh'' and ''Nature'' 47:590&ndash;3</ref> 1893)
:If a 4-algebra on basis <math>\{1,\,i,\,j,\,k\}</math> is associative and off-diagonal products are given by Hamilton's rules, then <math>i^2=-\!1=j^2=k^2</math>.

'''Proof:'''
:<math>j = ki = (-ji)i = -j(ii)</math>, so <math>i^2 = -1</math>. Cycle the letters <math>i</math>, <math>j</math>, <math>k</math> to obtain <math>i^2=-1=j^2=k^2</math>. ''QED''.

This theorem needed statement to justify resistance to the call of the physicists and the ''Electrician''. The quasigroup stimulated a considerable stir in the 1890s: the journal ''[[Nature (journal)|Nature]]'' was especially conducive to an exhibit of what was known by giving two digests of Knott's work as well as those of several other vector theorists. Michael J. Crowe devotes chapter six of his book ''[[A History of Vector Analysis]]'' to the various published views, and notes the hyperbolic quaternion:

:''Macfarlane constructed a new system of vector analysis more in harmony with Gibbs–Heaviside system than with the quaternion system. ...he...defined a full product of two vectors which was comparable to the full quaternion product except that the scalar part was positive, not negative as in the older system.''<ref>Crowe (1967) 191</ref>

In 1899 [[Charles Jasper Joly]] noted the hyperbolic quaternion and the non-associativity property<ref>Hamilton (1899) ''Elements of Quaternions'',2nd edition, edited by C.J. Joly; page 163</ref> while ascribing its origin to Oliver Heaviside.

The hyperbolic quaternions, as the ''Algebra of Physics'', undercut the claim that ordinary quaternions made on physics. As for mathematics, the hyperbolic quaternion is another [[hypercomplex number]], as such structures were called at the time. By the 1890s [[Richard Dedekind]] had introduced the [[ring (mathematics)|ring]] concept into commutative algebra, and the [[vector space]] concept was being abstracted by Peano. In 1899 [[Alfred North Whitehead]] promoted [[Universal algebra]], advocating for inclusivity. The concepts of quasigroup and [[algebra over a field]] are examples of [[mathematical structure]]s describing hyperbolic quaternions.

==MacFarlane's hyperbolic quaternion paper of 1900==
The ''Proceedings of the Royal Society at Edinburgh'' published "Hyperbolic Quaternions" 
in 1900, a paper in which Macfarlane regains associativity for multiplication by reverting 
to [[biquaternion|complexified quaternions]]. While there he used some expressions later 
made famous by [[Wolfgang Pauli]]: where Macfarlane wrote
:<math>ij=k\sqrt{-1}</math> 
:<math>jk=i\sqrt{-1}</math>
:<math>ki=j\sqrt{-1}</math>,
the [[Pauli matrices]] satisfy
:<math>\sigma_1\sigma_2=\sigma_3\sqrt{-1}</math>
:<math>\sigma_2\sigma_3=\sigma_1\sqrt{-1}</math>
:<math>\sigma_3\sigma_1=\sigma_2\sqrt{-1}</math>
while referring to the same complexified quaternions.

The opening sentence of the paper is "It is well known that quaternions are intimately connected with [[spherical trigonometry]] and in fact they reduce the subject to a branch of algebra." This statement may be verified by reference to the contemporary work ''[[Vector Analysis]]'' which works with a reduced quaternion system based on [[dot product]] and [[cross product]]. In Macfarlane's paper there is an effort to produce "trigonometry on the surface of the equilateral hyperboloids" through the algebra of hyperbolic quaternions, now re-identified in an associative ring of eight real dimensions. The effort is reinforced by a plate of nine figures on page 181.  They illustrate the descriptive power of his "space analysis" method. For example, figure 7 is the 
common [[Minkowski diagram]] used today in [[special relativity]] to discuss change of velocity of a frame of reference and [[relativity of simultaneity]].

On page 173 Macfarlane expands on his greater theory of quaternion variables. By way of contrast he notes that [[Felix Klein]] appears not to look beyond the theory of [[Quaternions and spatial rotation]].

==References==
{{reflist}}
*A. Macfarlane (1891) "Principles of the Algebra of Physics" ''Proceedings of the American Association for the Advancement of Science'' 40:65–117.
* A. Macfarlane (1894) ''[http://www.archive.org/details/principlesalgeb01macfgoog Papers on Space Analysis]'', B. Westerman, New York, weblink from [[archive.org]]. See paper 2, "The Imaginary of the Algebra".
* A. Macfarlane (1900a) ''[http://digital.lib.lehigh.edu/cdm4/eb_viewer.php?DMTHUMB=1&ptr=1088 Space-Analysis: a brief of twelve lectures]'', delivered at College Hall, [[Lehigh University]], February 5 to March 2.
*A. Macfarlane (1900b) "Hyperbolic Quaternions" ''Proceedings of the Royal Society at Edinburgh'', 1899–1900 session, pp. 169&ndash;181.
*J.W. Gibbs and E.B. Wilson (1901) ''[[Vector Analysis]]'', Yale.
* [[Oliver Heaviside]] (1892) "On the forces, stresses, and fluxes of energy in the electromagnetic field", ''Philosophical Transactions of the Royal Society of London'' A 183:423&ndash;80.
*M.J. Crowe (1967) ''[[A History of Vector Analysis]]'', University of Notre Dame
* [http://www.webcitation.org/5koDjWVLq Alexander Macfarlane and the Ring of Hyperbolic Quaternions]


[[Category:Non-associative algebra]]
[[Category:Quaternions]]
[[Category:Minkowski spacetime]]