Inabstract algebra, thealgebra ofhyperbolic quaternions is anonassociative algebra over thereal numbers with elements of the form

${\displaystyle q=a+bi+cj+dk,a,b,c,d\in \mathbb{R}}$

where the squares of i, j, and k are +1 and distinct elements of {i, j, k} multiply with theanti-commutative property.

The four-dimensional algebra of hyperbolic quaternions incorporates some of the features of the older and larger algebra ofbiquaternions. They both contain subalgebras isomorphic to thesplit-complex number plane. Furthermore, just as the quaternion algebraH can be viewed as aunion of complex planes, so the hyperbolic quaternion algebra is apencil of planes of split-complex numbers sharing the same real line.

It wasAlexander Macfarlane who promoted this concept in the 1890s as hisAlgebra of Physics, first through theAmerican Association for the Advancement of Science in 1891, then through his 1894 book of fivePapers in Space Analysis, and in a series of lectures atLehigh University in 1900.

Like thequaternions, the set of hyperbolic quaternions form avector space over thereal numbers ofdimension 4. Alinear combination

${\displaystyle q=a+bi+cj+dk}$

is ahyperbolic quaternion when${\displaystyle a,b,c,}$ and${\displaystyle d}$ are real numbers and the basis set${\displaystyle \{1,i,j,k\}}$ has these products:

${\displaystyle ij=k=-ji}$

${\displaystyle jk=i=-kj}$

${\displaystyle ki=j=-ik}$

${\displaystyle i^2=j^2=k^2=+1}$

Using thedistributive property, these relations can be used to multiply any two hyperbolic quaternions.

Unlike the ordinary quaternions, the hyperbolic quaternions are notassociative. For example,${\displaystyle (ij)j=kj=-i}$, while${\displaystyle i(jj)=i}$. In fact, this example shows that the hyperbolic quaternions are not even analternative algebra.

The first three relations show that products of the (non-real) basis elements areanti-commutative. Although this basis set does not form agroup, the set

${\displaystyle \{1,i,j,k,-1,-i,-j,-k\}}$

forms aloop, that is, aquasigroup with an identity element. One also notes that any subplane of the setM of hyperbolic quaternions that contains the real axis forms a plane ofsplit-complex numbers. If

${\displaystyle q^{\ast }=a-bi-cj-dk}$

is the conjugate of${\displaystyle q}$, then the product

${\displaystyle q(q^{\ast })=a^2-b^2-c^2-d^2}$

is thequadratic form used inspacetime theory. In fact, for eventsp andq, thebilinear form

${\displaystyle \eta (p,q)=-p_0{q}_0+p_1{q}_1+p_2{q}_2+p_3{q}_3}$

arises as the negative of the real part of the hyperbolic quaternion productpq*, and is used inMinkowski space.

Note that the set ofunits U = {q :qq* ≠ 0 } isnot closed under multiplication. See the references (external link) for details.

The hyperbolic quaternions form anonassociative ring; the failure ofassociativity in this algebra curtails the facility of this algebra in transformation theory. Nevertheless, this algebra put a focus on analytical kinematics by suggesting amathematical model:When one selects a unit vectorr in the hyperbolic quaternions, thenr² = +1. The plane${\displaystyle D_r=\{t+xr:t,x\in R\}}$ with hyperbolic quaternion multiplication is a commutative and associative subalgebra isomorphic to the split-complex number plane.Thehyperbolic versor${\displaystyle \exp (ar)=\cosh (a)+r\sinh (a)}$ transforms D_r by

Since the directionr in space is arbitrary, this hyperbolic quaternion multiplication can express anyLorentz boost using the parametera calledrapidity. However, the hyperbolic quaternion algebra is deficient for representing the fullLorentz group (seebiquaternion instead).

Writing in 1967 about the dialogue on vector methods in the 1890s, historianMichael J. Crowe 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.^[1]

Later, Macfarlane published an article in theProceedings of the Royal Society of Edinburgh in 1900. In it he treats a model forhyperbolic space H³ on thehyperboloid

${\displaystyle H^3=\{q\in M:q(q^{\ast })=1\}.}$

Thisisotropic model is called thehyperboloid model and consists of all thehyperbolic versors in the ring of hyperbolic quaternions.

The 1890s felt the influence of thecontinuous groups ofSophus Lie. An example of aone-parameter group is thehyperbolic versor with thehyperbolic angle parameter. This parameter is part of thepolar decomposition of a split-complex number. But it is a startling aspect of finite mathematics that makes the hyperbolic quaternion ring different:

The basis${\displaystyle \{1,i,j,k\}}$ of the vector space of hyperbolic quaternions is notclosed under multiplication: for example,${\displaystyle ji=-k}$. Nevertheless, the set${\displaystyle \{1,i,j,k,-1,-i,-j,-k\}}$ is closed under multiplication. It satisfies all the properties of an abstract group except the associativity property; being finite, it is aLatin square orquasigroup, a peripheralmathematical structure. Loss of the associativity property of multiplication as found in quasigroup theory is not consistent withlinear algebra since all linear transformations compose in an associative manner. Yet physical scientists were calling in the 1890s for mutation of the squares of${\displaystyle i}$,${\displaystyle j}$, and${\displaystyle k}$ to be${\displaystyle +1}$ instead of${\displaystyle -1}$ :TheYale University physicistWillard Gibbs had pamphlets with the plus one square in his three-dimensional vector system.Oliver Heaviside in England wrote columns in theElectrician, a trade paper, advocating the positive square. In 1892 he brought his work together inTransactions of the Royal Society A^[2] 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 inconvenientminus 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^[3] 1892)

If a 4-algebra on basis${\displaystyle \{1,i,j,k\}}$ is associative and off-diagonal products are given by Hamilton's rules, then${\displaystyle i^2=-1=j^2=k^2}$.

Proof:

${\displaystyle j=ki=(-ji)i=-j(ii)}$, so${\displaystyle i^2=-1}$. Cycle the letters${\displaystyle i}$,${\displaystyle j}$,${\displaystyle k}$ to obtain${\displaystyle i^2=-1=j^2=k^2}$.QED.

This theorem revealed a flaw inElements of Dynamic (1978) byW. K. Clifford who described products in 2x2 real matrices.^[4] In that system some basis squares are positive identity, yet Clifford had distinct basis elements multiplying as in the quaternion group rather that the symmetry group of a square. The theorem justified resistance to the call of the physicists and theElectrician. The quasigroup stimulated a considerable stir in the 1890s: the journalNature 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 bookA 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.^[1]

In 1899Charles Jasper Joly noted the hyperbolic quaternion and the non-associativity property^[5] while ascribing its origin to Oliver Heaviside.

The hyperbolic quaternions, as theAlgebra of Physics, undercut the claim that ordinary quaternions made on physics. As for mathematics, the hyperbolic quaternion is anotherhypercomplex number, as such structures were called at the time. By the 1890sRichard Dedekind had introduced thering concept into commutative algebra, and thevector space concept was being abstracted byGiuseppe Peano. In 1899Alfred North Whitehead promotedUniversal algebra, advocating for inclusivity. The concepts of quasigroup andalgebra over a field are examples ofmathematical structures describing hyperbolic quaternions.

TheProceedings of the Royal Society of Edinburgh published "Hyperbolic Quaternions" in 1900, a paper in which Macfarlane regains associativity for multiplication by reverting tocomplexified quaternions. While there he used some expressions later made famous byWolfgang Pauli: where Macfarlane wrote

${\displaystyle ij=k\sqrt{-1}}$

${\displaystyle jk=i\sqrt{-1}}$

${\displaystyle ki=j\sqrt{-1},}$

thePauli matrices satisfy

${\displaystyle {\sigma }_1{\sigma }_2={\sigma }_3\sqrt{-1}}$

${\displaystyle {\sigma }_2{\sigma }_3={\sigma }_1\sqrt{-1}}$

${\displaystyle {\sigma }_3{\sigma }_1={\sigma }_2\sqrt{-1}}$

while referring to the same complexified quaternions.

The opening sentence of the paper is "It is well known that quaternions are intimately connected withspherical trigonometry and in fact they reduce the subject to a branch of algebra." This statement may be verified by reference to the contemporary workVector Analysis which works with a reduced quaternion system based ondot product andcross 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 commonMinkowski diagram used today inspecial relativity to discuss change of velocity of a frame of reference andrelativity of simultaneity.

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

• Non-associative algebra
• Historical treatment of quaternions
• Minkowski spacetime

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