Hyperbolic Quaternion

In the abstract algebra of algebras over a field, the hyperbolic quaternion

is a mutated quaternion where

instead of the usual −1.

The four-dimensional algebra of hyperbolic quaternions incorporates some of the features of the older and larger algebra of biquaternions. They both contain subalgebras isomorphic to the split complex number plane. Furthermore, just as the quaternion algebra H can be viewed as a 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).

The hyperbolic quaternions form a non-associative 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 2 = +1. The plane with hyperbolic quaternion multiplication is a commutative and associative subalgebra isomorphic to the split-complex number plane. The hyperbolic versor transforms D_r by

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.