Eduard Study - Hypercomplex Numbers

Hypercomplex Numbers

In 1898 Eduard Study was the author of an article on hypercomplex numbers in the Klein's encyclopedia. This 34 page article was expanded to 138 pages in 1908 by Élie Cartan, who surveyed the hypercomplex systems in Encyclopédie des sciences mathématiques pures et appliqueés. Cartan acknowledged Eduard Study’s priority in his title with the words "after Eduard Study".

In the 1993 biography of Cartan by Akivis and Rosenfeld, one reads:

defined the algebra °H of ‘semiquaternions’ with the units 1, i, ε, η having the properties

Semiquaternions are often called ‘Study’s quaternions’.

In 1985 Helmut Karzel and Günter Kist developed "Study's quaternions" as the kinematic algebra corresponding to the group of motions of the Euclidean plane. These quaternions arise in "Kinematic algebras and their geometries" alongside ordinary quaternions and the ring of 2 × 2 real matrices which Karzel and Kist cast as the kinematic algebras of the elliptic plane and hyperbolic plane respectively. See the "Motivation and Historical Review" at page 437 of Rings and Geometry, R. Kaya editor.

Thus in the study of classical associative algebras over R there are two special ones: Study’s quaternions (4D) and Study’s biquaternions (8D).