The paper under review is a survey, addressed to a wide audience of mathematicians and physicists, on the study of Hamiltonian quaternions, namely, the associative division algebra \(\mathbb{H}\) over the field \(\mathbb{R}\) of real numbers, with a basis \(1,i,j,k\), and multiplication defined by the relations \(i^2=j^2=k^2=ijk=-1\). This earliest noncommutative number system was introduced by Sir W. Hamilton in October 1843.
The survey is presented in seven sections, in which the author introduces the reader both to the historical and the mathematical aspects of the research in this area. He sheds light on the role of Euler and Gauss as predecessors of Hamilton’s discovery, and recalls that Hamilton had great expectations for its wide applicability to mathematics, physics and astronomy. At the same time, it is pointed out that this opinion had a number of opponents, and similarly to the case of non-Euclidean geometry, the controversy over the usefulness of quaternions lasted for many years after Hamilton’s death.
The author presents the classical matrix realization (the Cayley model) of \(\mathbb{H}\), as well as some basic facts concerning the structure of the multiplicative group \(\mathbb{H}^*\) of \(\mathbb{H}\) (like Moivre’s formula), and gives an idea of the present place of quaternions in some areas of topology and theoretical physics. The paper contains information on several major mathematical achievements related to Hamilton’s discovery, such as the Cayley (division) \(\mathbb{R}\)-algebra of octaves, the Frobenius classification of finite-dimensional associative division \(\mathbb{R}\)-algebras, the Gel’fand-Mazur theorem for commutative Banach division algebras, Hopf’s theorem on commutative (but not necessarily associative) finite-dimensional division \(\mathbb{R}\)-algebras, the development of vector analysis, the Hurwitz-Radon theorem on compositions of sums of squares (and its applications to topology), and the fundamental theorem of algebra for quaternions due to Eilenberg and Niven. Also, it clarifies the role of quaternions in understanding and representing rotations in low-dimensional Euclidean spaces (for example, the normal structure of the special orthogonal groups \(\text{SO}(n)\): \(n\in\mathbb{N}\)), and reproduces Coxeter’s description of finite subgroups of \(\mathbb{H}^*\).
For the entire collection see [Zbl 1052.00009].