Finite dimensional algebras over the real field (or systems of hypercomplex numbers, as they were known at the beginning of the 20th century) have been extensively studied, particularly those algebras equipped with bilinear products whose invariance under certain operations define a geometry on the underlying vector space. As is well-known, a higher dimensional algebra generalizing the field of complex numbers cannot retain all the algebraic properties of the operations of the complex field, e.g., the product will not be commutative or the cancellation law will fail and zero-divisors appear. The first example of such an algebra, discovered by Hamilton, is the non-commutative 4-dimensional division algebra of real quaternions.

In this book, we find an exhaustive and detailed study of some commutative 2-dimensional algebras and their corresponding geometries. The main example is an analogue of the complex field where the multiplication gives rise to an inner product of signature (+1,–1). The corresponding non-Euclidean geometry of the underlying hyperbolic plane is developed in full detail, including the associated hyperbolic trigonometric functions. (The authors should have been more careful in the “definition,” on p. 96, of a “hyperbolic logarithmic function” since there is no such thing as “the inverse of the hyperbolic exponential function.”)

In later chapters, this example is generalized to consider multiplications where the corresponding quadratic form has a discriminant of the form *D* = *y*^{2} + 4*x.* This divides the *xy*-plane into 3 sectors: the parabola D = 0, “inside the parabola” (D < 0), and “outside the parabola” (D > 0). In the last two cases, the author study the corresponding geometry.

Complex numbers, quaternion algebras, and Clifford algebras have found applications in the physical sciences, some of which we may call classical. Studying the hyperbolic plane, which is a 2-dimensional analogue of Minkowski 4-dimensional space, allows the authors to show that a hyperbolic rotation corresponds to a Lorentz transformation. They go on to write the equations of uniform or accelerated motion in the hyperbolic plane and use them to give a solution to classical paradoxes such as the “twin paradox” of special relativity.

In the appendices the authors sketch the construction of some 4-dimensional commutative algebras that are no longer division algebras, i.e., they have zero divisors, and suggest some properties of the corresponding geometries.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fzc@oso.izt.uam.mx.