Cross product on \mathbb{R}^{n}, normed algebras and H–spaces

Sa？etak In this survey article, we present a construction of the cross product on \mathbb{R}^{n} , following the paper by P. F. McLoughlin, arXiv:1212.3515. We show that a naturally defined cross product exists only for n = 0, 1, 3, 7 (here \mathbb{R}^{0} denotes the zero vector space over R). We study the relationship between the cross product and Hurwitz’s theorem on the existence of normed algebras only for dimensions n = 1, 2, 4, 8, and the relationship with Adams’ theorem on continuous multiplications on spheres. Furthermore, we consider the possibilities of a generalization of the cross product on \mathbb{R}^{0} , from the function of two variables to the function of several variables