Smith, Jonathan D. H. A left loop on the 15-sphere. (English) Zbl 0841.17004 J. Algebra 176, No. 1, 128-138 (1995). It is well known that the sphere \(S^n\) does not admit a continuous multiplication for \(n\not\in \{0, 1, 3, 7\}\). For the allowed values of \(n\) examples are provided by restricting the multiplication of a normal real division algebra to the respective unit sphere. In the paper under review the author constructs a multiplication on \(\mathbb{R}^{16}\) which is multiplicative with respect to some norm \(|\cdot |: \mathbb{R}^{16}\to \mathbb{R}\), i.e. one has \(|z\cdot w|= |z|\cdot|w|\) for all \(z,w\in \mathbb{R}^{16}\). The left multiplications are linear and left division is always possible. Thus \(S^{15}\) with the induced multiplication becomes a left loop. Moreover, it is a right loop almost everywhere, i.e. the closure of the set of elements of \(S^{15} \times S^{15}\) for which right division is not possible has measure zero. Of course the multiplication on \(S^{15}\) cannot be continuous, however it is smooth enough to allow the explicit construction of 8 linearly independent vector fields on \(S^{15}\). Reviewer: N.Knarr (Braunschweig) Cited in 4 Documents MSC: 17A75 Composition algebras 20N05 Loops, quasigroups 22A30 Other topological algebraic systems and their representations 54H13 Topological fields, rings, etc. (topological aspects) 57R25 Vector fields, frame fields in differential topology Keywords:normed division semi-algebra; left loop PDFBibTeX XMLCite \textit{J. D. H. Smith}, J. Algebra 176, No. 1, 128--138 (1995; Zbl 0841.17004) Full Text: DOI