A left loop on the 15-sphere. (English) Zbl 0841.17004

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 \(S^{15}\).


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
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