Abstract

We give a short and elementary proof for the classification of the real vector product algebras. A real vector product algebra is a Euclidean space V = (V, 〈 〉) together with an anti-symmetric bilinear map V × V → V, (u, v) ↦ → uv having the property that the set {u, v, uv} is orthonormal whenever {u, v} is. A morphism between vector product algebras V and W is a linear map ϕ: V → W such that ϕ(uv) = ϕ(u)ϕ(v) and 〈ϕ(u), ϕ(v) 〉 = 〈u, v 〉 for all u, v ∈ V. Two vector product algebras V and W are isomorphic if there exists a bijective morphism ϕ: V → W. Note that in our definition, vector product algebras are not required to be finite-dimensional. This, however, follows from the theory. In the classification theorem, our Theorem 3, it is established that real vector product algebras exist in dimension 0, 1, 3 and 7 only, and form one isomorphism class in each

Citations