Eight-dimensional real quadratic division algebras (Q2701752)

It is known that every alternative real division algebra (with dimension 8) is quadratic. A dissident triple \((V,\xi,\eta)\) consists of a finite-dimensional Euclidean vector space \(V\), a linear form \(\xi: V\wedge V\to R\) and dissident map \(\eta:V\wedge V\to V\). The category \({\mathcal D}\) of all dissident triples and the category \({\mathcal A}\) of all quadratic real division algebras are related by a functor \({\mathcal G} (V,\xi,\eta)=R\times V\) which is an equivalence of categories as the author proved earlier. The problem of classifying all real quadratic division algebras is reduced to the problem of classifying all 8-dimensional real quadratic division algebras and so to the classification of all dissident maps \(\eta:R^7\wedge R^7\to R^7\). NEWLINENEWLINENEWLINEA linear map \(\varphi:V\wedge V\to V\) is called weak vector product iff \(\langle \varphi(u\wedge v),w\rangle= \langle u,\varphi(v\wedge w)\rangle\) for all \((u,v,w)\in V^3\) and \(\varphi(v\wedge w)\neq 0\). For each dissident map \(\eta:R^7\wedge R^7\to R^7\) there exist a weak vector product \(\rho: R^7\wedge R^7\to R^7\) and a positive-definite endomorphism \(\varepsilon: R^7\to R^7\) such that \(\eta= \varepsilon\rho\). It is shown that even weak factorization suffices to accomplish the complete classification of all 8-dimensional real quadratic division algebras.