Construction method for some real division algebras with \(\mathfrak{su}(3)\) as derivation algebra
Let \(F\) be a field of characteristic different from \(2\), and \(D\) a finite-dimensional, unital associative \(F\)-algebra, equipped with an involution \(\sigma:x\mapsto \bar{x}\), satisfying \(\bar{x}x\in F1\) for all \(x\in D\). Moreover, let \(P\simeq D^s\) be a free right \(D\)-module, \(g:P\times P\to D\) a sesquilinear (with respect to \(\sigma\)) form, and \(\times:P\times P \to P\) an anti-symmetric bilinear map. An algebra structure is defined on the vector space \(A=D\oplus P\) over \(F\) by \[ (a,u)(b,v) = (ab - g(v,u) \,,\, va +u\bar{b} + v\times u ) \,. \] The resulting algebra \(A\), denoted by \((D,P,g,\times)\) is not necessarily associative, but has an identity element \(1_A=(1_D,0)\). This construction can be seen as a generalisation both of the classical Cayley-Dickson doubling process, and of some more recent methods, such as the construction of quadratic division algebras from so-called dissident maps (see e.g. \textit{E. Dieterich} [Colloq. Math. 82, No. 1, 13--23 (1999; Zbl 0941.17002)]). However, it is not known in which cases this more general method produces a division algebra. The article under review focuses on the case where \(D\) is an associative Hurwitz algebra and \(g=ch\), where \(c\in D\smallsetminus F1\) and \(h:P\times P\to D\) is Hermitean and satisfies \(h(u,u)\in F1\) for all \(u\in P\). It is shown that, in this situation, if \((D,P,h,\times)\) is a division algebra and \(h(v\times u,u)=0\) for all \(u,v\in P\), then \((D,P,ch,\times)\) is also a division algebra. Criteria are given for when two algebras of this type are isomorphic. Specifying \(F=\mathbb{R}\) and \(A=(\mathbb{C},\mathbb{C}^3,c\langle\,,\,\rangle,\times)\), where \(\langle\,,\,\rangle\) and \(\times\) are the standard inner product and the standard cross product on \(\mathbb{C}^3\) respectively, and \(c\in\mathbb{C}\smallsetminus\mathbb{R}\), one gets a new class of unital division algebras of dimension \(8\) over the real numbers. It is shown that the derivation algebras of these algebras are isomorphic to \(\mathfrak{su}(3)\). Moreover, by taking certain Albert isotopes of \(A\), a class of non-unital division algebras with derivation algebras isomorphic to \(\mathfrak{su}(3)\) is obtained.
scientific article; zbMATH DE number 6105492
scientific article