The structure group of an alternative algebra
Let \(A\) be an alternative algebra over a field \(F\), and let \(A'=A+F1\) be the algebra obtained by adjoining an identity element 1 to \(A\). Let \(u\) and \(v\) be two elements in \(A'\). The \((u,v)\)-homotope of \(A\) is the algebra \(\overline A\) obtained by introducing the multiplication \(x \circ y=(xu)(vy)\) into the vector space \(A\). If \(u\) and \(v\) are invertible elements of \(A'\), \(\overline A\) is said to be a \((u,v)\)-isotope of \(A\). \textit{K. McCrimmon} [Math. Ann. 191, 253-262 (1971; Zbl 0203.33802)] showed that \(\overline A\) is again alternative and also that \((u,v)\)-isotopy preserves simplicity of an algebra and invertibility of an element of the algebra. In the present paper, the structure group \(\text{Str}(A)\) of an alternative algebra \(A\) with identity element 1 is defined as the totality of triples \((\eta,u,v)\) where \(\eta\) is an isomorphism from \(A\) onto its \((u,v)\)-isotope. The author proves that for any alternative algebra, the structure group is an extension, in the group-theoretical sense, of the narrow structure group (defined as in the Jordan case) by the unit group of the nucleus. It is proved that this extension is split if \(A\) is an associative algebra, and if \(A\) is the octonion algebra over an algebraically closed field it is not split in the category of algebraic groups. The author shows that the structure group is connected with the group of Albert autotopies and also that elements of the structure group play an important role in Albert's classical approach to the construction of Albert division algebras and Loos' treatment of alternative pairs with invertible elements. Using the principle of triality, natural sets of generators for the structure group and for the group of Tits and Weiss are exhibited in the case where the underlying algebra is an octonion algebra.
scientific article; zbMATH DE number 1868552