Isotropic subspaces in symmetric composition algebras and Kummer subspaces in central simple algebras of degree 3
A composition algebra is a (non-associative) algebra \((C, \ast )\) over a field \(F\), endowed with a non-degenerate quadratic ``norm form'' \(N: C \to F\), such that \(N(a \ast b) = N(a)N(b)\), \(a,b\in C\). Two special classes of these algebras have been fully classified. In the first place, it is known that a composition \(F\)-algebra with a unit is one of the following: the field \(F\); an étale quadratic extension of \(F\); a quaternion algebra; a Cayley algebra. In these cases, the norm form is a Pfister form, whence it is either anisotropic or hyperbolic. Secondly, a composition algebra is symmetric, if the bilinear form associated to the norm is associative. These include the Okubo algebras constructed from central simple \(F\)-algebras of degree \(3\). The maximal isotropic spaces in the split Cayley algebra were described by \textit{F. van der Blij} and \textit{T. A. Springer} [Nieuw Arch. Wiskd., III. Ser. 8, 158--169 (1960; Zbl 0127.11804)]. The paper under review translates this classification to arbitrary composition algebras. The authors take as a starting point Kaplansky's theorem stating that any composition algebra can be twisted into a unital one (see Proposition 33.27 in [\textit{M.-A. Knus} et al., The book of involutions. Providence, RI: AMS (1998; Zbl 0955.16001)]). This, applied to the van der Blij-Springer description, enables them to classify the maximal isotropic spaces in an arbitrary composition algebra, and their intersections in case the algebra is symmetric. In the rest of the paper, the authors consider symmetric \(F\)-algebras \((C, \ast )\). Analyzing intersection properties of maximal isotropic spaces in \(C\), they obtain two triality results: one for \(2\)-dimensional isotropic spaces, and another for isotropic vectors and maximal isotropic spaces. The authors bound the distance between isotropic spaces of various dimensions, and study the strong orthogonality relation on isotropic vectors, with its own bound on the distance. The results are used for classifying maximal \(3\)-central subspaces in central simple algebras of degree \(3\). Various linkage properties of maximal \(3\)-central spaces and \(3\)-central elements are obtained. Such results are also obtained for symmetric \(3\)-central elements with respect to an involution of the second kind inverting a cubic root of unity.
scientific article; zbMATH DE number 6006488