Isotropic subspaces in symmetric composition algebras and Kummer subspaces in central simple algebras of degree 3

Abstract

The maximal isotropic subspaces in split Cayley algebras were classified by van der Blij and Springer in Nieuw Archief voor Wiskunde VIII(3):158–169, 1960. Here we translate this classification to arbitrary composition algebras. We study intersection properties of such spaces in a symmetric composition algebra, and prove two triality results: one for two-D isotropic spaces, and another for isotropic vectors and maximal isotropic spaces. We 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 to classify maximal p-central subspaces in central simple algebras of degree p = 3. We prove various linkage properties of maximal p-central spaces and p-central elements. Analogous results are obtained for symmetric p-central elements with respect to an involution of the second kind inverting a third root of unity.