Quadratic forms, compositions and triality over F1

According to Tits, the quadric of dimension 6 over the “field” F1 with one element is a set of 8 points structured by a permutation of order 2 without fixed points. Subsets that are disjoint from their image under the permutation are the subspaces of the quadric. As in the classical case of hyperbolic quadrics over an arbitrary field, maximal subspaces come in two different types. We define a geometric triality to be a permutation of order 3 of the set consisting of points and maximal subspaces, carrying points to maximal subspaces of one type and maximal subspaces of the other type to points while preserving the incidence relations. We show analogues over F1 of the one-to-one correspondence between geometric trialities, trialitarian automorphisms of algebraic groups of type D4, and symmetric composition algebras of dimension 8. Here, the algebraic groups of type D4 are replaced by their Weyl group, which is the semi-direct product S23⋊S4, and composition algebras by a certain type of binary operation on a quadric to which a 0 is adjoined. As in the classical case, we show that there are two types of trialities, one related to octonions and the other to Okubo algebras.