The geometry of trilinear forms

Abstract

Abstract All of us at this conference are familar with projective geometry on a vector space Vover a field F,and the refinements of projective geometry induced by a bilinear form on V.This note discusses the analogous geometries induced by trilinear forms on V.There does not seem to be much in the literature on this subject. In [4], Cohen and Helminck determine all alternating trilinear forms of dimension at most 7 over fields of cohomological dimension at most 1. In (1], [2], [3] there is some discussion of alternating and symmetric trilinear forms. There is of course an extensive literature on Lie algebras and Jordan algebras which is of some relevance. In the case of bilinear forms, the study of alternating forms is easier than the study of symmetric forms. The same seems to be true of trilinear forms. Nevertheless I will restrict my discussion to symmetric trilinear forms, as I have done more work with symmetric forms than alternating forms. However much of the discussion carries over to alternating forms.