AUTOTOPISMS AND ISOTOPISMS OF TRILINEAR ALTERNATING FORMS

For a trilinear alternating form f on a vector space V, a generalization of the group of automorphisms group of autotopisms Atp(f), is introduced. An autotopism of f is a triple (alpha, beta, gamma) of automorphisms of V satisfying f(u, v, w) = f(alpha(u), beta(v), gamma(w)) for all u, v, w is an element of V. Basic results concerning this group are presented, and it is shown that the subgroup of Atp(f) containing autotopisms with identity in one coordinate is Abelian and that a mapping in this group has no fixed points if and only if its order is not a power of two. Moreover, the notion of equivalence of two trilinear alternating forms is generalized in a similar way, and a partial result is given. Examples of forms with both trivial (Atp(f) = Aut(f)) and nontrivial groups of autotopisms are presented.