Homogeneous tri-additive forms and derivations

Abstract

Gleason [A.M. Gleason, The definition of a quadratic form, Amer. Math. Monthly 73 (1966) 1049–1066] determined all functionals Q on K-vector spaces satisfying the parallelogram law Q ( x + y ) + Q ( x - y ) = 2 Q ( x ) + 2 Q ( y ) and the homogeneity Q ( λ x ) = λ 2 Q ( x ) . Associated with Q is a unique symmetric bi-additive form S such that Q ( x ) = S ( x , x ) and 4 S ( x , y ) = Q ( x + y ) - Q ( x - y ) . Homogeneity of Q corresponds to that of S: S ( λ x , λ y ) = λ 2 S ( x , y ) . The associated S is not necessarily bi-linear. Let V be a vector space over a field K, char ( K ) ≠ 2 , 3 . A tri-additive form T on V is a map of V 3 into K that is additive in each of its three variables. T is homogeneous of degree 3 if T ( λ x , λ y , λ z ) = λ 3 T ( x , y , z ) for all λ ∈ K , x , y , z ∈ V . We determine the structure of tri-additive forms that are homogeneous of degree 3. One of the keys to this investigation is to find the general solution of the functional equation F ( t ) + t 3 G ( 1 / t ) = 0 , where F : K → K is additive and G : K → K is quadratic. It is shown that T is not necessarily tri-linear, even if it is supposed in addition that T is symmetric.