(n + 1)-Ary derivations of simple n-ary algebras

Abstract

By a δ-derivation we mean a linear map φ such that φ(xy) = δ(φ(x)y + xφ(y)) for some fixed element δ of the ground field. For more details about δ-derivations, we ask the reader to consult [1-6]. A natural generalization of δ-derivations to n-ary algebras is a generalized derivation of an n-ary algebra. By an (n+1)-ary derivation of an n-ary algebra A we mean a tuple (f0, f1, . . . , fn) ∈ End(A)n+1 such that for arbitrary x1, . . . , xn ∈ A, it is true that