Ternary Hom-Jordan algebras induced by Hom-Jordan algebras

The purpose of this paper is to study the relationships between a Hom-Jordan algebras and its induced ternary Hom-Jordan algebras. We give some properties of the alpha(k)-generalized derivation algebra GDer(J) of a ternary Hom-Jordan algebras. In particular, we prove that GDer(J) = QDer(J) + Q Gamma(J), the sum of the alpha(k)-quasiderivation algebra and the ak-quasicentroid. We also prove that QDer(J) can be embedded as derivations in a larger ternary Hom-Jordan algebras. General results on alpha(k)-centroids of ternary Hom-Jordan algebras are also developed in the paper.