Centroid Hom-associative Algebras and Centroid Hom-Lie Algebras

In this article, we construct free centroid hom-associative algebras and free centroid hom-Lie algebras. We also construct some other relatively free centroid hom-associative algebras by applying the Gröbner–Shirshov basis theory for (unital) centroid hom-associative algebras. Finally, we prove that the “Poincaré–Birkhoff–Witt theorem” holds for certain type of centroid hom-Lie algebras over a field of characteristic 0, namely, every centroid hom-Lie algebra such that the eigenvectors of the map β linearly generates the whole algebra can be embedded into its universal enveloping centroid hom-associative algebra, and the linear basis of the universal enveloping algebra does not depend on the multiplication table of the centroid hom-Lie algebra under consideration.