Diagonal abelian extensions and morphism-fixed formal deformations of Hom-pre-Lie algebras

In this paper we use the cohomological approach to study abelian extensions and formal deformations of Hom-pre-Lie algebras. We show that diagonal abelian extensions of Hom-pre-Lie algebras can be classified by stable points in the second cohomology group under the action of an isomorphism. The infinitesimals of two equivalent formal deformations of a Hom-pre-Lie algebra A are in the same cohomological class in H-2(A; A), and if H-2(A; A) is trivial, then the Hom-pre-Lie algebra A is rigid.