Deformations and abelian extensions on anti-pre-Lie algebras

In this paper, we introduce the representation of anti-pre-Lie algebras and give the second cohomology group of anti-pre-Lie algebras. As applications, first, we study linear deformations of anti-pre-Lie algebras. The notion of a Nijenhuis operator on an anti-pre-Lie algebra is introduced which can generate a trivial linear deformation of an anti-pre-Lie algebra. Then, we study formal deformations of anti-pre-Lie algebras. We show that the infinitesimal of a formal deformation is a 2-cocycle with the coefficients in the regular representation and depends only on its cohomology class. Moreover, if the second cohomology group $H^2(A;A)$ is trivial, then the anti-pre-Lie algebra is rigid. Finally, we introduce the notion of abelian extensions. We show that abelian extensions are classified by the second cohomology group $H^2(A;V)$.