Deformations and abelian extensions of compatible pre-Lie superalgebras

In this paper, we give cohomologies and deformations theory, as well as abelian extensions for compatible pre-Lie superalgebras. Explicitly, we first introduce the notation of a compatible pre-Lie superalgebra and its representation. We also construct a new bidfferential graded Lie superalgebra whose Maurer–Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie superalgebra which controls deformations of a compatible pre-Lie superalgebra. Then, we introduce a cohomology of a compatible pre-Lie superalgebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie superalgebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We further give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We study formal deformations of compatible pre-Lie algebras. If the second cohomology group [Formula: see text] is trivial, then the compatible pre-Lie superalgebra is rigid. Additionally, we give a cohomology of a compatible pre-Lie superalgebra with coefficients in arbitrary representation and study abelian extensions of compatible pre-Lie superalgebras using this cohomology. We show that abelian extensions are classified by the second cohomology group. Finally, we study the relation between compatible Lie superalgebras and compatible pre-Lie superalgebras and we give some examples of compatible pre-Lie superalgebras.