Pre-Lie analogues of Poisson-Nijenhuis structures and Maurer-Cartan equations

In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce ON-structures on bimodules over pre-Lie algebras. We show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. We study solutions of the strong Maurer-Cartan equation on the twilled pre-Lie algebra associated to an O-operator, which gives rise to a pair of ON-structures which are naturally in duality. We show that KVN-structures and HN-structures on a pre-Lie algebra g are corresponding to ON-structures on the bimodule $({\mathfrak{g}}^{\ast };{\mathrm{a}\mathrm{d}}^{\ast },-R^{\ast })$, and $KV\mathrm{\Omega }$-structures are corresponding to solutions of the strong Maurer-Cartan equation on a twilled pre-Lie algebra associated to an $s$-matrix.