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 N -structures on bimodules over pre-Lie algebras. We show that an N -structure gives rise to a hierarchy of pairwise compatible -operators. We study solutions of the strong Maurer–Cartan equation on the twilled pre-Lie algebra associated to an -operator, which gives rise to a pair of N -structures which are naturally in duality. We show that KVN-structures and HN-structures on a pre-Lie algebra are corresponding to N -structures on the bimodule ( ∗ ; ad ∗ , − R ∗) , and KVB-structures are corresponding to solutions of the strong Maurer–Cartan equation on a twilled pre-Lie algebra associated to an -matrix. [ABSTRACT FROM AUTHOR]