Nijenhuis operators on pre-Lie algebras

First we use a new approach to define a graded Lie algebra whose Maurer-Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket, we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between O-operators, Rota-Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator "connects" two O-operators on a pre-Lie algebra whose any linear combination is still an O-operator in certain sense and hence compatible L-dendriform algebras appear naturally as the induced algebraic structures. For the case of the dual representation of the regular representation of a pre-Lie algebra, there is a geometric interpretation by introducing the notion of a pseudo-Hessian-Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian-Nijenhuis structures. Another application of Nijenhuis operators on pre-Lie algebras in geometry is illustrated by introducing the notion of a para-complex structure on a pre-Lie algebra and then studying para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras.