Nijenhuis operators on pre-Lie algebras

TL;DR

This paper introduces Nijenhuis operators on pre-Lie algebras using a new graded Lie algebra approach, explores their connections with other operators, and discusses geometric interpretations and applications in algebraic structures.

Contribution

It defines Nijenhuis operators on pre-Lie algebras via a novel graded Lie algebra framework and investigates their relationships with O-operators, Rota-Baxter operators, and geometric structures.

Findings

Nijenhuis operators generate trivial deformations of pre-Lie algebras.

Connections established between Nijenhuis, O-operators, and Rota-Baxter operators.

Introduction of geometric structures like pseudo-Hessian-Nijenhuis and para-complex structures.

Abstract

First we use a new approach to give 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 paracomplex quadratic pre-Lie algebras and paracomplex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras.