Nijenhuis pre-Lie bialgebras, Nijenhuis Lie bialgebras and \sss-equation

TL;DR

This paper introduces new methods to construct Nijenhuis operators on pre-Lie algebras and coalgebras, explores their bialgebraic structures, and links them to ext{ extbackslash sss}-equations and Lie bialgebras, advancing the understanding of algebraic deformations.

Contribution

It presents novel constructions of Nijenhuis operators on pre-Lie structures and establishes their connections with bialgebras and ext{ extbackslash sss}-equations, enriching the theory of algebraic deformations.

Findings

Constructed Nijenhuis operators via pseudo-Hessian pre-Lie algebras.

Defined Nijenhuis operators on pre-Lie coalgebras and their constructions.

Established that Nijenhuis balanced pre-Lie bialgebras yield Nijenhuis Lie bialgebras.

Abstract

Two aspects on the important notion of pre-Lie algebras are pre-Lie bialgebras (or left-symmetric bialgebras) with motivation from para-K\"ahler Lie algebras, and Nijenhuis operators on pre-Lie algebras arising from their deformation theory. In this paper, we present a method to construct Nijenhuis operators on a pre-Lie algebras via pseudo-Hessian pre-Lie algebras. Next, we introduce the notion of Nijenhuis operators on pre-Lie coalgebras and give their constructions, one from a linearly compatible pre-Lie coalgebra structure, and one from pre-Lie bialgebras. We then obtain a bialgebraic structure on Nijenhuis pre-Lie algebras by using dual representations and study their relations with \sss-equations and $\mathcal{O}$-operators. Finally we prove that a Nijenhuis balanced pre-Lie bialgebra produces a Nijenhuis Lie bialgebra.