Nijenhuis operators on 2D pre-Lie algebras and 3D associative algebras

TL;DR

This paper classifies Nijenhuis operators on 2D pre-Lie and 3D associative algebras and applies these to find solutions of the classical Yang-Baxter equation on related Lie algebras.

Contribution

It provides a complete description of Nijenhuis operators on specific low-dimensional algebras and connects these to solutions of the classical Yang-Baxter equation.

Findings

All Nijenhuis operators on 2D complex pre-Lie algebras identified.

All Nijenhuis operators on 3D complex associative algebras identified.

Solutions to the classical Yang-Baxter equation derived from these operators.

Abstract

In this paper, we describe all Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras. As an application, using these operators, we obtain solutions of the classical Yang-Baxter equation on the corresponding sub-adjacent Lie algebras.