Classical Yang-Baxter equations and Nijenhuis operators for Lie algebras

TL;DR

This paper explores the conditions under which Lie algebras are Nijenhuis, characterizes Nijenhuis operators on sl_2, and investigates the connections between classical Yang-Baxter equations and Nijenhuis operators within Lie bialgebra theory.

Contribution

It provides a comprehensive analysis of Nijenhuis operators on Lie algebras, especially on sl_2, and establishes new relations between Yang-Baxter equations and Nijenhuis structures.

Findings

Characterization of Nijenhuis operators on sl_2

Conditions for Lie algebras to be Nijenhuis

Relations between Yang-Baxter equations and Nijenhuis operators

Abstract

In this paper the conditions that when a Lie algebra is Nijenhuis are investigated. Furthermore all the Nijenhuis operators on $\mathfrak{sl}_2$ under the standard Cartan-Weyl basis are given. On the other hand, the relations between the classical Yang-Baxter equation and Nijenhuis operators $N$ on a Lie algebra and $P$ on a Lie coalgebra are derived by means of the bialgebraic theory for Nijenhuis Lie algebras.