Cohomology theory of Nijenhuis Lie algebras and (generic) Nijenhuis Lie bialgebras

TL;DR

This paper develops a cohomology theory for Nijenhuis Lie algebras, explores homotopy Nijenhuis operators, and extends the framework to Nijenhuis Lie bialgebras, linking them to classical Yang-Baxter solutions.

Contribution

It introduces a cohomology framework for Nijenhuis Lie algebras and extends the concept to Nijenhuis Lie bialgebras, including their relation to classical Yang-Baxter equations.

Findings

Defined cohomology for Nijenhuis Lie algebras.

Connected homotopy Nijenhuis operators to third cocycles.

Established equivalences between Nijenhuis Lie bialgebras, matched pairs, and Manin triples.

Abstract

The aim of this paper is twofold. In the first part, we define the cohomology of a Nijenhuis Lie algebra with coefficients in a suitable representation. Our cohomology of a Nijenhuis Lie algebra governs the simultaneous deformations of the underlying Lie algebra and the Nijenhuis operator. Subsequently, we define homotopy Nijenhuis operators on $2$-term $L_\infty$-algebras and show that in some cases they are related to third cocycles of Nijenhuis Lie algebras. In another part of this paper, we extend our study to (generic) Nijenhuis Lie bialgebras where the Nijenhuis operators on the underlying Lie algebras and Lie coalgebras need not be the same. In due course, we introduce matched pairs and Manin triples of Nijenhuis Lie algebras and show that they are equivalent to Nijenhuis Lie bialgebras. Finally, we consider the admissible classical Yang-Baxter equation whose antisymmetric solutions yield Nijenhuis Lie bialgebras.