Equivariant Nijenhuis Lie Algebras: Extensions to Classical Lie-Theoretic Structures

TL;DR

This paper introduces equivariant Nijenhuis Lie algebras, extending classical Lie structures with Nijenhuis operators satisfying an equivariance condition, and explores their bialgebra, Rota-Baxter, and pre-Lie generalizations.

Contribution

It develops the theory of equivariant Nijenhuis Lie algebras, including their bialgebra structures, classical Yang-Baxter solutions, and connections to Rota-Baxter and pre-Lie algebras.

Findings

Defined ENL bialgebras and their structures.

Characterized EN $r$-matrices via an equivariant classical Yang-Baxter equation.

Connected Rota-Baxter operators to solutions of the classical Yang-Baxter equation.

Abstract

We develop a theory of equivariant Nijenhuis Lie algebras (ENL algebras), namely Lie algebras equipped with Nijenhuis operators satisfying an equivariance condition with respect to the adjoint representation. This compatibility condition allows classical Lie bialgebra constructions to extend naturally to the operator-equipped setting. Within this framework, we define ENL bialgebras and establish the associated notions of matched pairs, Manin triples, and Drinfel'd doubles. We show that coboundary ENL bialgebras are characterized by EN $r$-matrices satisfying an equivariant classical Yang-Baxter equation. We further introduce EN-relative Rota-Baxter operators and prove that they provide an operator-theoretic realization of such $r$-matrices, leading to descendant ENL algebras and to solutions of the classical Yang-Baxter equation on semidirect ENL algebras. In the quadratic case, this construction recovers Rota-Baxter operators of weight zero. Finally, we extend the EN framework to pre-Lie algebras and show that pre-ENL algebras naturally induce associated ENL structures.