NL bialgebras

TL;DR

This paper introduces weak NL bialgebras, combining Lie bialgebras with Nijenhuis structures, and explores their properties, hierarchies, and applications to integrable systems like the Euler-top.

Contribution

It defines weak NL bialgebras, analyzes their hierarchy generation, and connects them to the algebraic structure of the Euler-top system.

Findings

Weak NL bialgebras generalize Lie bialgebras with Nijenhuis structures.

They generate compatible hierarchies of bialgebras.

The Euler-top system's algebraic structure is a weak NL bialgebra.

Abstract

In this paper, we introduce the concept of (weak) NL bialgebras. These structures consist of a Lie bialgebra $(\g,[\cdot,\cdot],\delta)$ equipped with a Nijenhuis structure on the Lie algebra $(\g,[\cdot,\cdot])$, satisfying specific compatibility conditions. This construction is analogous to Poisson-Nijenhuis structures studied in the context of integrable systems. We further investigate NL bialgebras that generate a compatible hierarchy of bialgebras, both on the original Lie algebra and its deformed versions, through the Nijenhuis structure of any order. Additionally, we demonstrate that the underlying algebraic structure of a particular case of the Euler-top system is a weak NL bialgebra.