
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 equipped with a Nijenhuis structure on the Lie algebra , 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.
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Taxonomy
TopicsAdvanced Topics in Algebra · Mobile Agent-Based Network Management
