Analogues of Poisson-Nijenhuis Structures: From Manifolds to Lie Bialgebras

Poisson-Nijenhuis structures arise from the interplay between a Poisson structure and a Nijenhuis tensor on a manifold, subject to specific compatibility conditions. These conditions generate a hierarchy of pairwise compatible Poisson structures, playing a central role in the theory of integrable systems. I will introduce an analogue of Poisson–Nijenhuis structures in the context of Lie bialgebras, which we call NL bialgebras. This framework captures the interaction between a Nijenhuis operator and a Lie bialgebra structure on a Lie algebra, under suitable compatibility conditions. I will explore when such a structure gives rise to a hierarchy of pairwise compatible Lie bialgebras. As an application, I present an example in which a weak NL bialgebra naturally underlies the algebraic framework of a well-known dynamical system--namely, a special case of the Euler top.