A new proof of the existence of hierarchies of Poisson-Nijenhuis structures
The paper under review provides a new approach to the phenomenon of hierarchies of Poisson-Nijenhuis structures in the sense of the paper by \textit{Y.~Kosmann-Schwarzbach} and \textit{F.~Magri} [Ann. Inst. Henri Poincaré, Phys. Théor. 53, No. 1, 35--81 (1990; Zbl 0707.58048)]. Thus, let \(P\colon T^*M\to TM\) be a Poisson tensor field and \(N\colon TM\to TM\) a Nijenhuis tensor field on the manifold \(M\). For every \(t\in{\mathbb R}\) define \(\Phi_t=\exp(tN)\colon TM\to TM\). One of the key statements (Proposition~3) of the paper says that the pair \((P,N)\) is a Poisson-Nijenhuis structure if and only if so are all the pairs \((P,\Phi_t)\) whenever \(t\in{\mathbb R}\). And one of the consequences of this fact (Corollary~3) is that, if \((P,N)\) is a Poisson-Nijenhuis structure then the pair \((\Phi_sP,\Phi_t)\) is a Poisson-Nijenhuis structure and moreover \(\Phi_sP+\Phi_tP\) is a Poisson tensor field for arbitrary \(s,t\in{\mathbb R}\). Under the same hypothesis, for all \(k,l\in{\mathbb N}\), the pair \((N^kP,N^l)\) is a Poisson-Nijenhuis structure and \(N^kP+N^lP\) is a Poisson tensor field (Corollary~5). An illustrating example concludes this well-written paper.
scientific article; zbMATH DE number 2149496