Poisson-Nijenhuis structures and the Vinogradov bracket
The authors study the compatibility conditions for a Poisson bivector and a Nijenhuis tensor to become a Poisson-Nijenhuis structure on a differentiable manifold. They express the condition by means of the graded Lie bracket defined by A. M. Vinogradov [see \textit{A. M. Vinogradov}, Mat. Zametki 47, No. 6, 138-140 (1990; Zbl 0712.58059) and \textit{A. Cabras} and \textit{A. M. Vinogradov}, J. Geom. Phys. 9, No. 1, 75- 100 (1992; Zbl 0748.58008)]. The authors show that the compatibility conditions are precisely the vanishing conditions for the higher order part of the bracket and characterize the Poisson-Nijenhuis structures for which the bracket vanishes. The construction allows one to define a differential bicomplex for Poisson-Nijenhuis manifolds. Finally this approach is applied to bi-Hamiltonian manifolds.
scientific article; zbMATH DE number 762850
scientific article