Poisson-Nijenhuis structures and the Vinogradov bracket
type: Item
instance of: scholarly article
title: Poisson-Nijenhuis structures and the Vinogradov bracket
zbMATH Open document ID: 0828.58015
DOI: 10.1007/BF02108287
published in: Annals of Global Analysis and Geometry
publication date: 1996-01-14T00:00:00Z
review text: 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.
reviewed by: Yuri E. Gliklikh
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 58A05
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 58A50
zbMATH DE Number: 762850
zbMATH Keywords: Vinogradov bracket
zbMATH Keywords: Poisson-Nijenhuis structure
author: José V. Beltrán
author: Juan Monterde
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 762850