Reduction of bihamiltonian manifolds and construction of the recursion operator
type: Item
instance of: scholarly article
title: Reduction of bihamiltonian manifolds and construction of the recursion operator
zbMATH Open document ID: 0770.58014
DOI: 10.1016/0393-0440(92)90029-Z
author: Mohamad Mehdi
published in: Journal of Geometry and Physics
publication date: 1992-11-23T00:00:00Z
review text: Since Magri's work, the study of completely integrable systems from a geometrical point of view is based on differentiable manifolds \(M\) with compatible Poisson tensors \(P\) and \(Q\) (i.e. such that their Schouten bracket equals zero). When \(Q\) is invertible, the field of endomorphisms \(h = PQ^{-1}\) is called a ``recursion'' operator. The study of the integrability of a system begins with the investigation of the possibility of reduction of the system to a Magri manifold with a recursion operator. In this paper a sufficient condition for such a reduction is obtained: if \((M,P,Q)\) is a Magri manifold such that \(P(\text{Ker }Q)\subset Q(\text{Ker }P)\), then exists a procedure of reduction which enables the construction of a recursion operator on certain quotient manifolds. This procedure is applied in the paper to the example of the heat equation.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 70175
zbMATH Keywords: bihamiltonian manifolds
zbMATH Keywords: Schouten bracket
zbMATH Keywords: reduction
reviewed by: B. Valiño-Alonso
MaRDI profile type: Publication
cites work: Q3989671
cites work: Q3244523
cites work: Hyperbolic systems of conservation laws II
cites work: Les variétés de Poisson et leurs algèbres de Lie associees
cites work: Reduction of Poisson manifolds
cites work: Q3803878
cites work: The local structure of Poisson manifolds
description: scientific article; zbMATH DE number 70175