Poisson Quasi-Nijenhuis Manifolds and the Toda System
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Poisson Quasi-Nijenhuis Manifolds and the Toda System G. Falqui1 · I. Mencattini2 · G. Ortenzi1 · M. Pedroni3 Received: 17 March 2020 / Accepted: 24 June 2020 / © Springer Nature B.V. 2020
Abstract The notion of Poisson quasi-Nijenhuis manifold generalizes that of PoissonNijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to integrability. In this note, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems. Generically (as we show by a class of examples with 3 degrees of freedom) the Poisson quasiNijenhuis structure is largely too general to ensure Liouville integrability of a system. However, we present a general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis manifolds, and we give sufficient conditions such that the spectral invariants of the “quasi-Nijenhuis recursion operator” of a Poisson quasi-Nijenhuis manifold (obtained by deforming a Poisson-Nijenhuis structure) are in involution. Then we prove that the closed (or periodic) n-particle Toda lattice, along with its relation with the open (or non periodic) Toda system, can be framed in such a geometrical structure.
M. Pedroni
[email protected] G. Falqui [email protected] I. Mencattini [email protected] G. Ortenzi [email protected] 1
Dipartimento di Matematica e Applicazioni, Universit`a di Milano-Bicocca, Milano, Italy
2
Instituto de Ciˆencias Matem´aticas e de Computac¸a˜ o, Universidade de S˜ao Paulo, S˜ao Paulo, Brazil
3
Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione, Universit`a di Bergamo, Bergamo, Italy
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Math Phys Anal Geom
(2020) 23:26
Keywords Integrable systems · Toda lattices · Poisson quasi-Nijenhuis manifolds · Bi-Hamiltonian manifolds Mathematics Subject Classification (2010) 37J35 · 53D17 · 70H06
1 Introduction It is well known that Poisson-Nijenhuis (PN) manifolds [10, 12] are an important notion in the theory of integrable systems. Roughly speaking, they are Poisson manifolds (M, π ) endowed with a tensor field of type (1, 1), say N : T M → T M, which is torsionless and compatible (see Section 2) with the Poisson tensor π . They turn out to be bi-Hamiltonian manifolds, with the traces of the powers of N satisfying the Lenard-Magri relations and thus being in involution with respect to the Poisson brackets induced by the Poisson tensors. An example of integrable system that can be studied in the context of PN manifolds is the open (or non periodic) n-particle Toda lattice. (For both the periodic and the non periodic Toda system, see [15] and references therein; see also [3, 13, 14].) The PN structure of the open Toda lattice was presented in [4]. Its Poisson tensor is non degenerate, so that the PN manifold is a symplectic manifold (sometimes it is called an ωN-manifold). This kind of geometrical structure was shown to play an important role in the bi-Hamiltonian
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