Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures

Abstract

We consider multicomponent local Poisson structures of the form P 3 + P 1 , under the assumption that the third order term P 3 is Darboux-Poisson and non-degenerate, and study the Poisson compatibility of two such structures. We give an algebraic interpretation of this problem in terms of Frobenius algebras and reduce it to classification of Frobenius pencils, i.e. of linear families of Frobenius algebras. Then, we completely describe and classify Frobenius pencils under minor genericity conditions. In particular we show that each such Frobenuis pencil is a subpencil of a certain maximal pencil. These maximal pencils are uniquely determined by some combinatorial object, a directed rooted in-forest with vertices labeled by natural numbers whose sum is the dimension of the manifold. These pencils are naturally related to certain (polynomial, in the most nondegenerate case) pencils of Nijenhuis operators. We show that common Frobenius coordinate systems admit an elegant invariant description in terms of the Nijenhuis pencil.

Applications of Nijenhuis geometry IV: Multicomponent KdV and Camassa–Holm equations

“…for arbitrary vector fields ν, η. In our recent paper [7], we constructed all non-degenerate pencils of compatible ∞-dimensional Poisson structures of type P 3 + P 1 , where the Poisson structure P 1 has order 1 and P 3 is a Darboux-Poisson structure of order 3. Magri-Lenard scheme applied to these pencils leads to certain integrable bi-Hamiltonian systems.…”

Integrating Nijenhuis Structures

“…But they also appear elsewhere (see, e.g., references in [1] for a partial list). Recently, Bolsinov, Konyaev and Matveev, in a series of papers [1,21,3,4,5] initiated a project consisting in systematically studying Nijenhuis operators in their own. For instance, in the first paper of the series [1] they discuss local normal forms, in the same spirit as Weinstein splitting theorem for Poisson structures [33], while in the second paper of the series [21] Konyaev discusses the linearization problem.…”