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

Abstract

We consider multicomponent local Poisson structures of the form ${\mathcal{P}}_3+{\mathcal{P}}_1$, under the assumption that the third order term ${\mathcal{P}}_3$ 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 edges labeled by numbers ${\lambda }_{\alpha }$'s and 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.

In Version v2, Theorem 4 and its proof are improved, and a mistake in Theorem 5 is corrected. In Version v3 we changed the signs in certain formulas for cosmetical reasons, to avoid multiple use of $(-1{)}^n$, and to make the paper better compatible with arXiv:2212.01605, and also updated the references

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• The algebraic classification of nilpotent algebras
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Cited by in corpus (4)

• Applications of Nijenhuis Geometry IV: multicomponent KdV and Camassa-Holm equations
• Projective geometry of homogeneous second order Hamiltonian operators
• Orthogonal separation of variables for spaces of constant curvature
• Bi-Hamiltonian structures of KdV type, cyclic Frobenius algebrae and Monge metrics