Kronecker webs, bihamiltonian structures, and the method of argument translation

Abstract

We show that manifolds which parameterize values of first integrals of integrable finite-dimensional bihamiltonian systems carry a geometric structure which we call a {\em Kronecker web}. We describe two functors between Kronecker webs and integrable bihamiltonian structures, one is left inverse to another one. Conjecturally, these two functors are mutually inverse (for ``small'' open subsets). The above conjecture is proven provided the bihamiltonian structure allows an antiinvolution of a particular form. This implies the conjecture of \cite{GelZakh99Web} that on a dense open subset the bihamiltonian structure on ${\mathfrak g}^{*}$ is flat if ${\mathfrak g}$ is semisimple, or if ${\mathfrak g}={\mathfrak G}\ltimes \operatorname{ad}_{{\mathfrak G}}$ and ${\mathfrak G}$ is semisimple, and for some other Lie algebras of mappings.