Kronecker webs, bihamiltonian structures, and the method of argument translation
Abstract
Description
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.
54 pages, In addition to cosmetic changes, revision II of this paper contains a major simplification of arguments in Section 7, adds Remarks 2.5 and 2.6. Section 14 is expanded beyond Corollary 14.24 by adding the discussion of the case of groups of mappings. Revision III reworked the introduction. The numeration of statements did not change
54 pages, In addition to cosmetic changes, revision II of this paper contains a major simplification of arguments in Section 7, adds Remarks 2.5 and 2.6. Section 14 is expanded beyond Corollary 14.24 by adding the discussion of the case of groups of mappings. Revision III reworked the introduction. The numeration of statements did not change