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

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

1999-08-09

arxiv.org/abs/math/9908034v3

Mathematics

Symplectic Geometry

54 pages, In addition to cosmetic changes, revision II of this paper contains a major simplification of arguments in Section 7

Scientific paper

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.

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