Kronecker网、双哈密顿结构和变元转换方法
Kronecker webs, bihamiltonian structures, and the method of argument translation
摘要：
我们表明，流形的参数值的首次积分的可积有限维双哈密顿系统进行的几何结构，我们称之为{ \em克罗内克网}。我们描述了两个函子之间的克罗内克网和可积双哈密顿结构，一个是左逆的另一个。从猜想上讲，这两个函子是相互逆的（对于“小”开子集）。证明了上述猜想，只要双哈密顿结构允许一个特定形式的反对合。这暗示了\cite{GelZakh99Web}的猜想：在稠密开子集上，如果${\mathfrak g}$是半单的，或者如果${\mathfrak g}={\mathfrak G}\ltimes \operatorname{ad}_{\mathfrak G}$和${\mathfrak G}$是半单的，以及对于某些其它的映射的李代数，${\mathfrak g}^{*}$上的双哈密顿结构是平坦的.
英文摘要：
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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