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

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Abstract

We show that manifolds which parameterize values of first integrals of integrable finite-dimensional bihamiltonian systems carry a geometric structure which we call aKronecker web. We describe two opposite direction functors between Kronecker webs and integrable bihamiltonian structures: one is left inverse to the other. Conjecturally, these two functors are mutually inverse (for “small” open subsets of the manifolds in question).

The conjecture above is proven here when the bihamiltonian structure allows an anti-involution of a particular form. This implies the conjecture of [15] that on a dense open subset the bihamiltonian structure on\(\mathfrak{g}^* \) is flat if\(\mathfrak{g}\) is semisimple.

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  1. Department of Mathematics, Ohio State University, 231 W. 18 Ave, 43210, Columbus, OH, USA

    Ilya Zakharevich

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Zakharevich, I. Kronecker webs, bihamiltonian structures, and the method of argument translation. Transformation Groups 6, 267–300 (2001). https://doi.org/10.1007/BF01263093

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