Kronecker webs, bihamiltonian structures, and the method of argument translation (Q5959061)

This paper starts with an interesting review on some topics about integrability of bi-Hamiltonian systems. Then the author shows that manifolds which parametrize values of first integrals of finite-dimensional bi-Hamiltonian systems carry a geometric structure, called Kronecker web. There are two opposite functors between Kronecker webs and integrable bi-Hamiltonian strucures having the property that one is the left inverse of the other. It is also proven that if the bi-Hamiltonian strucure admits an appropriate anti-involution structure then the functors are in fact mutually inverse, for ``small'' open subsets. This implis the conjecture of \textit{I. M. Gelfand} and \textit{I. Zakharevich} [Sel. Math., New Ser. 6, 131-183 (2000; Zbl 0986.37060)] that on a dense open subset the bi-Hamiltonian structure of \(\mathfrak{g}^\star\) is flat if \(\mathfrak g\) is semisimple.