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Completely integrable Hamiltonian systems and deformations of Lie algebras. (Systèmes hamiltoniens complètement intégrables et déformations d’algèbres de Lie.) (French) Zbl 0843.58063

In the Adler-Kostant-Symes theorem, as applied to completely integrable systems, two Lie algebras \(G\) and \(G_0\) appear, with the same underlying vector space. This paper states that in the cases that \(G\) is a finite-dimensional, semi-simple Lie algebra, or \(G= {\mathfrak {sl}}_2 (\mathbb{C}) \otimes \mathbb{C} [t,t^{-1} ]\), there exists a Lie algebra which is a deformation (of order 1) of \(G_0\), and which itself deforms to \(G\).
The proof (not fully included) is along cohomological reasonings.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q58 Other completely integrable PDE (MSC2000)
17B56 Cohomology of Lie (super)algebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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