Projections of Jordan bi-Poisson structures that are Kronecker, diagonal actions, and the classical Gaudin systems

Abstract

We propose a method of constructing completely integrable systems based on reduction of bihamiltonian structures. More precisely, we give an easily checkable necessary and sufficient conditions for the micro-kroneckerity of the reduction (performed with respect to a special type action of a Lie group) of micro-Jordan bihamiltonian structures whose Nijenhuis tensor has constant eigenvalues. The method is applied to the diagonal action of a Lie group $G$ on a direct product of $N$ coadjoint orbits $Ø=O_1\times ...\times O_N$ endowed with a bihamiltonian structure whose first generator is the standard symplectic form on $Ø$. As a result we get the so called classical Gaudin system on $Ø$. The method works for a wide class of Lie algebras including the semisimple ones and for a large class of orbits including the generic ones and the semisimple ones.

24 p

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Cited by in corpus (6)

• Symmetries and Integrability
• Gaudin Models and Bending Flows: a Geometrical Point of View
• Integrability of Invariant Geodesic Flows on n-Symmetric Spaces
• Invariant Nijenhuis Tensors and Integrable Geodesic Flows
• Dirac brackets and reduction of invariant bi-Poisson structures
• Integrability of geodesic flows for metrics on suborbits of the adjoint orbits of compact groups