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 O=O 1×⋯×O N⊂ g ∗×⋯× g ∗ endowed with a bihamiltonian structure whose first generator is the standard symplectic form on O . As a result we get the so-called classical Gaudin system on O . 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.