Projections of Jordan bi-Poisson structures that are Kronecker, diagonal actions, and the classical Gaudin systems
✍ Scribed by Andriy Panasyuk
- Publisher
- Elsevier Science
- Year
- 2003
- Tongue
- English
- Weight
- 185 KB
- Volume
- 47
- Category
- Article
- ISSN
- 0393-0440
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✦ Synopsis
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
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.