Poisson pencils: reduction, exactness, and invariants

Abstract

We study the invariants (in particular, the central invariants) of suitable Poisson pencils from the point of view of the theory of bi-Hamiltonian reduction, paying a particular attention to the case where the Poisson pencil is exact. We show that the exactness is preserved by the reduction. In the Drinfeld-Sokolov case, the same is true for the characteristic polynomial of the pencil, which plays a crucial role in the definition of the central invariants. We also discuss the bi-Hamiltonian structures of a generalized Drinfeld-Sokolov hierarchy and of the Camassa-Holm equation.

26 pages. Minor changes

References in corpus (1)

• A simple way of making a Hamiltonian system into a bi-Hamiltonian one

Cited by in corpus (2)

• Opers for higher states of quantum KdV models
• On integrability of transverse Lie-Poisson structures at nilpotent elements