Singularities of bi-Hamiltonian systems

2015-06-16T10:59:53Z (GMT) by Alexey V. Bolsinov Anton Izosimov
We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with a fixed 2-cocycle. In terms of such linearizations, we give a criterion for non-degeneracy of singular points of bi-Hamiltonian systems and describe their types. © 2014 Springer-Verlag Berlin Heidelberg.