Symplectic structures associated to Lie-Poisson groups

TL;DR

This paper generalizes the Kirillov symplectic form to Lie-Poisson groups by classifying symplectic leaves and describing symplectic forms on manifolds analogous to cotangent bundles and coadjoint orbits.

Contribution

It introduces a framework for symplectic structures on Lie-Poisson groups, extending classical constructions to a broader class of geometric objects.

Findings

Classification of symplectic leaves in Lie-Poisson manifolds

Explicit description of symplectic forms on these leaves

Generalization of the Kirillov symplectic form

Abstract

The Lie-Poisson analogues of the cotangent bundle and coadjoint orbits of a Lie group are considered. For the natural Poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are described. Thus the construction of the Kirillov symplectic form is generalized for Lie-Poisson groups.