Symplectic realizations of bihamiltonian structures

TL;DR

This paper introduces a method for constructing bihamiltonian structures through complex symplectic form reduction, providing conditions for their realization and illustrating with examples related to semisimple Lie algebras.

Contribution

It presents a novel reduction technique for generating bihamiltonian structures from complex symplectic forms, expanding the understanding of their geometric properties.

Findings

Conditions for bihamiltonian structure realization established

Reduction method applied to examples involving semisimple Lie algebras

Generalization of bihamiltonian structures in odd-dimensional manifolds

Abstract

A method of constructing a class of bihamiltonian structures is presented. Elements of this class are generalizations of the so-called bihamiltonian structures of general position on odd-dimensional manifolds. The method consists in a simultaneous reduction of both the real and imaginary parts of a complex symplectic form. Necessary and sufficient conditions of getting a bihamiltonian structure from the mentioned class are obtained. The second part of the paper is devoted to a series of examples of such a reduction related to semisimple Lie algebras.