Poly-Poisson structures

TL;DR

This paper introduces poly-Poisson structures as a higher-order generalization of Poisson structures, explores their properties, and discusses their behavior under group actions, providing new examples and theoretical insights.

Contribution

It defines poly-Poisson structures, proves their relation to polysymplectic foliations, and analyzes reduction under Lie group actions, extending Poisson geometry.

Findings

Poly-Poisson structures are equipped with a polysymplectic foliation.

Reduced spaces under polysymplectic Lie group actions are poly-Poisson.

Several new examples of poly-Poisson manifolds are provided.

Abstract

In this paper we introduce poly-Poisson structures as a higher-order extension of Poisson structures. It is shown that any poly-Poisson structure is endowed with a polysymplectic foliation. It is also proved that if a Lie group acts polysymplectically on a polysymplectic manifold then, under certain regularity conditions, the reduced space is a poly-Poisson manifold. In addition, some interesting examples of poly-Poisson manifolds are discussed.