Poly-symplectic groupoids and poly-Poisson structures

TL;DR

This paper introduces poly-symplectic groupoids and poly-Poisson structures, extending symplectic geometry concepts, and explores their properties, symmetries, and reductions, providing new insights and revisiting existing theories.

Contribution

It defines poly-symplectic groupoids and poly-Poisson structures, offering equivalent descriptions and connecting them with AV-Dirac structures, advancing the understanding of poly-symplectic geometry.

Findings

Defined poly-symplectic groupoids and poly-Poisson structures

Established equivalent descriptions including AV-Dirac structures

Analyzed symmetries and reduction in the poly-symplectic context

Abstract

We introduce poly-symplectic groupoids, which are natural extensions of symplectic groupoids to the context of poly-symplectic geometry, and define poly-Poisson structures as their infinitesimal counterparts. We present equivalent descriptions of poly-Poisson structures, including one related with AV-Dirac structures. We also discuss symmetries and reduction in the setting of poly-symplectic groupoids and poly-Poisson structures, and use our viewpoint to revisit results and develop new aspects of the theory initiated by D.Iglesias, J.C Marrero and M. Vaquero.