A-Poisson structures

TL;DR

This paper introduces the concept of A-Poisson structures on manifolds of infinitely near points, extending Poisson and symplectic structures to these generalized manifolds and exploring their properties.

Contribution

It defines A-Poisson manifolds on M^{A} and shows their relation to classical Poisson and symplectic structures, extending geometric frameworks.

Findings

M^{A} inherits an A-Poisson structure from M when M is Poisson.

The A-Poisson structure on M^{A} coincides with the prolongation of the Poisson structure from M when M is symplectic.

The paper establishes foundational properties of A-Poisson structures on manifolds of near points.

Abstract

Let M be a paracompact differentiable manifold, A a local algebra and M^{A} a manifold of infinitely near points on M of kind A. We define the notion of A-Poisson manifold on M^{A}. We show that when M is a Poisson manifold, then M^{A} is an A-Poisson manifold. We also show that if (M,) is a symplectic manifold, the structure of A-Poisson manifold on M^{A} defined by ^{A} coincide with the prolongation on M^{A} of the Poisson structure on M defined by the symplectic form.