How do we define a Poisson map between two Poisson vector space?

Question

I was reading a paper "(Co) isotropic Pairs in Poisson and Presymplectic Vector Spaces " by Jonathan Lorand and Alan weinstein and I stumble across the term Poisson map. I tried to find the source of materials from where I could get a definition of a Poisson map, but i couldn't. So, if anyone has studied about Poisson map please tell me something. Thankyou

Answer 1

See e.g. Lectures on Poisson Geometry, Homework 2.3 (p. 34):

A linear Poisson map between two Poisson vector spaces $(V,\pi_V)$ and $(W,\pi_W)$ is a linear map $\Phi: V \to W$ such that: $$\pi_W^{\#} = \Phi \circ \pi_V^{\#} \circ \Phi^*.$$ Show that if $\Phi: (V,\pi_V) \to (W,\pi_W)$ is a linear Poisson map, then:

1. If $C \subset V$ is a coisotropic subspace then so is $\Phi(C) \subset W$.
2. If $C \subset W$ is a coisotropic subspace then so is $\Phi^{-1}(C) \subset V$.

Here $\pi_V : V^* \times V^* \to \mathbb{R}$ is a skew-symmetric bilinear form, and $\pi_V^{\#} : V^* \to V$ by $\xi \mapsto \pi(\xi,\cdot)$.