 Poisson map

A smooth map $r:M\to N$ between Poisson manifolds is a Poisson map when $\{f\circ r,g\circ r{\}}_M=\{f,g{\}}_N\circ r$ for all smooth functions on $N$. When a Lie group acts by Poisson automorphisms and the quotient is smooth, brackets of invariant functions descend and make the quotient projection Poisson. The quotient can be Poisson without being symplectic.