The generalized Weyl Poisson algebras and their Poisson simplicity criterion

A new large class of Poisson algebras, the class of generalized Weyl Poisson algebras, is introduced. It can be seen as Poisson algebra analogue of generalized Weyl algebras or as giving a Poisson structure to (certain) generalized Weyl algebras. A Poisson simplicity criterion is given for generalized Weyl Poisson algebras, and an explicit description of the Poisson centre is obtained. Many examples are considered (e.g. the classical polynomial Poisson algebra in 2n variables is a generalized Weyl Poisson algebra).