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Poisson Polynomial Rings

Let A be a Poisson algebra with Poisson bracket {·, ·} A and let α,δ be linear maps from A into itself. Here we find a necessary and sufficient condition for the pair (α,δ) such that the polynomial ring A[x] has the Poisson bracket for all a, b ∈ A and construct a class of Poisson algebras including the coordinate rings of Poisson 2 × 2-matrices and Poisson symplectic 4-space.