Recall that the commutative algebra \(B\) equipped with a bracket \(\{\cdot,\cdot\}\) is called a Poisson algebra if the bracket makes \(B\) a Lie algebra and is assumed to be an associative algebra derivation in each argument. In the paper under review the author studies polynomial identities of Poisson algebras over a field of characteristic 0.
First he considers the free Poisson algebra \(P(X)\) which is isomorphic to the symmetric algebra \(S(L(X))\) of the free Lie algebra \(L(X)\). He distinguishes the class of customary polynomials which are commutative polynomials of \(\{x_i,x_j\}\). An important example of a customary polynomial is the Poisson standard polynomial \(s_{2n}\) of degree \(2n\). The author gives a characterization of the multilinear customary polynomials as the multilinear elements of \(P(X)\) which are associative derivations in each appearing variable. It turns out that any Poisson algebra satisfying some polynomial identity, satisfies also some customary identity.
Finally, the author shows that a regular affine domain \(A\) of Krull dimension \(2n\) which is also a Poisson algebra is symplectic if and only if it satisfies the Poisson standard identity \(s_{2n+2}\) and \(s_{2n}(A)\) lies in no proper associative ideal of \(A\). This may be considered as a Poisson analogue of the Artin-Procesi Theorem.