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Characterizations of Poisson algebras. (English) Zbl 0841.17012
A Poisson algebra $$R$$ is an associative algebra with 1 furnished with a Lie bracket $$\{\;,\;\}$$ being a biderivation for the associative structure, i.e. satisfying the Leibniz rule $$\{x, yz\}= \{x, y\}z+ y\{x, z\}$$. Every associative algebra $$R$$ is a Poisson algebra with respect to the commutator bracket. If $$R$$ has additionally a filtration $$R_0 \subset R_1 \subset \dots$$ then we have a Poisson structure on the associated graded commutative algebra $$gr (R)$$ with the Poisson bracket obtained from the commutator. The most prominent example of this type is the cotangent algebra $$gr(D(A))$$ of a commutative algebra $$A$$ obtained from the algebra $$D(A)$$ of differential operators on $$A$$ with the natural filtration given by rank.
Every element $$x$$ of a Poisson algebra $$R$$ defines the associated Hamiltonian vector field $$H_x\in \text{Der} (R)$$ given by $$H_x (y)= \{x, y\}$$ and the family $$\text{Ham} (R)$$ of all Hamiltonian vector fields is a Lie subalgebra in the Lie algebra $$\text{Der} (R)$$ of derivations. A commutative with respect to the associative structure Poisson algebra is called symplectic if $$\text{Der} (R)= R\cdot \text{Ham} (R)$$, i.e. $$\text{Der} (R)$$ is as an $$R$$-module generated by Hamiltonian vector fields.
The paper is concerned with characterizations of symplectic algebras. One of the results states that every regular symplectic domain $$R$$ with gradation $$R= R_0\oplus R_1\oplus \dots$$ is actually the cotangent algebra $$gr(D (R_0))$$. Other results address the relationship between “symplectic” and Poisson simple in the sense that there are no proper associative ideals in $$R$$ being also Lie ideals. Although Poisson simplicity does not yield a characterization of symplectic algebras, the relationship is very tight under some extra natural hypotheses.

##### MSC:
 17B66 Lie algebras of vector fields and related (super) algebras 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 17B70 Graded Lie (super)algebras
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##### References:
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