Characterizations of Poisson algebras.

*(English)*Zbl 0841.17012A 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.

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.

Reviewer: J.Grabowski (Warszawa)

##### 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 |

##### Keywords:

graded algebra; ideal; Poisson algebra; cotangent algebra; Hamiltonian vector field; characterizations of symplectic algebras; Poisson simplicity
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\textit{D. R. Farkas}, Commun. Algebra 23, No. 12, 4669--4686 (1995; Zbl 0841.17012)

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