Abstract
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).
Highlights
The GWA-construction turns out to be a useful one. Using it for large classes of algebras, all the simple modules were classified, explicit formulae were found for the global and Krull dimensions, their elements were classified in the sense of Dixmier [5], etc
The key idea of the proof is to introduce another class of Poisson algebras, elements of which are denoted by D[X, Y ; ∂, α], for which existence problem has an easy solution and to show that each generalized Weyl Poisson algebras (GWPAs) is a factor algebra of some D[X, Y ; ∂, α]
We show that many classical Poisson algebras are GWPAs
Summary
Two new classes of Poisson algebras are introduced and their existence is proved. 4. Let D = K [C, H ] be a Poisson polynomial algebra with trivial Poisson bracket, a ∈ D and ∂ is a derivation of the algebra D. The GWPA A = D[X , Y ; a, ∂} of rank 1 is a generalization of some Poisson algebras that are associated with U (sl2), see the example. The associated graded algebra gr(U ) with respect to the filtration F = {Fi }i∈N that is determined by the total degree of the elements X , Y and H is a Poisson polynomial algebra K [X , Y , H ] where.
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