The generalized Weyl Poisson algebras and their Poisson simplicity criterion

A new class of Poisson algebras, the class of {\em 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 explicit descriptions of the Poisson centre and the absolute Poisson centre are obtained. Many examples are considered.


Introduction
the centre Z (D) of D such that σ i (a j ) = a j for all i = j.T h egeneralized Weyl algebra A = D[X , Y ; σ, a] (briefly GWA) of rank n is a ring generated by D and 2n indeterminates X 1 , ..., X n , Y 1 , ..., Y n subject to the defining relations: where [x, y]=xy − yx. We say that a and σ are the sets of defining elements and automorphisms of the GWA A, respectively.
The n'th Weyl algebra A n = A n (K ) over a field (a ring) K is an associative Kalgebra generated by 2n elements X 1 , ..., X n , Y 1 , ..., Y n , subject to the relations: where δ ij is the Kronecker delta function. The Weyl algebra A n is a generalized Weyl algebra A = D[X , Y ; σ ; a] of rank n where D = K [H 1 , ..., H n ] is a polynomial ring in n variables with coefficients in K , σ = (σ 1 ,...,σ n ) where σ i (H j ) = H j − δ ij and a = (H 1 ,...,H n ).Themap is an algebra isomorphism (notice that Y i X i → H i ).
It is an experimental fact that many quantum algebras of small Gelfand-Kirillov dimension are GWAs (e.g. U (sl 2 ), U q (sl 2 ), the quantum Weyl algebra, the quantum plane, the Heisenberg algebra and its quantum analogues, the quantum sphere and many others).
The GWA-construction turns out to be a useful one. Using it for large classes of algebras (including the mentioned ones above), 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 generalized Weyl Poisson algebra D[X , Y ; a,∂}. Our aim is to introduce a Poisson algebra analogue of generalized Weyl algebras. An associative commutative algebra A is called a Poisson algebra if it is a Lie algebra (A, {·, ·}) such that {a, xy}= {a, x}y +x{a, y} for all elements a, x, y ∈ D.Let A be a Poisson algebra with Poisson bracket {·, ·},PZ(A) := {a ∈ A |{a, x}=0 for all x ∈ A} be its Poisson centre and PDer K (A) be the set of derivations of the Poisson algebra A (see Sect. 2 for details).
Definition Let D be a Poisson algebra, ∂ = (∂ 1 ,...,∂ n ) ∈ PDer K (D) n be an n-tuple of commuting derivations of the Poisson algebra D, a = (a 1 ,...,a n ) ∈ PZ(D) n be such that ∂ i (a j ) = 0 for all i = j. The generalized Weyl algebra = D[X 1 ,...,X n , Y 1 ,...,Y n ]/(X 1 Y 1 − a 1 ,...,X n Y n − a n ) admits a Poisson structure which is an extension of the Poisson structure on D and is given by the rule: For all i, j = 1,...,n and d ∈ D, The Poisson algebra is denoted by A = D[X , Y ; a,∂} and is called the generalized Weyl Poisson algebra of rank n (or GWPA, for short) where X = (X 1 ,...,X n ) and Existence of generalized Weyl Poisson algebras is proven in Sect. 2 (Lemma 2.1). The key idea of the proof is to introduce another class of Poisson algebras, elements of which are denoted by D[X , Y ; ∂,α] (see Sect. 2), for which existence problem has an easy solution and then to show that each GWPA is a factor algebra of some In Sect. 3, a proof is given of the following Poisson simplicity criterion for generalized Weyl Poisson algebras; see Proposition 3.1 for the notation. As a first step in the proof of Theorem 1.1, the following field criterion for the Poisson centre PZ(A) of a GWPA A = D[X , Y ; a,∂} of rank n is proven (in Sect. 3).

The generalized Weyl Poisson algebras
In this section, two new classes of Poisson algebras are introduced and their existence is proved. One of them is the class of generalized Weyl Poisson algebras (GWPAs). Examples are considered. At the end of the section, it is shown that some GWPAs are obtained from GWAs by a sort of quantization procedure (Proposition 2.3).
Poisson algebras. A commutative associative algebra D is called a Poisson algebra if it is a Lie algebra (D, {·, ·}) such that {a, xy}={a, x}y + x{a, y} for all elements a, x, y ∈ D.
For a K -algebra D,l e tD e r K (D) be the set of its K -derivations. If, in addition, Let D be a Poisson algebra, ∂ = (∂ 1 ,...,∂ n ) ∈ PDer K (D) n be an n-tuple of commuting derivations of the Poisson algebra D and X = (X 1 ,...,X n ) be an ntuple of commuting variables. The polynomial algebra D[X ]=D[X 1 ,...,X n ] with coefficients from D admits a Poisson structure which is an extension of the Poisson structure on D given by the rule (3) The Poisson algebra D[X ] is denoted by D[X ; ∂] and is called the Poisson Ore extension of D of rank n. Let G be a monoid. Suppose that the associative algebra D =⊕ g∈G D g is a Ggraded algebra (D g D h ⊆ D gh for all g, h ∈ G). If, in addition, D is a Poisson algebra and {D g , D h }⊆D gh for all g, h ∈ G, then we say that the Poisson algebra D is a G-graded Poisson algebra. The The Let us show that the Poisson structure on the polynomial algebra The Poisson structure on the algebra D[ For n ≥ 1, the Poisson algebra is an iteration of this construction n times. Consistency of the defining relations of generalized Weyl Poisson algebra follows from the next lemma.

Lemma 2.1 We keep the assumptions of the Definition of GWPA
where ∂(a) = (∂ 1 (a 1 ), . . . , ∂ n (a n )). Then, X 1 Y 1 − a 1 ,...,X n Y n − a n ∈ PZ(A) and the generalized Weyl Poisson algebra A = D[X , Y ; a,∂} is a factor algebra of the Poisson algebra A, Proof By the very definition, the element is The isomorphisms s I where I ⊆{ 1,...,n} of GWPAs of rank n.
is an isomorphism. Similarly, let A = D[X , Y ; a,∂} be a GWPA of rank n ≥ 1 and I be a subset of the set {1,...,n}.L e ts I be a bijection of the set X ∪ Y = {X 1 ,...,X 1 , Y 1 ,...,Y n } which is given by the rule Then, the Dhomomorphism of Poisson algebras is an isomorphism.
Recall that δ ij is the Kronecker delta function. The next proposition shows that the Poisson algebras D[X , Y ; ∂,α] are GWPAs.

Proposition 2.2 The Poisson algebra
Proof Consider the following elements of the polynomial algebra A = D[X , Y ], Examples of GWPAs 1. If D is a algebra with trivial Poisson bracket, then any choice of elements a = (a 1 ,...,a n ) and ∂ = (∂ 1 ,...,∂ n ) ∈ Der K (D) n such that ∂ i (a j ) = 0 for all i = j determines a GWPA D[X , Y ; a,∂} of rank n. If, in addition, n = 1, then there is no restriction on a 1 and ∂ 1 .
Let S be a multiplicative set of D. Then, is, in fact, isomorphic to a Poisson algebra in the paper of Cho and Oh [4] which is obtained as a quantization of a certain GWA with respect to the quantum parameter q.In [4,Theorem 3.7], a Poisson simplicity criterion is given for this Poisson algebra.
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.TheGWP AA = D[X , Y ; a,∂} of rank 1 is a generalization of some Poisson algebras that are associated with U (sl 2 ),seethe next example.
5. Let U = U (sl 2 ) be the universal enveloping algebra of the Lie algebra over a field K of characteristic zero. The associated graded algebra gr(U ) with respect to the filtration F ={F i } i∈N that is determined by the total degree of the elements X , is a GWPA of rank 1 where ∂ H := ∂ ∂ H . 6. Let U be the universal enveloping algebra of the Heisenberg Lie algebra Z is a Poisson central element .

Poisson simplicity criterion for generalized Weyl Poisson algebras
In this section, for generalized Weyl Poisson algebras, a proof of the Poisson simplicity criterion (Theorem 1.1) is given, an explicit description of their Poisson centre is obtained (Proposition 3.1) and a proof of the criterion for the Poisson centre being a field (Proposition 1.2)isgiven.
Let A be a Poisson algebra. An ideal I of the associative algebra A is called a Poisson ideal if {A, I }⊆I . A Poisson ideal is also called an ideal of the Poisson algebra. Suppose that D be a set of derivations of the associative algebra A. Then, the set A D := {a ∈ A | ∂(a) = 0 for all ∂ ∈ D} is a subalgebra of A which is called the algebra of D-constants (or the algebra of constants for D). An ideal J of the algebra The Poisson centre of a GWPA. Let A = D[X , Y ; a,∂} be a GWPA of rank n.For all elements λ, d ∈ D, α ∈ Z n and i = 1,...,n The next proposition describes the Poisson centre of a GWPA.  (16) and (17).
Since λv α is a unit, (λv α ) −1 = µv −α (since the algebra A is a Z n -graded algebra), and so 1 = λv α · µv −α = λµa |α| and 1 = µv −α · λv α = µλa |α| where a |α| := n i=1 a |α i | i ∈ PZ(A). Hence, a |α| is a unit in PZ(A); then, the elements λ and µ are units in D. Clearly, v := 1 + λv α ∈ PZ(A). The algebra A is a Z n -graded algebra. In particular, it is a Ze i -graded algebra (since Ze i ⊆ Z n ). Let l i be the length with respect to the Ze i -grading (which is a Z-grading). Then, for all nonzero elements u ∈ A, since the elements 1 and λ are units. This implies that the element u is not a unit. Therefore, D α = 0 for all α ∈ Z n \{0}, by Proposition 3.1. (⇐) Suppose that conditions 1 and 2 hold. Then, the implication follows from the Claim.
Claim. Suppose that conditions 1 and 2 hold. Then, every nonzero Poisson ideal of A intersects nontrivially PZ(A).
Let I be a nonzero Poisson ideal A. We have to show that I ∩ PZ(A) = 0. Let u = α∈Z n u α be a nonzero element of I where u α ∈ A α . The set supp(u) ={ α ∈ Z n | u α = 0} is called the support of u. Recall that, for α ∈ Z n , |α|=α 1 +···+α n . The additive group Z n admits the degree-by-lexicographic ordering ≤ where α<β iff either |α| < |β| or |α|=| β| and there exists an element i ∈{ 1,...,n} such that α j = β j for all j < i and α i <β i . Clearly, the inequalities α ≤ β and β ≤ α are equivalent to the equality α = β. The partially ordered set (Z n , ≤) is a linearly ordered set (for all distinct elements α, β ∈ Z n either α>βor α<β ) and α<β implies that α + γ<β+ γ for all γ ∈ Z n . Every nonzero element b = α∈Z n b α of A (where b α ∈ A α ) can be written as where α is the maximal element of supp(b) and the three dots denote smaller terms (i.e. the sum β<α b β ). The term b α = λ α v α is called the leading term of b, denoted lt(b), and the element λ α ∈ D is called the leading coefficient of b, denoted lc(b). Since the algebra A is a Z n -graded Poisson algebra, for all nonzero elements b, c ∈ A, provided lc(b)lc(c) = 0, and Up to isomorphism in (8) (i.e. interchanging some X i and Y i , if necessary), we can assume that the ideal I contains a nonzero element u = λ α X α + ··· where α 1 ≥ 0,...,α n ≥ 0. Then, the set of leading coefficients a ={λ α | u = λ α X α +··· ∈ I , all α i ≥ 0} of elements of I is a ∂-invariant ideal of the ring D since Therefore, by condition 1, there exists an element u = X α +··· ∈ I (i.e. λ α = 1). Then, using the equalities condition 2 and the fact that char(K ) = 0 (condition 3), we can assume that u = 1 +··· ∈ I , i.e. u = 1 + α<0 u α . For a finite set S, we denote by |S| the number of its elements. Let m = min{|supp(u)||u = 1 +··· ∈ I }.
We can assume that |supp(u)|=m. The Poisson algebra A is a