The generalized Weyl Poisson algebras and their Poisson simplicity criterion

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).


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V. V. Bavula the centre Z (D) of D such that σ i (a j ) = a j for all i = j. The generalized 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] = x y − 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 i j 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 i j and a = (H 1 , . . . , H n ). The map 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 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, x y} = {a, x}y + x{a, y} for all elements a, x, y ∈ D.
For a K -algebra D, let Der K (D) be the set of its K -derivations. If, in addition, (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 Therefore, Z i ∈ PZ(A). Now, the lemma is obvious. is the D-homomorphism of Poisson algebras 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}. Let s 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 i j 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 ], It follows from the defining relations of the Poisson algebras A and A that there is an epimorphism A → A of Poisson algebras given by the rule is clearly a bijection (it is the 'identity map' of associative algebras when we identify X i Y i with H i ).

By Proposition 2.2, the Poisson algebra
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, In the case n = 1, the Poisson algebra 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 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 .

An algebraic torus action on a GWPA Let
is an automorphism of the Poisson algebra A. The subgroup T n = {t λ | λ ∈ K * n } of Aut Pois (A) is an algebraic torus T n K * n , t λ → λ. For all α ∈ Z n and u α ∈ A α = Dv α , t λ (u α ) = λ α · u α where λ α = n i=1 λ α i i .
Associated graded algebra of a GWA is a GWPA. Let A = D[X , Y ; σ, a] be a GWA of rank n such that D = ∪ i∈N D i is a filtered algebra (D i D j ⊆ D i+ j for all i, j ∈ N; The associated graded algebra Let determines the Poisson structure on gr(A). For each i = 1, . . . , n, the map is a K -derivation of the commutative algebra gr(D). The derivations ∂ 1 , . . . , ∂ n commute since the automorphisms σ 1 , . . . , σ n commute. Notice that Therefore, the Poisson algebra gr(A) is a GWPA gr(D)[X , Y ; a, −∂} where a = (a 1 , . . . , a n ) and −∂ = (−∂ 1 , . . . , −∂ n ). So, we proved that the following proposition holds.

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) is given.
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).
The next corollary shows that, in general, the Poisson centre of a GWPA A is small. For an element α = (α 1 , . . . , α n ) ∈ Z n , the set supp(α) := {i | α i = 0} is called the support of α. Suppose that D α = 0 for some α = 0. Then, α i = 0 for some i. Fix a nonzero element of PZ(A) α = D α v α , say λv α where λ ∈ D α . 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).
We can assume that |supp(u)| = m. The Poisson algebra A is a  Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.