The generalized Weyl Poisson algebras and their Poisson simplicity criterion

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).

1 Introduction

In this paper, K is a field, algebra means a K-algebra (if it is not stated otherwise) and $K^*=K\backslash \{ 0\}$.

Generalized Weyl algebras, [1,2,3]. Let D be a ring, $\sigma =(\sigma _1,...,\sigma _n)$ be an n-tuple of commuting automorphisms of D, $a=(a_1,...,a_n)$ be an n-tuple of elements of the centre Z(D) of D such that $\sigma _i(a_j)=a_j$ for all $i\ne j$. The generalized Weyl algebra$A=D[X, Y; \sigma ,a]$ (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:

$$\begin{aligned} Y_iX_i= & {} a_i,\;\; X_iY_i=\sigma _i(a_i),\;\; X_id=\sigma _i(d)X_i,\;\; Y_id=\sigma _i^{-1}(d)Y_i\;\; (d \in D),\\ {[}X_i,X_j]= & {} [X_i,Y_j]=[Y_i,Y_j]=0, \;\; \mathrm{for \; all}\;\; i\ne j, \end{aligned}$$

where $[x, y]=xy-yx$. We say that a and $\sigma$ 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 K-algebra generated by 2n elements $X_1, ..., X_n,Y_1,..., Y_n$, subject to the relations:

$$\begin{aligned}{}[Y_i,X_i]=\delta _{ij}\;\; \mathrm{and}\;\; [X_i,X_j]=[Y_i,Y_j]=0\;\; \mathrm{for\,all}\; i,j, \end{aligned}$$

where $\delta _{ij}$ is the Kronecker delta function. The Weyl algebra $A_n$ is a generalized Weyl algebra $A=D[X, Y; \sigma ;a]$ of rank n where $D=K[H_1,...,H_n]$ is a polynomial ring in n variables with coefficients in K, $\sigma = (\sigma _1, \ldots , \sigma _n)$ where $\sigma _i(H_j)=H_j-\delta _{ij}$ and $a=(H_1, \ldots , H_n)$. The map

$$\begin{aligned} A_n\rightarrow A,\;\; X_i\mapsto X_i,\;\; Y_i \mapsto Y_i,\;\; i=1,\ldots ,n, \end{aligned}$$

is an algebra isomorphism (notice that $Y_iX_i\mapsto H_i$).

It is an experimental fact that many quantum algebras of small Gelfand-Kirillov dimension are GWAs (e.g. $U(\mathrm{sl}_2)$, $U_q(\mathrm{sl}_2)$, 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$D[X,Y; a, \partial \}$. 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 $(A, \{ \cdot , \cdot \})$ such that $\{ a, xy\}= \{ a, x\}y+x\{ a, y\}$ for all elements $a,x,y\in D$. Let A be a Poisson algebra with Poisson bracket $\{\cdot , \cdot \}$, $\mathrm{PZ} (A):=\{ a\in A\, | \, \{a, x\}=0$ for all $x\in A\}$ be its Poisson centre and $\mathrm{PDer}_K(A)$ be the set of derivations of the Poisson algebra A (see Sect. 2 for details).

Definition

Let D be a Poisson algebra, $\partial = (\partial _1, \ldots , \partial _n)\in \mathrm{PDer}_K(D)^n$ be an n-tuple of commuting derivations of the Poisson algebra D, $a=(a_1, \ldots , a_n)\in \mathrm{PZ}(D)^n$ be such that $\partial _i (a_j)=0$ for all $i\ne j$. The generalized Weyl algebra

$$\begin{aligned} A= & {} D[ X, Y; (\mathrm{id}_D, \ldots , \mathrm{id}_D), a]\\= & {} D[X_1, \ldots , X_n, Y_1, \ldots , Y_n]/(X_1Y_1-a_1, \ldots , X_nY_n-a_n) \end{aligned}$$

admits a Poisson structure which is an extension of the Poisson structure on D and is given by the rule: For all $i,j=1, \ldots , n$ and $d\in D$,

$$\begin{aligned} \{ Y_i, d\}= & {} \partial _i(d)Y_i, \;\; \{ X_i, d\}=-\,\partial _i(d)X_i \;\; \mathrm{and}\;\; \{ Y_i, X_i\} = \partial _i (a_i), \end{aligned}$$

(1)

$$\begin{aligned} \{ X_i, X_j\}= & {} \{ X_i, Y_j\}=\{ Y_i, Y_j\} =0\;\; \mathrm{for\; all}\;\; i\ne j. \end{aligned}$$

(2)

The Poisson algebra is denoted by $A =D[ X, Y; a, \partial \}$ and is called the generalized Weyl Poisson algebra of rank n (or GWPA, for short) where $X=(X_1, \ldots , X_n)$ and $Y= (Y_1, \ldots , Y_n)$.

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 $D[ X,Y; \partial , \alpha ]$ (see Sect. 2), for which existence problem has an easy solution and then to show that each GWPA is a factor algebra of some $D[ X,Y; \partial , \alpha ]$. The Poisson algebras $D[ X,Y; \partial , \alpha ]$ turn out to be also GWPAs (Proposition 2.2).

Poisson simplicity criterion for generalized Weyl Poisson algebras. A Poisson algebra is a simple Poisson algebra if the ideals 0 and A of the associative algebra A are the only ideals I such that $\{ A, I\} \subseteq I$. The ideal I is called a Poisson ideal of the Poisson algebra A. An ideal I of the ring D is called $\partial$-invariant, where $\partial = (\partial _1, \ldots , \partial _n)\in \mathrm{PDer}_K(D)^n$, if $\partial _i(I)\subseteq I$ for all $i=1, \ldots , n$. The set $D^\partial := \{ d\in D\, | \, \partial _1(d)=0, \ldots , \partial _n(d)=0\}$ is called the ring of $\partial$-constants of D.

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.

Theorem 1.1

Let $A=D[X,Y;a, \partial \}$ be a GWPA of rank n. Then, the Poisson algebra A is a simple Poisson algebra iff

• 1. the Poisson algebra D has no proper $\partial$-invariant Poisson ideals,

• 2. for all $i=1, \ldots , n$, $Da_i+D\partial _i(a_i) = D$, and

• 3. the algebra $\mathrm{PZ}(A)$ is a field, i.e. char$(K)=0$, $\mathrm{PZ}(D)^\partial$ is a field and $D_{\alpha }=0$ for all $\alpha \in \mathbb {Z}^n\backslash \{ 0\}$ (see the proposition below).

As a first step in the proof of Theorem 1.1, the following field criterion for the Poisson centre $\mathrm{PZ}(A)$ of a GWPA $A=D[X,Y;a, \partial \}$ of rank n is proven (in Sect. 3).

Proposition 1.2

Let $A= D[X,Y; a, \partial \}$ be a GWPA of rank n. Then, $\mathrm{PZ}(A)$ is a field iff char$(K)=0$, $\mathrm{PZ}(D)^\partial$ is a field and $D_{\alpha }=0$ for all $\alpha =(\alpha _1, \ldots , \alpha _n)\in \mathbb {Z}^n\backslash \{ 0\}$ where $D_\alpha = \{ \lambda \in D^\partial \, | \, \mathrm{pad }_\lambda := \{\lambda , \cdot \} = \lambda \sum _{i=1}^n \alpha _i \partial _i, \; \lambda \alpha _i\partial _i(a_i)=0$ for $i=1, \ldots , n\}$.

An explicit description of the Poisson centre is obtained (Proposition 3.1). Many examples are considered. We show that many classical Poisson algebras are GWPAs.

At the end of Sect. 2, we show that GWPAs appear as associated graded Poisson algebras of certain GWAs (Proposition 2.3). This is a sort of quantization procedure.

At the end of Sect. 3, examples of simple GWPAs (as Poisson algebras) are considered (Corollary 3.5). This family of simple Poisson algebras includes, as a particular case, the classical Poisson polynomial algebras$P_{2n} = K[X_1, \ldots , X_n, Y_1, \ldots , Y_n]$ ($\{ Y_i, X_j\} =\delta _{ij}$ and $\{X_i, X_j\} = \{ X_i, Y_j\} = \{ Y_i, Y_j\} =0$ for all $i\ne j$).

2 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 $(D, \{ \cdot , \cdot \})$ such that $\{ a, xy\}= \{ a, x\}y+x\{ a, y\}$ for all elements $a,x,y\in D$.

For a K-algebra D, let $\mathrm{Der }_K(D)$ be the set of its K-derivations. If, in addition, $(D, \{ \cdot , \cdot \} )$ is a Poisson algebra, then

$$\begin{aligned} \mathrm{PDer}_K(D):=\{ \delta \in \mathrm{Der }_K(D)\, | \, \delta (\{ a,b\})= \{ \delta (a),b\}+\{ a,\delta (b)\} \;\; \mathrm{for\; all}\;\; a,b\in D\} \end{aligned}$$

is the set of derivations of the Poisson algebra D. The vector space $\mathrm{Der }_K(D)$ is a Lie algebra, where $[\delta , \partial ]:= \delta \partial -\partial \delta$, and $\mathrm{PDer}_K(D)$ is a Lie subalgebra of $\mathrm{Der }_k(D)$. The set of inner derivations

$$\begin{aligned} \mathrm{IDer}_K(D):= \{ \mathrm{ad }_a\, | \, a\in D\}\;\; \mathrm{(where} \;\; \mathrm{ad }_a (b):= [a,b]:=ab-ba) \end{aligned}$$

is an ideal of the Lie algebra $\mathrm{Der }_K(D)$ (since $[\delta , \mathrm{ad }_a]= \mathrm{ad }_{\delta (a)}$ for all $\delta \in \mathrm{Der }_K(D)$ and $a\in D$). Similarly, the set of inner derivations of the Poisson algebraD

$$\begin{aligned} \mathrm{PIDer}_K(D):= \{ \mathrm{pad }_a\, | \, a\in D\}\;\; \mathrm{(where}\;\; \mathrm{pad }_a (b):= \{ a,b\} ) \end{aligned}$$

is an ideal of the Lie algebra $\mathrm{PDer}_K(D)$ (since $[\delta ,\mathrm{pad }_a]=\mathrm{pad }_{\delta (a)}$ for all $\delta \in \mathrm{PDer}_K(D)$ and $a\in D$). By the very definition, the Poisson algebra D is a Lie algebra with respect to the bracket $\{ \cdot , \cdot \}$. The map $D\rightarrow \mathrm{PIDer}_K(D)$, $a\mapsto \mathrm{pad }_a$, is an epimorphism of Lie algebras with kernel

$$\begin{aligned} \mathrm{PZ}(D):= \{ a\in D\, | \, \{ a, D\} =0\} \end{aligned}$$

which is called the centre of the Poison algebra (or the Poisson centre of D). So, the Poisson structure of the algebra D induces the ‘multiplicative structure’ on the Lie algebra $\mathrm{PIDer}_K(D)$, i.e. $\mathrm{pad }_{ab}(\cdot )= \mathrm{pad }_{a}(\cdot )\, b+a\, \mathrm{pad }_{b}(\cdot )$.

Notice that the centre$Z(D) :=\{ z\in D\, | \, zd=dz$ for all $d\in D\}$ of any associative algebra D is invariant under the action of $\mathrm{Der }_K(D)$: Let $z\in Z(D)$, $d\in D$ and $\partial \in \mathrm{Der }_K(D)$; then, applying the derivation $\partial$ to the equality $zd=dz$, we obtain the equality $\partial (z) d= d\partial (z)$, i.e. $\partial (z) \in Z(D)$. Similarly, the Poisson centre $\mathrm{PZ}(D)$ is invariant under the action of $\mathrm{PDer}_K(D)$: Let $z\in \mathrm{PZ}(D)$, $d\in D$ and $\partial \in \mathrm{PDer}_K(D)$; then, applying the derivation $\partial$ to the equality $\{ z,d\} =0$, we obtain the equality $\{ \partial (z) ,d\}=0$, i.e. $\partial (z) \in \mathrm{PZ}(D)$.

Let D be a Poisson algebra, $\partial = (\partial _1, \ldots , \partial _n)\in \mathrm{PDer}_K(D)^n$ be an n-tuple of commuting derivations of the Poisson algebra D and $X= (X_1, \ldots , X_n)$ be an n-tuple of commuting variables. The polynomial algebra $D[X]=D[X_1, \ldots , X_n]$ with coefficients from D admits a Poisson structure which is an extension of the Poisson structure on D given by the rule

$$\begin{aligned} \{ X_i, X_j\}=0\;\; \mathrm{and}\;\; \{ X_i, d\} = \partial _i(d) X_i\;\; \mathrm{for\;} 1\le i,j\le n\;\; \mathrm{and}\;\; d\in D. \end{aligned}$$

(3)

The Poisson algebra D[X] is denoted by $D[ X; \partial ]$ and is called the Poisson Ore extension of D of rank n.

Let G be a monoid. Suppose that the associative algebra $D= \oplus _{g\in G} D_g$ is a G-graded algebra ($D_gD_h\subseteq D_{gh}$ for all $g,h\in G$). If, in addition, D is a Poisson algebra and $\{ D_g, D_h\}\subseteq D_{gh}$ for all $g,h\in G$, then we say that the Poisson algebra D is a G-graded Poisson algebra.

The Poisson algebra$D[X,Y; \partial , \alpha ]$. Now, we introduce a class of Poisson algebras which is used in the proof of existence of GWPAs (Lemma 2.1).

Definition

Let D be a Poisson algebra, $\partial = (\partial _1, \ldots , \partial _n)\in \mathrm{PDer}_K(D)^n$ be an n-tuple of commuting derivations of the Poisson algebra D and $\alpha = (\alpha _1, \ldots , \alpha _n)\in \mathrm{PZ}(D)^n$. Then, the polynomial algebra $D[X,Y]=D[X_1, \ldots , X_n, Y_1, \ldots , Y_n]$ with coefficients in D admits a Poisson structure which is an extension of the Poisson structure on D given by the rule: For all $i,j=1, \ldots , n$ and $d\in D$,

$$\begin{aligned} \{ Y_i, d\}= & {} \partial _i(d)Y_i, \;\; \{ X_i, d\}=-\,\partial _i(d)X_i \;\; \mathrm{and}\;\; \{ Y_i, X_i\} = \alpha _i, \end{aligned}$$

(4)

$$\begin{aligned} \{ X_i, X_j\}= & {} \{ X_i, Y_j\}=\{ Y_i, Y_j\} =0\;\; \mathrm{for\; all}\;\; i\ne j. \end{aligned}$$

(5)

The Poisson algebra D[X, Y] is denoted by $\mathcal{A}=D[ X, Y; \partial , \alpha ]$ where $X=(X_1, \ldots , X_n)$ and $Y= (Y_1, \ldots , Y_n)$.

Let us show that the Poisson structure on the polynomial algebra D[X, Y] is well defined. Let $n=1$. The Poisson algebra $D[X_1, Y_1; \partial _1, \alpha _1]$ is an extension of the Poisson Ore extension $D[ X_1; -\partial _1]$ by adding a commuting variable $Y_1$ where the Poisson structure on the algebra $D[X_1][Y_1]$ is given by the rule

$$\begin{aligned} \{ Y_1, d\} = \partial _1(d) Y_1\;\; \mathrm{and}\;\; \{ Y_1, X_1\} = \alpha _1. \end{aligned}$$

The Poisson structure on the algebra $D[X_1][Y_1]$ is well defined as $\{ Y_1, \cdot \}$ respects the relation $\{ X_1, d\} = -\,\partial _1(d) X_1$ for all $d\in D$:

$$\begin{aligned} \{ Y_1, \{ X_1, d\}\}= & {} \{ \alpha _1, d\} +\{ X_1, \partial _1 (d) Y_1\}= 0 -\partial _1^2(d)X_1Y_1-\partial _1(d) \alpha _1\\= & {} -\,\{ Y_1, \partial _1(d) X_1\}. \end{aligned}$$

For $n\ge 1$, the Poisson algebra

$$\begin{aligned} D[ X, Y; \partial , \alpha ]=D[ X_1, Y_1; \partial _1, \alpha _1]\cdots [ X_n, Y_n; \partial _n, \alpha _n]. \end{aligned}$$

(6)

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 $A=D[X,Y; a, \partial \}$. Let $\mathcal{A}= D[X,Y; \partial , \partial (a)]$ where $\partial (a) = (\partial _1(a_1), \ldots , \partial _n(a_n))$. Then, $X_1Y_1-a_1, \ldots , X_nY_n-a_n\in \mathrm{PZ}(\mathcal{A})$ and the generalized Weyl Poisson algebra $A =D[ X, Y; a, \partial \}$ is a factor algebra of the Poisson algebra $\mathcal{A}$,

$$\begin{aligned} A\simeq \mathcal{A}/(X_1Y_1-a_1, \ldots , X_nY_n-a_n). \end{aligned}$$

Proof

By the very definition, the element $Z_i=X_iY_i-a_i\in \mathrm{PZ}(A)$: For all i, j such that $i\ne j$, $\{ X_j, Z_i\} = \partial _j(a_i) X_j=0$ and $\{ Y_j, Z_i\} = -\,\partial _j(a_i) Y_j=0$ (since $\partial _j(a_i)=0$ for all $i\ne j$). For all $d\in D$,

$$\begin{aligned} \{ Z_i, d\}= & {} \{ X_i, d\} Y_i+X_i\{ d, Y_i\} = -\,\partial _i(d) X_iY_i+X_i\partial (d) Y_i=0, \\ \{ X_i, Z_i\}= & {} X_i(- \partial _i(a_i))+\partial _i(a_i) X_i=0, \\ \{ Y_i, Z_i\}= & {} \partial _i(a_i)Y_i-\partial _i(a_i) Y_i=0. \end{aligned}$$

Therefore, $Z_i\in \mathrm{PZ}(\mathcal{A})$. Now, the lemma is obvious. $\square$

A$\mathbb {Z}^n$-grading of a GWPA$A= D[X,Y; a, \partial \}$. The GWPA of rank n,

$$\begin{aligned} A:= D[X,Y; a, \partial \} =\bigoplus _{\alpha \in \mathbb {Z}^n}A_\alpha , \end{aligned}$$

(7)

is a $\mathbb {Z}^n$-graded Poisson algebra where $A_\alpha = Dv_\alpha$, $v_\alpha =\prod _{i=1}^n v_{\alpha _i}(i)$ and

$$\begin{aligned} v_j(i)={\left\{ \begin{array}{ll} X_i^j&{}\quad \text {if}j>0,\\ 1&{}\quad \text {if}j=0,\\ Y_i^{|j|}&{}\quad \text {if }j<0.\\ \end{array}\right. } \end{aligned}$$

So, $A_\alpha A_\beta \subseteq A_{\alpha +\beta }$ and $\{ A_\alpha , A_\beta \}\subseteq A_{\alpha +\beta }$ for all elements $\alpha , \beta \in \mathbb {Z}^n$.

The isomorphisms$s_I$where$I\subseteq \{ 1, \ldots , n\}$of GWPAs of rankn. Let $A=D[X_1,Y_1; a_1, \partial _1\}$ be a GWPA of rank 1. Clearly, $A\simeq D[Y_1,X_1; a_1, -\partial _1\}$, i.e. the D-homomorphism of Poisson algebras

$$\begin{aligned}&s_1: A=D[X_1,Y_1; a_1, \partial _1 \}\rightarrow D[Y_1,X_1; a_1, -\partial _1\}, \;\;\nonumber \\&X_1\mapsto Y_1, \;\; Y_1\mapsto X_1, \;\; \nonumber \\&d\mapsto d\;\; (d\in D), \end{aligned}$$

(8)

is an isomorphism. Similarly, let $A= D[X,Y; a,\partial \}$ be a GWPA of rank $n\ge 1$ and I be a subset of the set $\{ 1, \ldots , n\}$. Let $s_I$ be a bijection of the set $X\cup Y=\{ X_1, \ldots , X_1, Y_1, \ldots , Y_n\}$ which is given by the rule

$$\begin{aligned} s_I(X_i)= {\left\{ \begin{array}{ll} Y_i&{} \text {if }i\in I,\\ X_i&{} \text {if }i\not \in I, \\ \end{array}\right. } \;\; \mathrm{and}\;\; s_I(Y_i)= {\left\{ \begin{array}{ll} X_i&{} \text {if }i\in I,\\ Y_i&{} \text {if }i\not \in I. \\ \end{array}\right. } \end{aligned}$$

Let $\mathrm{sign}(I)\partial := (\varepsilon _1 \partial _1, \ldots ,\varepsilon _n\partial _n )$ where $\varepsilon _i={\left\{ \begin{array}{ll} -1&{} \text {if }i\in I,\\ 1&{} \text {if }i\not \in I. \\ \end{array}\right. }$ Then, the D-homomorphism of Poisson algebras

$$\begin{aligned}&s_I: A\rightarrow D[s_I(X), s_I(Y); a, \mathrm{sign}(I)\partial \}, \;\; \nonumber \\&X_i\mapsto s_I(X_i), \;\; Y_i\mapsto s_I(Y_i), \;\; \nonumber \\&d\mapsto d \;\; (d\in D), \end{aligned}$$

(9)

is an isomorphism.

Recall that $\delta _{ij}$ is the Kronecker delta function. The next proposition shows that the Poisson algebras $D[X,Y; \partial , \alpha ]$ are GWPAs.

Proposition 2.2

The Poisson algebra $\mathcal{A}= D[X,Y; \partial , \alpha ]$ is a GWPA of rank n

$$\begin{aligned} D[H_1, \ldots , H_n] [X,Y; H, \partial \} \end{aligned}$$

where $D[H_1, \ldots , H_n]$ is a Poisson polynomial algebra over D such that $\{ H_i, D\} =0$ and $\{ H_i, H_j\} =0$ for all i, j, $H=(H_1, \ldots , H_n)$ and $\partial _i(H_j) =\delta _{ij}\alpha _jH_j$ for all i, j.

Proof

Consider the following elements of the polynomial algebra $\mathcal{A}=D[X,Y]$,

$$\begin{aligned} H_1=X_1Y_1, \ldots , H_n=X_nY_n. \end{aligned}$$

Then, $\{ H_i, D\}=0$ and $\{ H_i, H_j\} =0$ for all i, j. So, the elements $H_1, \ldots , H_n$ belong to the Poisson centre of the Poisson algebra $\mathcal{D}= D[H_1, \ldots , H_n]$. Let $A= D[H_1, \ldots , H_n] [X,Y; H, \partial \}$. It follows from the defining relations of the Poisson algebras $\mathcal{A}$ and A that there is an epimorphism $\mathcal{A}\rightarrow A$ of Poisson algebras given by the rule $X_i\mapsto X_i$, $Y_i\mapsto Y_i$, $d\mapsto d$ where $d\in D$ (since $X_iY_i\mapsto H_i$) which is clearly a bijection (it is the ‘identity map’ of associative algebras when we identify $X_iY_i$ with $H_i$). $\square$

By Proposition 2.2, the Poisson algebra $\mathcal{A}= D[X,Y; \partial , \alpha ]=\oplus _{\beta \in \mathbb {Z}^n} \mathcal{A}_\beta$ is $\mathbb {Z}^n$-graded ($\mathcal{A}_\beta \mathcal{A}_\gamma \subseteq \mathcal{A}_{\beta +\gamma }$ and $\{ \mathcal{A}_\beta , \mathcal{A}_\gamma \} \subseteq \mathcal{A}_{\beta +\gamma }$ for all $\beta , \gamma \in \mathbb {Z}^n$) where $\mathcal{A}_\beta = \mathcal{D}v_\beta$, $\mathcal{D}= D[H_1, \ldots , H_n]$ and $v_\beta = \prod _{i=1}^n v_{\beta _i}(i)$ where $v_j(i)={\left\{ \begin{array}{ll} X_i^j&{} \text {if }j>0,\\ 1&{} \text {if }j=0,\\ Y_i^{|j|}&{} \text {if }j<0.\\ \end{array}\right. }$

Examples of GWPAs 1. If D is a algebra with trivial Poisson bracket, then any choice of elements $a=(a_1, \ldots , a_n)$ and $\partial = (\partial _1, \ldots , \partial _n)\in \mathrm{Der }_K(D)^n$ such that $\partial _i(a_j) = 0$ for all $i\ne j$ determines a GWPA $D[X,Y;a, \partial \}$ of rank n. If, in addition, $n=1$, then there is no restriction on $a_1$ and $\partial _1$.

2. The classical Poisson polynomial algebra$P_{2n} = K[X_1, \ldots , X_n, Y_1, \ldots , Y_n]$ ($\{ Y_i, X_j\} =\delta _{ij}$ and $\{X_i, X_j\} = \{ X_i, Y_j\} = \{ Y_i, Y_j\} =0$ for all $i\ne j$) is a GWPA

$$\begin{aligned} P_{2n}=K[H_1, \ldots , H_n][X,Y; a, \partial \} \end{aligned}$$

(10)

where $K[H_1, \ldots , H_n]$ is a Poisson polynomial algebra with trivial Poisson bracket, $a=(H_1, \ldots , H_n)$, $\partial = (\partial _1, \ldots , \partial _n)$ and $\partial _i= \frac{\partial }{\partial _{H_i}}$ (via the isomorphism of Poisson algebras $P_{2n}\rightarrow K[H_1, \ldots , H_n][X,Y; a, \partial \}$, $X_i\mapsto X_i$, $Y_i\mapsto Y_i$).

3. $A= D[X,Y; a,\partial \}$ where $D=K[H_1, \ldots , H_n]$ is a Poisson polynomial algebra with trivial Poisson bracket, $a= (a_1, \ldots , a_n)\in K[H_1]\times \cdots \times K[H_n]$, $\partial = (\partial _1, \ldots , \partial _n)$ where $\partial _i=b_i\partial _{H_i}$ (where $\partial _{H_i}=\frac{\partial }{\partial H_i}$) and $b_i\in K[H_i]$. In particular, $D[X,Y; (H_1, \ldots , H_n), (\partial _{H_1}, \ldots , \partial _{H_n})\}=P_{2n}$ is the classical Poisson polynomial algebra.

Let S be a multiplicative set of D. Then, $S^{-1}A\simeq (S^{-1} D)[X,Y; a, \partial \}$ is a GWPA. In particular, for $S=\{ H^\alpha \, | \, \alpha \in \mathbb {Z}^n\}$, we have $K[H_1^{\pm 1}, \ldots , H_n^{\pm 1}][X,Y; a, \partial \}$. In the case $n=1$, the Poisson algebra

$$\begin{aligned} K[H_1^{\pm 1}][X_1, Y_1; a_1, -H_1\frac{d}{dH_1}\} \end{aligned}$$

where $a_1\in K[H_1^{\pm 1}]$ 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, $a\in D$ and $\partial$ is a derivation of the algebra D. The GWPA $A= D[X, Y; a, \partial \}$ of rank 1 is a generalization of some Poisson algebras that are associated with $U(\mathrm{sl}_2)$, see the next example.

5. Let $U=U(\mathrm{sl}_2)$ be the universal enveloping algebra of the Lie algebra

$$\begin{aligned} \mathrm{sl_2}=K\langle X,Y, H\, | \, [H, X]=X,\;\; [ H, Y]=-\,Y, \;\; [ X,Y]=2H\rangle \end{aligned}$$

over a field K of characteristic zero. The associated graded algebra $\mathrm{gr} (U)$ with respect to the filtration $\mathcal{F}= \{ \mathcal{F}_i\}_{i\in \mathbb {N}}$ that is determined by the total degree of the elements X, Y and H is a Poisson polynomial algebra K[X, Y, H] where

$$\begin{aligned} \{ H, X\} = X, \;\; \{ X, Y\} =-\,Y\;\; \mathrm{and}\;\; \{ X,Y\} = 2H. \end{aligned}$$

The element $C=XY+H^2$ belongs to the Poisson centre of the Poisson polynomial algebra $\mathrm{gr}(U)$. The Poisson algebra

$$\begin{aligned} \mathrm{gr}(U)=K[C,H][X,Y;a=C-H^2, \partial _H\} \end{aligned}$$

(11)

is a GWPA of rank 1 where $\partial _H:=\frac{\partial }{\partial H}$.

6. Let U be the universal enveloping algebra of the Heisenberg Lie algebra

$$\begin{aligned} \mathcal{H}_n= & {} K\langle X_1, \ldots , X_n, Y_1, \ldots , Y_n, Z\, | \, [X_i, Y_j]=\delta _{ij} Z,\; [X_i, X_j]=[Y_i, Y_j]=0\; \mathrm{for\; all}\;\; i,j;\\&Z\; \mathrm{is \; a \; Poisson \; central \; element}\rangle . \end{aligned}$$

The associated graded algebra $\mathrm{gr}(U)$ with respect to the filtration by the total degree of the canonical generators is a Poisson polynomial algebra $K[X_1, \ldots , X_n, Y_1, \ldots , Y_n, Z]$ where, for all i, j,

$$\begin{aligned} \{ X_i, Y_j\} = \delta _{ij}Z, \;\; \{ X_i, X_j\} = \{ Y_i, Y_j\}=0 \end{aligned}$$

and the element Z belongs to the Poisson centre of $\mathrm{gr}(U)$. Then, the polynomial algebra

$$\begin{aligned} \mathrm{gr}(U)= D[X,Y;a, \partial \} \end{aligned}$$

(12)

is a GWPA of rank n where $D= K[H_1, \ldots , H_n, Z]$ is a Poisson polynomial algebra with trivial Poisson bracket, $X= (X_1, \ldots , X_n)$, $Y=(Y_1, \ldots , Y_n)$, $a=(a_1=H_1, \ldots , a_n=H_n)$, $\partial = (Z\partial _{H_1}, \ldots , Z\partial _{H_n})$ and $\partial _{H_i}:=\frac{\partial }{\partial H_i}$.

Let $A_s= D_s[X_{(s)},Y_{(s)}; a_{(s)}, \partial _{(s)}\}$ be GWPAs of rank $n_s$ where $s=1, \ldots , m$. The tensor product of algebras

$$\begin{aligned} A=\bigotimes _{s=1}^m A_s=\bigg ( \bigotimes _{s=1}^m D_s\bigg ) [X,Y; a, \partial \} \end{aligned}$$

(13)

is a GWPA of rank $n_1+\cdots + n_m$ where $X=(X_{(1)}, \ldots , X_{(m)})$, $Y=(Y_{(1)}, \ldots , Y_{(m)})$, $a=(a_{(1)}, \ldots , a_{(m)})$ and $\partial =(\partial _{(1)}, \ldots , \partial _{(m)})$. The Poisson structure on A is a tensor product of Poisson structures on $A_s$, i.e. for all elements $u=\otimes _{s=1}^mu_s$, $v=\otimes _{s=1}^mv_s\in A$ (where $u_s, v_s\in A_s$),

$$\begin{aligned} \{ u,v\} = \sum _{s=1}^mu_1v_1\otimes \cdots \otimes \{ u_s, v_s\} \otimes \cdots \otimes u_mv_m. \end{aligned}$$

Example

The classical Poisson polynomial algebra $P_{2n}$ [see (10)] is the tensor product $P_2^{\otimes n}$ of n copies of the classical Poisson polynomial algebra $P_2$.

An algebraic torus action on a GWPA Let $A= D[X,Y; a, \partial \}$ be a GWPA of rank n and $\mathrm{Aut}_\mathrm{Pois}(A)$ be the group of automorphisms of the Poisson algebra A. Elements of $\mathrm{Aut}_\mathrm{Pois}(A)$ are called Poisson automorphisms of A. For each element $\lambda = (\lambda _1, \ldots , \lambda _n)\in K^{*n}$, the K-algebra homomorphism

$$\begin{aligned} t_\lambda : A\rightarrow A, \;\; X_i\mapsto \lambda _i X_i, \;\; Y_i\mapsto \lambda _i^{-1} Y_i, \;\; d\mapsto d\;\; (d\in D), \end{aligned}$$

is an automorphism of the Poisson algebra A. The subgroup $\mathbb {T}^n = \{ t_\lambda \, | \, \lambda \in K^{*n}\}$ of $\mathrm{Aut}_\mathrm{Pois}(A)$ is an algebraic torus$\mathbb {T}^n \simeq K^{*n}$, $t_\lambda \mapsto \lambda$. For all $\alpha \in \mathbb {Z}^n$ and $u_\alpha \in A_\alpha = Dv_\alpha$, $t_\lambda (u_\alpha ) = \lambda ^\alpha \cdot u_\alpha$ where $\lambda ^\alpha = \prod _{i=1}^n \lambda _i^{\alpha _i}$.

The subgroup

$$\begin{aligned} \mathrm{Aut}_\mathrm{Pois}(D)^{\partial , a}:=\{ \sigma \in \mathrm{Aut}_\mathrm{Pois}(D) \, | \, \sigma \partial _i = \partial _i \sigma \;\; \mathrm{and} \;\; \sigma (a_i) = a_i\;\; \mathrm{for}\;\; i=1, \ldots , n\} \end{aligned}$$

of $\mathrm{Aut}_\mathrm{Pois}(D)$ can be seen as a subgroup of $\mathrm{Aut}_\mathrm{Pois}(A)$ where each automorphism $\sigma \in \mathrm{Aut}_\mathrm{Pois}^{\partial , a}(D)$ trivially acts at X and Y, i.e. $\sigma (X_i) = X_i$ and $\sigma (Y_i) = Y_i$. Clearly,

$$\begin{aligned} \mathbb {T}^n\times \mathrm{Aut}_\mathrm{Pois}(D)^{\partial , a} \subseteq \mathrm{Aut}_\mathrm{Pois}(A). \end{aligned}$$

(14)

Associated graded algebra. of a GWA is a GWPA Let $A=D[X,Y; \sigma , a]$ be a GWA of rank n such that $D=\cup _{i\in \mathbb {N}}D_i$ is a filtered algebra ($D_iD_j\subseteq D_{i+j}$ for all $i,j\in \mathbb {N}$; $D_{-1}=0$),

$$\begin{aligned}{}[d_i, d_j]\in D_{i+j-\nu }\;\; \mathrm{for \; all}\;\; d_i\in D_i\;\; \mathrm{and}\;\; d_j\in D_j\;\; \mathrm{where}\;\; \nu \;\; \mathrm{is\; a \; positive \; integer;} \end{aligned}$$

$\sigma _i(D_j) = D_j$ and $(\sigma _i-1)(D_j) \subseteq D_{j-\nu }$ for all $i=1, \ldots , n$ and $j\in \mathbb {N}$. Suppose that $a_i\in D_{d_i}\backslash D_{d_i-1}$ for some $d_i\ge 1$. The algebra A admits a filtration $\{ A_s\}_{s\in \frac{1}{2}\mathbb {N}}$ where

$$\begin{aligned}&A_s=\sum _{i+d\cdot \alpha \le s}D_iv_\alpha , \;\; d=(d_1, \ldots , d_n), \;\; \alpha = (\alpha _1, \ldots , \alpha _n)\in \mathbb {Z}^n\;\; \mathrm{and} \;\; d\cdot \alpha \\&\quad = \frac{1}{2}\sum _{i=1}^n d_i|\alpha _i|. \end{aligned}$$

The associated graded algebra

$$\begin{aligned} \mathrm{gr} (A)= & {} \mathrm{gr}(D)[X, Y; (\mathrm{id}, \ldots , \mathrm{id}), \overline{a}]\\= & {} \mathrm{gr}(D)[X_1,\ldots , X_n, Y_1, \ldots , Y_n]/(X_1Y_1-\overline{a}_1, \ldots , X_nY_n-\overline{a}_n) \end{aligned}$$

is a commutative GWA where $\overline{a}_i = a_i+D_{d_i-1}\in D_{d_i}/D_{d_i-1}$. For all elements $u_s\in A_s$ and $u_t\in A_t$,

$$\begin{aligned}{}[u_s, u_t]\in A_{s+t-\nu }. \end{aligned}$$

(15)

Let $\overline{u}_s= u_s+A_{s-1}\in A_s/A_{s-1}$ and $\overline{u}_t= u_t+A_{t-1}\in A_t/A_{t-1}$. The bracket

$$\begin{aligned} \{ u_s, u_t\}:=\overline{[u_s, u_t]}:=[u_s, u_t]+ A_{s+t-\nu -1}\in A_{s+t-\nu }/A_{s+t-\nu -1} \end{aligned}$$

determines the Poisson structure on $\mathrm{gr}(A)$. For each $i=1, \ldots , n$, the map

$$\begin{aligned} \partial _i:= \overline{\sigma _i-1}:\mathrm{gr} (D)\rightarrow \mathrm{gr}(D),\;\; \mathrm{gr}(D)_j\ni \overline{b}_j\mapsto (\sigma _i-1) (b_j) +D_{j-\nu -1}\in \mathrm{gr}(D)_{j-\nu }, \end{aligned}$$

is a K-derivation of the commutative algebra $\mathrm{gr}(D)$. The derivations $\partial _1, \ldots , \partial _n$ commute since the automorphisms $\sigma _1, \ldots , \sigma _n$ commute. Notice that

$$\begin{aligned}{}[X_i, b_j]=(\sigma _i-1) (b_j) X_i\;\; \mathrm{and}\;\; [Y_i, b_j]=(\sigma _i^{-1}-1) (b_j) Y_i. \end{aligned}$$

Hence, $\{ X_i, \overline{b}_j\} = \partial _i(b_j) X_i$ and $\{ Y_i, \overline{b}_j\} = -\partial _i(b_j) Y_i$ since

$$\begin{aligned} (\sigma _i^{-1} -1) (b_j) = -\,(\sigma _i-1)\sigma _i^{-1} (b_j) \equiv -\partial _i(\overline{b}_j)\mod D_{j-\nu -1}. \end{aligned}$$

Therefore, the Poisson algebra $\mathrm{gr} (A)$ is a GWPA $\mathrm{gr}(D)[X, Y; \overline{a}, -\partial \}$ where $\overline{a}= (\overline{a}_1, \ldots , \overline{a}_n)$ and $-\partial = (-\partial _1, \ldots , -\partial _n)$. So, we proved that the following proposition holds.

Proposition 2.3

Let $A=D[X,Y; \sigma , a]$ be a GWA of rank n such that $D=\cup _{i\in \mathbb {N}}D_i$ is a filtered algebra; $[d_i, d_j]\in D_{i+j-\nu }$ for all $d_i\in D_i$ and $d_j\in D_j$ where $\nu$ is a positive integer; $\sigma _i(D_j) = D_j$ and $(\sigma _i-1)(D_j) \subseteq D_{j-\nu }$ for all $i=1, \ldots , n$ and $j\in \mathbb {N}$. Suppose that $a_i\in D_{d_i}\backslash D_{d_i-1}$ for some $d_i\ge 1$. Let $\{ A_s\}_{s\in \frac{1}{2}\mathbb {N}}$ be the filtration as above. The associated graded algebra $\mathrm{gr}(A)$ is a GWPA $\mathrm{gr}(D)[X, Y; \overline{a}, -\partial \}$ where $\overline{a}$ and $-\partial$ are defined above.

Example

1. The n’th Weyl algebra $A_n$ is a GWA $K[H_1, \ldots , H_n][X,Y; \sigma , a]$ where $\sigma _i(H_j) = H_j-\delta _{ij}$ and $a_i= H_i$ for $i,j=1, \ldots , n$. The polynomial algebra $D= K[H_1, \ldots , H_n]$ admits a natural filtration $\{ D_i\}_{i\in \mathbb {N}}$ by the total degree of the variables $H_1, \ldots , H_n$. The automorphisms $\sigma _1, \ldots , \sigma _n$ satisfy the conditions of Proposition 2.3 with $\nu = 1$, $d_1=\cdots = d_n=1$ and $\partial _1=-\,\frac{\partial }{\partial H_1}, \ldots ,\partial _n=-\,\frac{\partial }{\partial H_n}$. Notice that $\mathrm{gr}(D) = D$. By Proposition 2.3, the algebra

$$\begin{aligned} \mathrm{gr}(A_n) \simeq D[X,Y]/(X_1Y_1-H_1, \ldots , X_nY_n-H_n)\simeq K[X,Y]=P_{2n} \end{aligned}$$

is a GWPA $D[X,Y; (H_1, \ldots , H_n), (\frac{\partial }{\partial H_1}, \ldots ,\frac{\partial }{\partial H_n} )\}$ which is the classical Poisson algebra $P_{2n}$ with the canonical Poisson bracket ($\{ Y_i, X_j\} = \delta _{ij}$, $\{ X_i, X_j\} = \{ X_i, Y_j\} = \{ Y_i, Y_j\} =0$ for all i, j such that $i\ne j$).

2. The universal enveloping algebra $U= U(\mathrm{sl}_2)$ is the GWA $A= K[C,H][X,Y; \sigma , a]$ of rank 1 where $\sigma (H) = H-1$, $\sigma (C) = C$ and $a= C-H(H+1)$ (the element C is the Casimir element, $C= YX+H(H+1)$). The filtration $\mathcal{F}= \{ \mathcal{F}_i \}_{i\in \mathbb {N}}$ on U that was considered above (which is defined by the total degree of the canonical generators X, Y and H) induces a filtration $\{ D_i:= D\cap \mathcal{F}_i \}_{i\in \mathbb {N}}$ on the polynomial algebra $D=K[C,H]$. Clearly,

$$\begin{aligned} D_i = \bigoplus _{2s+t\le i}KC^sH^t\;\; \mathrm{for\; all}\;\; i\in \mathbb {N}. \end{aligned}$$

The automorphism $\sigma$ and the filtration $\{ D_i\}_{i\in \mathbb {N}}$ satisfy the conditions of Proposition 2.3 where $d_1=2$ and $\nu = -\,1$. The associated graded Poisson algebra $\mathrm{gr}(A)\simeq K[C, H][X, Y; C-H^2, \partial _H\}$ is canonically isomorphic to the associated graded Poisson algebra $\mathrm{gr}(U)$ as $\mathbb {N}$-graded Poisson algebra (since $\mathrm{gr}(A)_{\frac{1}{2}+i}=0$ for all $i\in \mathbb {N}$), see (11).

The filtration $\{ D'_i:=\oplus _{j\le i}K[C]H^i\}_{i\in \mathbb {N}}$ also satisfies the conditions of Proposition 2.3 where $d_1=2$ and $\nu = -\,1$ but the associated graded algebra $\mathrm{gr}'(A)$ is a GWPA $K[C,H][X,Y; -H^2, \partial _H\}$. The associated graded Poisson algebras $\mathrm{gr}(A)$ and $\mathrm{gr}'(A)$ are not isomorphic since the algebra $\mathrm{gr}(A)$ is smooth but the algebra $\mathrm{gr}'(A)\simeq K[C]\otimes K[X,Y](XY-H^2)$ is singular as the points $\{ (C,H,X,Y)=(\lambda , 0,0,0)\, | \, \lambda \in K\}$ are singular. So, the Poisson algebras $\mathrm{gr}(A)$ and $\mathrm{gr}'(A)$ are also not isomorphic.

3 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 $\{ A, I\} \subseteq I$. A Poisson ideal is also called an ideal of the Poisson algebra. Suppose that $\mathcal{D}$ be a set of derivations of the associative algebra A. Then, the set $A^\mathcal{D}:=\{ a\in A\, | \, \partial (a) =0$ for all $\partial \in \mathcal{D}\}$ is a subalgebra of A which is called the algebra of$\mathcal{D}$-constants (or the algebra of constants for $\mathcal{D}$). An ideal J of the algebra A is called a $\mathcal{D}$-invariant ideal if $\partial (J)\subseteq J$ for all $\partial \in \mathcal{D}$.

The Poisson centre of a GWPA. Let $A= D[X,Y; a, \partial \}$ be a GWPA of rank n. For all elements $\lambda , d\in D$, $\alpha \in \mathbb {Z}^n$ and $i=1, \ldots , n$

$$\begin{aligned} \{ d, \lambda v_\alpha \}= & {} (-\mathrm{pad }_\lambda +\lambda \sum _{i=1}^n \alpha _i\partial _i)(d) v_\alpha , \end{aligned}$$

(16)

$$\begin{aligned} \{ v_{\pm 1}(i), \lambda v_\alpha \}= & {} {\left\{ \begin{array}{ll} \mp \partial _i(\lambda ) v_{\alpha \pm e_i}&{} \text {if }\alpha _i=0 \;\; \mathrm{or}\;\; \mathrm{sign}(\alpha _i) = \pm ,\\ (\mp \partial _i(\lambda ) a_i +\lambda \alpha _i\partial _i(a_i) ) v_{\alpha \pm e_i}&{} \text {if } \mathrm{sign}(\alpha _i) = \mp .\\ \end{array}\right. }\nonumber \\ \end{aligned}$$

(17)

The next proposition describes the Poisson centre of a GWPA.

Proposition 3.1

Let $A= D[X,Y; a, \partial \}$ be a GWPA of rank n. Then, $\mathrm{PZ}(A) = \bigoplus _{\alpha \in \mathbb {Z}^n} \mathrm{PZ}(A)_\alpha$ is a $\mathbb {Z}^n$-graded (associative) algebra where $\mathrm{PZ}(A)_\alpha = D_\alpha v_\alpha$, $D_0=\mathrm{PZ}(D)^\partial$ and, for all $\alpha \ne 0$, $D_\alpha = \{ \lambda \in D^\partial \, | \, \mathrm{pad }_\lambda = \lambda \sum _{i=1}^n \alpha _i \partial _i, \; \lambda \alpha _i\partial _i(a_i)=0$ for $i=1, \ldots , n\}$.

Proof

The GWPA $A=\oplus _{\alpha \in \mathbb {Z}^n} A_\alpha$ is a $\mathbb {Z}^n$-graded Poisson algebra, hence so is its Poisson centre, i.e. $\mathrm{PZ}(A) = \oplus _{\alpha \in \mathbb {Z}^n} \mathrm{PZ}(A)_\alpha$ where $\mathrm{PZ}(A)_\alpha = \mathrm{PZ}(A)\cap A_\alpha$. Since $A_\alpha = Dv_\alpha$ for all $\alpha \in \mathbb {Z}^n$, statement 2 follows from (16) and (17). $\square$

The next corollary shows that, in general, the Poisson centre of a GWPA A is small.

Corollary 3.2

Let $A= D[X,Y;a,\partial \}$ be a GWPA of rank n. Suppose that char$(K)=0$ and the elements $\partial _1(a_1), \ldots , \partial _n(a_n)$ are nonzero divisors in the algebra D (e.g. D is a domain and $\partial _1(a_1)\ne 0, \ldots , \partial _n(a_n)\ne 0$). Then, $\mathrm{PZ}(A) = \mathrm{PZ}(D)^\partial$.

For an element $\alpha = (\alpha _1, \ldots , \alpha _n)\in \mathbb {Z}^n$, the set $\mathrm{supp }(\alpha ):=\{ i\, | \, \alpha _i\ne 0\}$ is called the support of $\alpha$.

Corollary 3.3

Let $A= D[X,Y;a,\partial \}$ be a GWPA of rank n. Suppose that char$(K)=0$. Then, for all elements $\alpha \in \mathbb {Z}^n\backslash \{ 0\}$, $D_\alpha \subseteq D^{\partial , \mathrm{pad }(\partial (a))}\cap \mathrm{ann}_D \{\partial _i(a_i)\, | \,$$i\in \mathrm{supp }(\alpha )\}$, i.e.

• 1. $\{ D_\alpha , a_i\} =0$ for $i=1, \ldots , n$, and

• 2. $D_\alpha \partial _i (a_i)=0$ for all i such that $\alpha _i\ne 0$.

Proof

By Proposition 3.1.(3), $D_\alpha \partial _i (a_i)=0$ for all $i\in \mathrm{supp }(\alpha )$ (since char$(K)=0$). Then, for all $\lambda \in D_\alpha$ and $i=1, \ldots , n$, $\{ \lambda , a_i\} = \mathrm{pad }_\lambda (a_i) = \sum _{i=1}^n \lambda \alpha _i \partial _i(a_i) =0$, i.e. $\{ D_\alpha , a_i\} =0$ for $i=1, \ldots , n$. $\Box$

Let $A=\bigoplus _{i\in \mathbb {Z}} A_i$ be a $\mathbb {Z}$-graded (associative) algebra. Each element $a\in A$ is a unique sum $a=\sum _{i\in \mathbb {Z}} a_i$ where $a_i\in A_i$. The lengthl(a) of the element a is equal to $-\infty$ if $a=0$, and, for $a\ne 0$, $l(a):= n-m$ where $n=\max \{ i \, | \, a_i\ne 0\}$ and $m=\min \{ i \, | \, a_i\ne 0\}$.

Let A be a Poisson algebra and $z\in \mathrm{PZ}(A)$. The zA is a Poisson ideal of A. If the Poisson algebraAis simple, then necessarily the Poisson centre$\mathrm{PZ}(A)$is a field.

Proof of Proposition 1.2

$(\Rightarrow )$ Suppose that $p=\mathrm{char}(K)\ne 0$. Then, by Proposition 3.1, the element $1+X^p$ of $\mathrm{PZ}(A)$ is not invertible. Therefore, we must have $p=0$. The algebras A and $\mathrm{PZ}(A)$ are $\mathbb {Z}^\alpha$-graded algebras and $\mathrm{PZ}(A)_0= \mathrm{PZ}(D)^\partial$. Therefore, $\mathrm{PZ}(D)^\partial$ must be a field.

Suppose that $D_{\alpha }\ne 0$ for some $\alpha \ne 0$. Then, $\alpha _i\ne 0$ for some i. Fix a nonzero element of $\mathrm{PZ}(A)_\alpha =D_{\alpha }v_\alpha$, say $\lambda v_\alpha$ where $\lambda \in D_{\alpha }$. Since $\lambda v_\alpha$ is a unit, $(\lambda v_\alpha )^{-1}=\mu v_{-\alpha }$ (since the algebra A is a $\mathbb {Z}^n$-graded algebra), and so

$$\begin{aligned} 1= \lambda v_\alpha \cdot \mu v_{-\alpha }=\lambda \mu a^{|\alpha |}\;\; \mathrm{and}\;\; 1=\mu v_{-\alpha }\cdot \lambda v_\alpha = \mu \lambda a^{|\alpha |} \end{aligned}$$

where $a^{|\alpha |}:=\prod _{i=1}^na_i^{|\alpha _i|}\in \mathrm{PZ}(A)$. Hence, $a^{|\alpha |}$ is a unit in $\mathrm{PZ}(A)$; then, the elements $\lambda$ and $\mu$ are units in D. Clearly, $v:=1+\lambda v_\alpha \in \mathrm{PZ}(A)$. The algebra A is a $\mathbb {Z}^n$-graded algebra. In particular, it is a $\mathbb {Z}e_i$-graded algebra (since $\mathbb {Z}e_i \subseteq \mathbb {Z}^n$). Let $l_i$ be the length with respect to the $\mathbb {Z}e_i$-grading (which is a $\mathbb {Z}$-grading). Then, for all nonzero elements $u\in A$,

$$\begin{aligned} l_i(uv) = l_i(u) +l_i(v) \ge l_i(v)=|\alpha _i|>0, \end{aligned}$$

since the elements 1 and $\lambda$ are units. This implies that the element u is not a unit. Therefore, $D_{\alpha }=0$ for all $\alpha \in \mathbb {Z}^n \backslash \{ 0\}$, by Proposition 3.1.(3).

$(\Leftarrow )$ By Proposition 3.1, $\mathrm{PZ}(A) = \mathrm{PZ}(D)^\partial$ is a field. $\square$

An ideal I of an algebra A is called a proper ideal if $I\ne 0, A$.

Proof of Theorem 1.1

$(\Rightarrow )$ Suppose that $\mathfrak {a}$ is a proper $\partial$-invariant Poisson ideal of the Poisson algebra D, then $\mathfrak {a}A=\oplus _{\alpha \in \mathbb {Z}^n}\mathfrak {a}v_\alpha$ is a proper ideal of the Poisson algebra A. So, the first condition holds.

Suppose that $\mathfrak {b}:= Da_i+D\partial _i(a_i) \ne D$ for some i. Then,

$$\begin{aligned} I=\bigoplus _{\alpha \in \mathbb {Z}^n, \alpha _i\ne 0}Dv_\alpha \oplus \bigoplus _{\alpha \in \mathbb {Z}^n, \alpha _i = 0} \mathfrak {b}v_\alpha \end{aligned}$$

is a proper ideal of the Poisson algebra A. So, the second condition holds.

The third condition obviously holds. (If a nonzero element z of $\mathrm{PZ}(A)$ is also a nonunit, then zA is a proper Poisson ideal of A).

$(\Leftarrow )$ 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 $\mathrm{PZ}(A)$ .

Let I be a nonzero Poisson ideal A. We have to show that $I\cap \mathrm{PZ}(A) \ne 0$. Let $u=\sum _{\alpha \in \mathbb {Z}^n} u_\alpha$ be a nonzero element of I where $u_\alpha \in A_\alpha$. The set $\mathrm{supp }(u) = \{ \alpha \in \mathbb {Z}^n\, | \, u_\alpha \ne 0\}$ is called the support of u. Recall that, for $\alpha \in \mathbb {Z}^n$, $|\alpha |=\alpha _1+\cdots +\alpha _n$. The additive group $\mathbb {Z}^n$ admits the degree-by-lexicographic ordering$\le$ where $\alpha <\beta$ iff either $|\alpha | <|\beta |$ or $|\alpha | =|\beta |$ and there exists an element $i\in \{ 1, \ldots , n\}$ such that $\alpha _j= \beta _j$ for all $j<i$ and $\alpha _i<\beta _i$. Clearly, the inequalities $\alpha \le \beta$ and $\beta \le \alpha$ are equivalent to the equality $\alpha = \beta$. The partially ordered set $(\mathbb {Z}^n, \le )$ is a linearly ordered set (for all distinct elements $\alpha , \beta \in \mathbb {Z}^n$ either $\alpha >\beta$ or $\alpha <\beta$) and $\alpha < \beta$ implies that $\alpha +\gamma <\beta +\gamma$ for all $\gamma \in \mathbb {Z}^n$. Every nonzero element $b=\sum _{\alpha \in \mathbb {Z}^n} b_\alpha$ of A (where $b_\alpha \in A_\alpha$) can be written as

$$\begin{aligned} b= b_\alpha +\cdots \end{aligned}$$

where $\alpha$ is the maximal element of $\mathrm{supp }(b)$ and the three dots denote smaller terms (i.e. the sum $\sum _{\beta <\alpha }b_{\beta }$). The term $b_\alpha = \lambda _\alpha v_\alpha$ is called the leading term of b, denoted $\mathrm{lt}(b)$, and the element $\lambda _\alpha \in D$ is called the leading coefficient of b, denoted $\mathrm{lc}(b)$. Since the algebra A is a $\mathbb {Z}^n$-graded Poisson algebra, for all nonzero elements $b,c\in A$,

$$\begin{aligned} \mathrm{lt}(bc) = \mathrm{lt}(b) \mathrm{lt}(c) \end{aligned}$$

(18)

provided $\mathrm{lc}(b) \mathrm{lc}(c)\ne 0$, and

$$\begin{aligned} \mathrm{lt}(\{ b,c\} ) = \{ \mathrm{lt}(b), \mathrm{lt}(c)\} \end{aligned}$$

(19)

provided $\{ \mathrm{lt}(b), \mathrm{lt}(c)\} \ne 0$.

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 $u=\lambda _\alpha X^\alpha +\cdots$ where $\alpha _1\ge 0, \ldots , \alpha _n\ge 0$. Then, the set of leading coefficients

$$\begin{aligned} \mathfrak {a}= \{ \lambda _\alpha \, | \, u = \lambda _\alpha X^\alpha +\cdots \in I, \;\ \mathrm{all}\;\; \alpha _i\ge 0\} \end{aligned}$$

of elements of I is a $\partial$-invariant ideal of the ring D since

$$\begin{aligned} d_1ud_2= & {} d_1\lambda _\alpha d_2X^\alpha +\cdots \;\;\;\;\; \;\;\; \; \mathrm{if}\; d_1\lambda _\alpha d_2\ne 0\;\; (d_1, d_2\in D),\\ uX^\beta= & {} \lambda _\alpha X^{\alpha +\beta }+\cdots , \\ \{ u, X_i\}= & {} \partial _i(\lambda _\alpha ) X^{\alpha +e_i}+\cdots \;\;\;\;\;\; \mathrm{if}\;\; \partial _i(\lambda _\alpha )\ne 0. \end{aligned}$$

Therefore, by condition 1, there exists an element $u=X^\alpha +\cdots \in I$ (i.e. $\lambda _\alpha =1$). Then, using the equalities

$$\begin{aligned} Y_iX_i^\alpha = a_iX^{\alpha -e_i}\;\; \mathrm{and}\;\;\{ Y_i, X^\alpha \} =\alpha _i\partial _i (a_i) X^{\alpha _i-e_i}, \end{aligned}$$

condition 2 and the fact that char$(K)=0$ (condition 3), we can assume that $u=1+\cdots \in I$, i.e. $u=1+\sum _{\alpha <0}u_\alpha$. For a finite set S, we denote by |S| the number of its elements. Let

$$\begin{aligned} m = \min \{ |\mathrm{supp }(u)|\, | \, u=1+\cdots \in I\}. \end{aligned}$$

We can assume that $|\mathrm{supp }(u)|=m$. The Poisson algebra A is a $\mathbb {Z}^n$-graded Poisson algebra. Hence, by the choice of m, for all elements $d\in D$ and $i=1, \ldots , n$:

$$\begin{aligned} 0= & {} \{ d,u\} =\sum _{\alpha<0} \{ d, u_\alpha \},\;\;\;\;\;\;\;\, \mathrm{i.e.}\;\; \{ d,u_\alpha \}=0, \\ 0= & {} \{ X_i,u\} =\sum _{\alpha<0} \{ X_i, u_\alpha \},\;\;\; \mathrm{i.e.}\;\; \{ X_i,u_\alpha \}=0, \\ 0= & {} \{ Y_i,u\} =\sum _{\alpha <0} \{ Y_i, u_\alpha \},\;\;\;\;\; \mathrm{i.e.}\;\; \{ Y_i,u_\alpha \}=0, \end{aligned}$$

i.e. all $u_\alpha \in \mathrm{PZ}(A)$, and so $0\ne u\in \mathrm{PZ}(A)$, as required. $\square$

Corollary 3.4

Let $A=D[X,Y;a, \partial \}$ be a GWPA of rank n. Suppose that the conditions 1 and 2 of Theorem 1.1 hold. Then, every nonzero Poisson ideal of A intersects $\mathrm{PZ}(A)$ nontrivially.

Proof

The corollary is precisely the Claim in the proof of Theorem 1.1. $\square$

Corollary 3.5

Let $D=K[H_1, \ldots , H_n]$ be a Poisson polynomial algebra with trivial Poisson bracket, $a=(a_1, \ldots , a_n)$ where $a_i\in K[H_i]$ and $\partial = (b_1\partial _{H_1}, \ldots , b_n\partial _{H_n})$ where $b_i\in K[H_i]$. Then, the GWPA $A=D[X,Y;a, \partial \}$ of rank n is a simple Poisson algebra iff char$(K)=0$, $b_1, \ldots , b_n\in K^*:=K\backslash \{ 0\}$ and $K[H_i]a_i+K[H_i]\frac{da_i}{dH_i}=K[H_i]$ for $i=1, \ldots , n$.

Proof

The corollary follows from Theorem 1.1. In more detail, condition 2 of Theorem 1.1 is equivalent to the conditions $K[H_i]a_i+K[H_i]\frac{da_i}{dH_1}=K[H_i]$ for $i=1, \ldots , n$ (since $a_i\in K[H_i]$). Condition 1 of Theorem 1.1 is equivalent to the condition char$(K)=0$ and $b_1, \ldots , b_n\in K^*:=K\backslash \{ 0\}$ (since $b_iD$ is a $\partial$-invariant ideal of D). If conditions 1 and 2 hold, then condition 3 of Theorem 1.1 holds automatically since $D^\partial =K=\mathrm{PZ}(D)$ (then $D_{\alpha }=0$ for all $\alpha \in \mathbb {Z}^n\backslash \{ 0\}$). $\square$

By Corollary 3.5, the classical Poisson polynomial algebra

$$\begin{aligned} P_{2n}\simeq K[H_1, \ldots , H_n][X,Y;(H_1, \ldots , H_n), (\partial _{H_1}, \ldots , \partial _{H_n})\} \end{aligned}$$

is a simple Poisson algebra.