# The Poisson enveloping algebra and the algebra of Poisson differential
operators of a generalized Weyl Poisson algebra

V. V. Bavula

###### Abstract

For a generalized Weyl Poisson algebra $A$, explicit sets of generators and
defining relations are presented for its Poisson enveloping algebra ${\cal
U}(A)$. Simplicity criteria are given for the algebra ${\cal U}(A)$ and
algebra of Poisson differential operators $P{\cal D}(A)$ on $A$. The Gelfand-
Kirillov dimensions of the algebras ${\cal U}(A)$ and $P{\cal D}(A)$ are
calculated. It is proven that the algebra ${\cal U}(A)$ is a domain provided
that the coefficient ring $D$ of the generalized Weyl Poisson algebra $A$ is a
domain of essentially finite type over a perfect field.

For the algebra $A$, the set of its minimal primes and the prime radical are
described and an equidimensionality criterion is given. For the
equidimensional algebra $A$ of essentially finite type, two regularity
criteria are presented.

${\bf Contents}$

  1. 1.

Introduction.

  2. 2.

Regularity and equidimensionality criteria for commutative generalized Weyl
algebras.

  3. 3.

Generators and defining relations of the Poisson enveloping algebra of a
generalized Weyl Poisson algebra.

  4. 4.

A simplicity criterion for the Poisson enveloping algebra of a generalized
Weyl Poisson algebra.

Mathematics subject classification 2020: 17B63, 17B65, 17B20, 16S32, 16D30,
14F10, 16P90, 13N15, 14J17, 14B05.

##  1 Introduction

In this paper, $K$ is a field of arbitrary characteristic,
$P_{n}=K[x_{1},\ldots,x_{n}]$ is a polynomial algebra in $n$ variables over
the field $K$, $D$ is a $K$-algebra and

$A=D[X,Y;a]=D[X_{1},\ldots,X_{N},Y_{1},\ldots,Y_{N}]/(X_{1}Y_{1}-a_{1},\ldots,X_{N}Y_{N}-a_{N})$

is a commutative generalized Weyl algebra (GWA) of rank $N$ where
$X=(X_{1},\ldots,X_{N})$, $Y=(Y_{1},\ldots,Y_{N})$ and
$a=(a_{1},\ldots,a_{N})\in D^{N}$.

The set of minimal primes and an equidimensionality criterion for a GWA $A$.
For an algebra $R$, we denote by $\min(R)$ the set of its minimal primes.
Proposition 2.1 gives an explicit description of the set of minimal primes of
a commutative generalized Weyl algebra $A$ via the set of minimal primes of
the Noetherian algebra $D$, it also describes the prime radical of the algebra
$A$ via the prime radical of the algebra $D$. Proposition 2.1 also gives an
equidimensionality criterion for the GWA $A$.

Regularity criteria for the GWA $A$. In the case when the algebra $D$ is of
essentially finite type of pure dimension $d<\infty$ over a perfect field $K$,
two regularity criteria are given for the GWA $A=D[X,Y;a]$ of rank $N$,
Theorem 2.2 and Theorem 2.3. The first criterion, Theorem 2.2, is given via an
explicit ideal of the algebra $D$. The second criterion, Theorem 2.3, is given
in terms of a new concept introduced in the paper - a homological sequence of
algebra (every regular sequence is a homological sequence but not the other
way round, in general). Proposition 2.5 collects some of the properties of
homological sequences.

The generalized Weyl Poisson algebras and their Poisson simplicity criterion.
In [6], a new large class of Poisson algebras - the class of generalized Weyl
Poisson algebras - is introduced.

An associative commutative algebra $A$ is called a Poisson algebra if it is a
Lie algebra $(A,\\{\cdot,\cdot\\})$ that satisfies the Leibniz’s rule: For all
elements $a,x,y\in D$,

$\\{a,xy\\}=\\{a,x\\}y+x\\{a,y\\}.$

The set ${\rm PZ}(A):=\\{a\in A\,|\,\\{a,x\\}=0$ for all $x\in A\\}$ is called
the Poisson centre of $A$, it is a subalgebra of $A$. Let ${\rm Der}_{K}(A)$
be the set of $K$-derivations of the associative algebra $A$. Then

${\rm PDer}_{K}(D):=\\{\delta\in{\rm
Der}_{K}(D)\,|\,\delta(\\{a,b\\})=\\{\delta(a),b\\}+\\{a,\delta(b)\\}\;\;{\rm
for\;all}\;\;a,b\in D\\}$

is the set of derivations of the Poisson algebra $D$. Elements of ${\rm
PDer}(A)$ are called Poisson derivations of $A$.

Definition, [6]. Let $D$ be a Poisson algebra,
$\partial=(\partial_{1},\ldots,\partial_{n})\in{\rm PDer}_{K}(D)^{n}$ be an
$n$-tuple of commuting derivations of the Poisson algebra $D$,
$a=(a_{1},\ldots,a_{n})\in{\rm PZ}(D)^{n}$ be such that
$\partial_{i}(a_{j})=0$ for all $i\neq j$. The commutative generalized Weyl
algebra

$A=D[X,Y;a]=D[X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n}]/(X_{1}Y_{1}-a_{1},\ldots,X_{n}Y_{n}-a_{n})$

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

$\\{Y_{i},d\\}=\partial_{i}(d)Y_{i},\;\;\\{X_{i},d\\}=-\partial_{i}(d)X_{i}\;\;{\rm
and}\;\;\\{Y_{i},X_{i}\\}=\partial_{i}(a_{i}),$ (1)
$\\{X_{i},X_{j}\\}=\\{X_{i},Y_{j}\\}=\\{Y_{i},Y_{j}\\}=0\;\;{\rm
for\;all}\;\;i\neq j.$ (2)

The Poisson algebra is denoted by $A=D[X,Y;a,\partial\\}$ and is called the
generalized Weyl Poisson algebra of degree/rank $n$ (or GWPA, for short) where
$X=(X_{1},\ldots,X_{n})$ and $Y=(Y_{1},\ldots,Y_{n})$.

Examples of GWPAs are given in Section 3. A Poisson simplicity criterion is
given in [6, Theorem 1.1], see Theorem 4.2. This result is used in several
theorems of this paper.

Generators and defining relations of the Poisson enveloping algebra ${\cal
U}({\cal A})$ of a GWPA $A$. For an arbitrary Lie algebra ${\cal G}$, every
${\cal G}$-module is a module over its universal enveloping algebra $U({\cal
G})$, which is an associative algebra, and vice versa. Furthermore, the
defining relations of the algebra $U({\cal G})$ are precisely the defining
relations of the Lie algebra ${\cal G}$ where the Lie bracket
$[\cdot,\cdot]_{\cal G}$ of ${\cal G}$ is ‘replaced’ by the algebra
commutator: If $\\{X_{i}\\}_{i\in I}$ is a $K$-basis of the Lie algebra ${\cal
G}$ then the algebra $U({\cal G})$ is generated by the elements
$\\{X_{i}\\}_{i\in I}$ that subject to the defining relations:

$X_{i}X_{j}-X_{j}X_{i}=[X_{i},X_{j}]_{\cal G}\;\;{\rm for\;all}\;\;i,j\in I.$

Similarly, for each Poisson algebra ${\cal P}$ there is a concept of Poisson
module over ${\cal P}$, and every Poisson module over ${\cal P}$ is a module
over, the so-called, Poisson enveloping algebra ${\cal U}({\cal P})$, which is
an associative algebra, and vice versa. In [7], for each Poisson algebra
${\cal P}$ that is defined by generators abd defining relations, explicit sets
of generators and defining relations are given for its Poisson enveloping
algebra ${\cal U}({\cal P})$, [7, Theorem 2.2], see Theorem 3.1. Using this
result, for each GWPA $A$, Theorem 3.2 gives explicit sets of generators and
defining relations for the algebra ${\cal U}(A)$. Examples are considered in
Proposition 3.3, Corollary 3.4 and Corollary 3.5.

The Poincaré-Birkhof-Witt Theorem for Poisson algebras. The (classical)
Poincaré-Birkhof-Witt Theorem states that for each Lie algebra ${\cal G}$
there is a natural isomorphism of graded algebras,

${\rm gr}\,U({\cal G})\simeq{\rm Sym}({\cal G}),$

where ${\rm gr}\,U({\cal G})$ is the associated graded algebra of the
universal enveloping algebra $U({\cal G})$ of the Lie algebra ${\cal G}$ and
${\rm Sym}({\cal G})$ is the symmetric algebra of ${\cal G}$. For a smooth
Poisson algebra ${\cal P}$, a similar result holds [18, Theorem 3.1]:

${\rm gr}\,{\cal U}({\cal P})\simeq{\rm Sym}_{\cal P}(\Omega_{\cal P}),$

where ${\rm gr}\,{\cal U}({\cal P})$ is the associated graded algebra of the
Poisson enveloping algebra ${\cal U}({\cal P})$ of the Poisson algebra ${\cal
P}$ and ${\rm Sym}_{\cal P}(\Omega_{\cal P})$ is the symmetric algebra of the
${\cal P}$-module $\Omega_{\cal P}$ of Kähler differentials of the associative
algebra ${\cal P}$. In fact, [18, Theorem 3.1] holds in sightly more general
situation, namely, for the universal enveloping algebra of the Lie-Reinhatr
algebra. The pair $({\cal P},\Omega_{\cal P})$ is an example of a Lie-Reinhart
algebra and its universal enveloping algebra $V({\cal P},\Omega_{\cal P})$ is
isomorphic to the Poisson enveloping algebra (PEA) ${\cal U}({\cal P})$, [18,
Section 2, p.197] (see also [11]). Recently, it was proven that the PBW
Theorem holds for certain singular Poisson hypersurfaces, [12, Theorem 3.7].
[7, Theorem 3.5] states that the Poincaré-Birkhof-Witt Theorem holds for all
Poisson algebras (over an arbitrary field).

In [16], the Poisson enveloping algebra of a Poisson algebra was introduced as
a universal object in a certain category and an alternative to Reinhart’s
proof of its existence was given. For certain classes of Poisson algebras
explicit descriptions of their Poisson enveloping algebras were presented in
[17, 19, 20, 13, 14, 12].

Simplicity criterion for the algebra $P{\cal D}(A)$ of Poisson differential
operators on the generalized Weyl Poisson algebra $A$. Corollary 4.3 is a
simplicity criterion for the algebra of Poisson differential operators $P{\cal
D}(A)$ on a generalized Weyl Poisson algebra $A$.

Simplicity criterion for the Poisson enveloping algebra ${\cal U}(A)$ of the
generalized Weyl Poisson algebra $A$. Corollary 4.5 is a simplicity criterion
for the algebra ${\cal U}(A)$ where $A$ is a generalized Weyl Poisson algebra.
If in addition, the coefficient ring $D$ of the generalized Weyl Poisson
algebra $A=D[X,Y;a,\partial\\}$ is a regular domain of essentially finite type
over a field of characteristic zero, Theorem 4.9 is an explicit simplicity
criterion for the algebra ${\cal U}(A)$. Theorem 4.9.(4) is a very efficient
tool in proving/disproving simplicity of the algebra ${\cal U}(A)$.

The Gelfand-Kirillov dimension of the algebras ${\cal U}(A)$, $P{\cal D}(A)$,
${\rm gr}\,{\cal U}(A)$ and ${\rm Sym}_{A}(\Omega_{A})$. Let
$A=D[X,Y;a,\partial\\}$ be a generalized Weyl Poisson algebra where $D$ is a
domain of essentially finite type over a perfect field. Let $\Omega_{A}$ be
the $A$-module of Kähler differential of the algebra $A$ and ${\rm
Sym}_{A}(\Omega_{A})$ be its symmetric algebra. Corollary 4.11 (resp.,
Corollary 4.13) gives an exact figure for the Gelfand-Kirillov dimension of
the algebras ${\cal U}(A)$, ${\rm gr}\,{\cal U}(A)$ and ${\rm
Sym}_{A}(\Omega_{A})$ (resp., $P{\cal D}(A)$, char$(K)=0$).

The algebra ${\cal U}(A)$ is a domain when $A$ is a regular domain of
essentially finite type. Let $A=D[X,Y;a,\partial\\}$ be a generalized Weyl
Poisson algebra where $D$ is a domain of essentially finite type over a
perfect field. Corollary 4.15 states that the algebra ${\cal U}(A)$ is a
domain provided the algebra $A$ is a regular domain.

##  2 Regularity and equidimensionality criteria for commutative generalized
Weyl algebras

The aim of this section is to give regularity criteria for a commutative
generalized Weyl algebra $A=D[X,Y;a]$ of rank $N$ where $D$ is a commutative
algebra of essentially finite type of pure dimension $d<\infty$ over a perfect
field $K$. The first criterion, Theorem 2.2, is given in terms of an explicit
ideal of the ring $D$. The second criterion, Theorem 2.3, is more conceptual
and is given via a new concept - a homological sequence of algebra. One of the
key results in proving Theorem 2.3 is Proposition 2.1 which describes the set
of minimal primes of the algebra $A$ via the set of minimal primes of the
algebra $D$ and gives an equidimensionality criterion for the algebra $A$.

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 $i\neq j$.
The generalized Weyl algebra $A=D[X,Y;\sigma,a]$ (briefly GWA) of degree/rank
$n$ is a ring generated by $D$ and $2n$ indeterminates
$X_{1},...,X_{n},Y_{1},...,Y_{n}$ subject to the defining relations:

$Y_{i}X_{i}=a_{i},\;\;X_{i}Y_{i}=\sigma_{i}(a_{i}),\;\;X_{i}d=\sigma_{i}(d)X_{i},\;\;Y_{i}d=\sigma_{i}^{-1}(d)Y_{i}\;\;(d\in
D),$ $[X_{i},X_{j}]=[X_{i},Y_{j}]=[Y_{i},Y_{j}]=0,\;\;{\rm for\;all}\;\;i\neq
j,$

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:

$[Y_{i},X_{i}]=\delta_{ij}\;\;{\rm
and}\;\;[X_{i},X_{j}]=[Y_{i},Y_{j}]=0\;\;{\rm for\,all}\;i,j,$

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

$A_{n}\rightarrow A,\;\;X_{i}\mapsto X_{i},\;\;Y_{i}\mapsto
Y_{i},\;\;i=1,\ldots,n,$

is an algebra isomorphism (notice that $Y_{i}X_{i}\mapsto H_{i}$).

It is an experimental fact that many quantum algebras of small Gelfand-
Kirillov dimension are GWAs (eg, $U({\rm sl}_{2})$, $U_{q}({\rm sl}_{2})$, 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 mention 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, [9], etc.

A ${\mathbb{Z}}^{n}$-grading on a GWPA. The GWPA of rank $n$,

$A:=D[X,Y;a,\partial\\}=\bigoplus_{\alpha\in{\mathbb{Z}}^{n}}A_{\alpha},$ (3)

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

$v_{j}(i)=\begin{cases}X_{i}^{j}&\text{if }j>0,\\\ 1&\text{if }j=0,\\\
Y_{i}^{|j|}&\text{if }j<0.\\\ \end{cases}$

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 set of minimal primes of a generalized Weyl algebra. For an algebra $R$,
$\min(R)$ is the set of its minimal primes and $\mathfrak{n}_{R}$ is its prime
radical. We say that a commutative algebra $R$ has pure dimension $d$ if the
Krull dimension of all factor algebras $R/\mathfrak{p}$, where
$\mathfrak{p}\in\min(R)$, is $d$. We also say the the algebra $R$ is
equidimensional. Let ${\cal C}_{R}$ be the set of all regular elements of the
algebra $R$, (i.e. the set of non-zero-divisors).

Poposition 2.1 gives an explicit description of the set of minimal primes of a
commutative generalized Weyl algebra $A$ via the set of minimal primes of the
Noethirian algebra $D$, it also describes the prime radical of the algebra $A$
via the prime radical of the algebra $D$.

###### Proposition 2.1

Let $D$ be a commutative Noetherian ring, $A=D[X,Y;a]$ be a commutative GWA of
rank $N$, for each $\mathfrak{p}\in\min(D)$, let

${\rm in}(\mathfrak{p})=\\{j\,|\,a_{j}\in\mathfrak{p},1\leq j\leq N\\}\;\;{\rm
and}\;\;{\rm out}(\mathfrak{p})=\\{j\,|\,a_{j}\not\in\mathfrak{p},1\leq j\leq
N\\}.$

Then

  1. 1.

$\min(A)=\coprod_{\mathfrak{p}\in\min(D)}\min(A,\mathfrak{p})$ where

$\displaystyle\min(A,\mathfrak{p})$ $\displaystyle:=$
$\displaystyle\\{P\in\min(A)\,|\,\mathfrak{p}\subseteq
P\\}=\\{P_{\mathfrak{p},\varepsilon}\,|\,\varepsilon=(\varepsilon_{1},\ldots,\varepsilon_{|{\rm
in}(\mathfrak{p})|})\in\\{\pm\\}^{{|{\rm in}(\mathfrak{p})|}}\\},$
$\displaystyle P_{\mathfrak{p},\varepsilon}$ $\displaystyle:=$
$\displaystyle\begin{cases}(\mathfrak{p},v_{\varepsilon_{1}}(i_{1}),\ldots,v_{\varepsilon_{|{\rm
in}(\mathfrak{p})|}}(i_{|{\rm in}(\mathfrak{p})|}))&\text{if }{\rm
in}(\mathfrak{p})\neq\emptyset,\\\ (\mathfrak{p})&\text{if }{\rm
in}(\mathfrak{p})=\emptyset,\\\ \end{cases}$

$v_{+}(j)=X_{j}$ and $v_{-}(j)=Y_{j}$,

    1. (a)

for all $P\in\min(A,\mathfrak{p})$, $P\cap D=\mathfrak{p}$,

    2. (b)

$|\min(A,\mathfrak{p})|=2^{|{\rm in}(\mathfrak{p})|}$ and
$\min(A)|\geq|\min(D)|$,

    3. (c)

$\min(A)|=|\min(D)|$ iff ${\rm in}(\mathfrak{p})=\emptyset$ for all
$\mathfrak{p}\in\min(D)$ iff
$\min(A)=\\{A\mathfrak{p}\,|\,\mathfrak{p}\in\min(D)\\}$.

  2. 2.

$\mathfrak{n}_{A}=A\mathfrak{n}_{D}$ where $\mathfrak{n}_{A}$ and
$\mathfrak{n}_{D}$ are the prime radicals of the rings $A$ and $D$,
respectively. In particular, the ring $A$ is reduced iff the ring $D$ is so.

  3. 3.

For each minimal prime $\mathfrak{p}\in\min(D)$, the set
$\min(A,\mathfrak{p})$ is the set of all the minimal primes over the ideal
$A\mathfrak{p}$ of $A$, and the ideal
$A\mathfrak{p}=\bigcap_{P\in\min(A,\mathfrak{p})}P$ is a semiprime ideal of
$A$.

  4. 4.

$A/P_{\mathfrak{p},\varepsilon}=D/\mathfrak{p}[v_{-\varepsilon_{1}}(i_{1}),\ldots,v_{-\varepsilon_{|{\rm
in}(\mathfrak{p})|}}(i_{|{\rm
in}(\mathfrak{p})|})][(X_{j_{1}},\ldots,X_{j_{|{\rm
out}(\mathfrak{p})|}}),(Y_{j_{1}},\ldots,Y_{j_{|{\rm out}(\mathfrak{p})|}});$
$a=(a_{j_{1}},\ldots,a_{j_{|{\rm out}(\mathfrak{p})|}})]$ is a commutative GWA
of rank $|{\rm out}(\mathfrak{p})|$ with coefficients in the polynomial
algebra

$D/\mathfrak{p}[v_{-\varepsilon_{1}}(i_{1}),\ldots,v_{-\varepsilon_{|{\rm
in}(\mathfrak{p})|}}(i_{|{\rm in}(\mathfrak{p})|})]$

in $|{\rm in}(\mathfrak{p})|$ variables with coefficients in $D/\mathfrak{p}$.
In particular, the field of fractions of the algebra
$A/P_{\mathfrak{p},\varepsilon}$ is

$Q(A/P_{\mathfrak{p},\varepsilon})=Q(D/\mathfrak{p})(X_{1},\ldots,X_{N})$

which is the field of rational functions in the variables $X_{1},\ldots,X_{N}$
over the field of fractions $Q(D/\mathfrak{p})$ of the algebra
$D/\mathfrak{p}$, and ${\rm tr.deg}_{K}Q(A/P_{\mathfrak{p},\varepsilon})={\rm
tr.deg}_{K}Q(D/\mathfrak{p})+N$.

  5. 5.

If, in addition, the algebra $D$ has pure dimension $d$ then the algebra $A$
has pure dimension $d+N$, and vice versa.

Proof. 4. Statement 4 is obvious (use the ${\mathbb{Z}}^{N}$-grading of the
GWA $A$).

1\. (i) All ideals $P_{\mathfrak{p},\varepsilon}$ of $A$ are distinc prime
ideals: This follows from statement 4.

(ii) $P_{\mathfrak{p},\varepsilon}\cap D=\mathfrak{p}$: This follows from the
fact that the ideal $P_{\mathfrak{p},\varepsilon}$ is a homogeneous ideal of
the GWA $A$ (with respect of the ${\mathbb{Z}}^{N}$-grading) and definitions
of the sets ${\rm in}(\mathfrak{p})$ and ${\rm out}(\mathfrak{p})$.

(iii) The set $\min(A)$ contains precisely all the ideals
$P_{\mathfrak{p},\varepsilon}$: Let $P$ be a prime ideal of $A$. Then the
intersection $D\cap P$ is a prime ideal of the algebra $D$. Hence,
$\mathfrak{p}\subseteq D\cap P$ for some $\mathfrak{p}\in\min(D)$. Let ${\rm
in}(\mathfrak{p})=\\{i_{1},\ldots,i_{l}\\}$ and ${\rm
out}(\mathfrak{p})=\\{j_{1},\ldots,j_{m}\\}$. Then the factor algebra

$A/A\mathfrak{p}=D/\mathfrak{p}[X_{\rm out}(\mathfrak{p}),Y_{\rm
out}(\mathfrak{p});a_{\rm out}(\mathfrak{p})][X_{\rm in}(\mathfrak{p}),Y_{\rm
in}(\mathfrak{p});0]$

where $X_{\rm out}(\mathfrak{p})=(X_{j_{1}},\ldots,X_{j_{m}})$, $Y_{\rm
out}(\mathfrak{p})=(Y_{j_{1}},\ldots,Y_{j_{m}})$, $X_{\rm
in}(\mathfrak{p})=(X_{i_{1}},\ldots,X_{i_{l}})$, $Y_{\rm
in}(\mathfrak{p})=(Y_{i_{1}},\ldots,Y_{i_{l}})$ and $a_{\rm
out}(\mathfrak{p})=(\overline{a_{j_{1}}},\ldots,\overline{a_{j_{m}}})$ where
$\overline{a_{\nu}}=a_{\nu}+\mathfrak{p}$. Since

$X_{i_{1}}Y_{i_{1}}=0,\ldots,X_{i_{l}}Y_{i_{l}}=0,$

we must have $P_{\mathfrak{p},\varepsilon}\subseteq P$ for some $\varepsilon$.
So, every prime ideal of $A$ contains some prime ideal
$P_{\mathfrak{p},\varepsilon}$ and all the ideals
$P_{\mathfrak{p},\varepsilon}$ are distinct. Now, the statement (iii) follows,
and as a result we have statement 1.

3\. By statement 1, $A\mathfrak{p}=\bigcap_{P\in\min(A,\mathfrak{p})}P$, and
statement 3 follows.

2\. Statement 2 follows from statements 1 and 3:

$\displaystyle A\mathfrak{n}_{D}$ $\displaystyle=$ $\displaystyle
A\Big{(}\bigcap_{\mathfrak{p}\in\min(D)}\mathfrak{p}\Big{)}=\bigcap_{\mathfrak{p}\in\min(D)}A\mathfrak{p}\stackrel{{\scriptstyle{\rm
st}.3}}{{=}}\bigcap_{\mathfrak{p}\in\min(D)}\bigcap_{P\in\min(A,\mathfrak{p})}P$
$\displaystyle\stackrel{{\scriptstyle{\rm st}.1}}{{=}}$
$\displaystyle\bigcap_{P\in\min(A)}P=\mathfrak{n}_{A}.$

The rest is obvious.

5\. Statement 5 follows from statement 4: For each minimal prime
$P_{\mathfrak{p},\varepsilon}$ of $A$,

$A/P_{\mathfrak{p},\varepsilon}=D/\mathfrak{p}[X_{\rm
out}(\mathfrak{p}),Y_{\rm out}(\mathfrak{p});a_{\rm
out}(\mathfrak{p})][v_{-\varepsilon_{1}}(i_{1}),\ldots,v_{-\varepsilon_{l}}(i_{l})]$

is a polynomial algebra in the variables
$v_{-\varepsilon_{1}}(i_{1}),\ldots,v_{-\varepsilon_{l}}(i_{l})$ withe
coefficients in the GWA $B=D/\mathfrak{p}[X_{\rm out}(\mathfrak{p}),Y_{\rm
out}(\mathfrak{p});a_{\rm out}(\mathfrak{p})]$ of degree $m=|{\rm
out}(\mathfrak{p})|$ and all the coordinates of the vector $a_{\rm
out}(\mathfrak{p})=(a_{j_{1}},\ldots,a_{j_{m}})$ are nonzero (hence regular)
elements of the domain $D/\mathfrak{p}$. Hence,
$Q(A/P_{\mathfrak{p},\varepsilon})=Q(D/\mathfrak{p})(X_{1},\ldots,X_{m})$.
Now,

${\rm Kdim}(A/P_{\mathfrak{p},\varepsilon})={\rm Kdim}(B)+l.$

Claim. ${\rm Kdim}(B)={\rm Kdim}(D/\mathfrak{p})+m=d+m$.

By the Claim,

${\rm Kdim}(A/P_{\mathfrak{p},\varepsilon})=d+m+l=d+n,$

as required (since $m+l=n$).

To prove the Claim we use induction on $m$. The result is obvious if $m=0$.
Suppose that $m>0$ and that the Claim is true for all $m^{\prime}<m$. Then

$B=B_{m-1}[X_{m},Y_{m};a_{m}]=B_{m-1}[X_{m},Y_{m}]/(X_{m}Y_{m}-a_{m}).$

The polynomial algebra $B_{m-1}[X_{m},Y_{m}]$ is a Noetherian domain of Krull
dimension ${\rm Kdim}(B_{m-1})+2$. By the Krull’s Principal Ideal Theorem, the
algebra $B$ is a Noetherian algebra of pure dimension

${\rm Kdim}(B_{m-1})+2-1={\rm Kdim}(B_{m-1})+1=d+m-1+1=d+m$

since, by induction, ${\rm Kdim}(B_{m-1})=d+m-1$, and the Claim follows.
$\Box$

Homological sequences.

Definition. Let $D$ be a regular algebra of essentiallu finite type. A
sequence of elements $a_{1},\ldots,a_{N}$ in $D$ is called a homological
sequence of length $N$ if the following two conditions hold:

1\. the element $a_{1}$ is a homological sequence of length 1, that is (by
definition) the element $a_{1}$ is a regular element of $D$ such that the
factor algebra $D/(a_{1})$ is regular provided the element $a_{1}$ is not a
unit, and

2\. the images $\overline{a}_{2},\ldots,\overline{a}_{N}$ of the elements
$a_{2},\ldots,a_{N}$ in the ring

$D_{1}:=\begin{cases}D&\text{if }a_{1}\in D^{\times},\\\ D/(a_{1})&\text{if
}a_{1}\not\in D^{\times}\cup\\{0\\},\\\ \end{cases}$

is a homological sequence of length $N-1$. The algebra $D_{1}$ is called the
algebra of the element $a_{1}$ in the homological sequence $s$:
$a_{1},\ldots,a_{N}$ in $D$. Similarly, the algebra $D_{2}$ of the element
$\overline{a}_{2}$ in the homological sequence
$\overline{a}_{2},\ldots,\overline{a}_{N}$ in $D_{1}$ is called the algebra of
the element $a_{2}$ in the homological sequence $s$ in $D$. Finally, the
algebra $D_{N}$ of the element $a_{N}$ in the homological sequence $s$ is
defined. The sequence of algebras $D_{1},\ldots,D_{N}$ is called the sequence
of algebras of the homological sequence $s$.

Notice that the algebras $D_{1},\ldots,D_{N}$ are regular algebras of
essentially finite type and $\overline{a}_{i}\in{\cal C}_{D_{i-1}}$ for all
$i=1,\ldots,N$. Let

${\cal U}(s):=\\{a_{i}\,|\,\overline{a}_{i}\in D_{i-1}^{\times}\\}\;\;{\rm
and}\;\;{\cal N}(s):=\\{a_{i}\,|\,\overline{a}_{i}\not\in
D_{i-1}^{\times}\\}.$

The set ${\cal U}(s)$ (resp., ${\cal N}(s)$) is called the set of relative
units (resp., regular non-units) of $s$. If ${\cal
N}(s)=\\{a_{i_{1}},\ldots,a_{i_{\nu}}\\}$ where $1\leq
i_{1}<\cdots<a_{i_{\nu}}\leq N$ then

$D_{i_{\mu}}=D/(a_{i_{1}},\ldots,a_{i_{\nu}})\;\;{\rm
for\;all}\;\;\mu=1,\ldots,\nu.$ (4)

Furthermore, there is a chain of equalities and epimorphisms :

$D_{0}=D_{1}=\cdots=D_{i_{1}-1}\rightarrow
D_{i_{1}}=\cdots=D_{i_{2}-1}\rightarrow\cdots
D_{i\mu}=\cdots=D_{i_{\mu}-1}\rightarrow\cdots\rightarrow
D_{i_{\nu}}=\cdots=D_{N}$

where the arrows are epimorphisms (eg, $D_{i_{\mu}-1}\rightarrow
D_{i_{\mu}}\simeq D_{i_{\mu}-1}/(a_{\mu}))$ and $D_{0}:=D$. The elements in
${\cal N}(s)$ are a regular sequence of the algebra $D$. The algebra $D_{i}$
has pure dimension $d-u_{i}$ where $u_{i}=\\#\\{j\,|\,a_{j}\in{\cal
N}(s),j\leq i\\}$.

Regularity criteria for a GWA $A$. Theorem 2.2 is a first regularity criterion
for the a GWA $A$.

###### Theorem 2.2

Let $D=S^{-1}(P_{n}/I)$ be a commutative algebra of essentially finite type of
pure dimension $d$ $(=n-r$ where $P_{n}=K[x_{1},\ldots,x_{n}]$,
$I=(f_{1},\ldots,f_{m})$ and $r=r(\frac{\partial f_{i}}{\partial x_{j}})$)
over a perfect field $K$ and $A=D[X,Y;a]$ be a GWA of rank $N$. Then the
algebra $A$ is regular iff $\sum_{I}\mathfrak{b}_{I}=D$ where the sum is taken
over all subsets $I\subseteq\\{1,\ldots,N\\}$ with $|I|\geq N-d$,
$CI=\\{1,\ldots,N\\}\backslash I$ and
$\mathfrak{b}_{I}:=\mathfrak{a}_{CI}a_{I}$ is an ideal of $D$ where
$a_{I}:=\prod_{i\in I}a_{i}$, $a_{\emptyset}:=1$, and the ideal
$\mathfrak{a}_{CI}$ of $D$ is generated by all the minors of size $r+|CI|$ of
the $(m+|CI|)\times n$ matrix with coefficients in $D$ where the $m+|CI|$ rows
are ${\rm grad}(f_{1}),\ldots,{\rm grad}(f_{m}),{\rm
grad}(a_{j_{1}}),\ldots,{\rm grad}(a_{j_{|CI|}})$ where
$CI=\\{j_{1}<\cdots<j_{|CI|}\\}$.

Proof. By Theorem 2.1.(5), the algebra $A$ is an algebra of essentially finite
type of pure dimension $d+N$ over a perfect field $K$. By the Jacobian
Criterion of Regularity, the algebra $A$ is regular iff the Jacobian ideal
$\mathfrak{a}=\mathfrak{a}(A)_{d+N}$ of $A$ is equal to $A$. The strategy of
the proof of the theorem is as follows. First, we find explicit set of
generators of the ideal $\mathfrak{a}$. Then we show that
$D\cap\mathfrak{a}=\sum_{I}\mathfrak{b}_{I}$, and the theorem follows. Notice
that

$A=S^{-1}K[x_{1},\ldots,x_{n},X_{1},Y_{1},\ldots,X_{N},Y_{N}]/(f_{1},\ldots,f_{m},a_{1}-X_{1}Y_{1},\ldots,a_{N}-X_{N}Y_{N}).$

The Jacobian matrix of $A$ is of size $(m+N)\times(n+2N)$ with $m+N$ rows as
follows

$\displaystyle({\rm grad}(f_{1}),0,\ldots,0),$ $\displaystyle\ldots,$
$\displaystyle({\rm grad}(f_{m}),0,\ldots,0),$ $\displaystyle({\rm
grad}(a_{1}),-X_{1},-Y_{1},0,\ldots,0),$ $\displaystyle\ldots,$
$\displaystyle({\rm grad}(a_{N}),0,\ldots,0,-X_{N},-Y_{N}).$

It follows that

$\mathfrak{a}=\sum_{I,\varepsilon}\mathfrak{a}_{CI}v_{I,\varepsilon}$

where $v_{I,\varepsilon}=\prod_{i\in I}v_{\varepsilon_{i}}(i)$,
$\varepsilon=(\varepsilon_{i})_{i\in I}\in\\{\pm\\}^{I}$ and $v_{+}(i)=X_{i}$
and $v_{-}1(i)=Y_{i}$ and the set $I$ runs through all the subsets of
$\\{1,\ldots,N\\}$ such that $r+N-|I|\leq n$ $(\Leftrightarrow|I|\geq N-d$).
The ideal $\mathfrak{a}$ is a sum of homogeneous ideals w.r.t. the
${\mathbb{Z}}^{N}$-grading of the GWA $A$. So,

$1\in\mathfrak{a}\;\;{\rm iff}\;\;1\in\mathfrak{a}\cap
D=\sum_{I,\varepsilon}\mathfrak{a}_{CI}v_{I,\varepsilon}v_{I,-\varepsilon}=\sum_{I}\mathfrak{a}_{CI}a_{I},$

as required. $\Box$

Let $S_{N}$ be the symmetric group, i.e. the group of all permutations of the
set $\\{1,\ldots,N\\}$. Theorem 2.3 is the second regularity criterion for
GWAs.

###### Theorem 2.3

Let $D$ be a commutative algebra of essentially finite type of pure dimension
$d$ over a perfect field $K$ and $A=D[X,Y;a]$ be a GWA of rank $N$. Then the
following statements are equivalent:

  1. 1.

The algebra $A$ is a regular algebra.

  2. 2.

The algebra $D$ is a regular algebra and the sequence $a_{1},\ldots,a_{N}$ is
a homological sequence in $D$ of length $N$.

  3. 3.

The algebra $D$ is a regular algebra and every permutation of the sequence
$a_{1},\ldots,a_{N}$ is a homological sequence in $D$ of length $N$.

If one of the equivalent conditions hold then the algebras
$D_{1},\ldots,D_{N}$ of the homological sequence $a_{1},\ldots,a_{N}$ are
commutative regular equidimensional algebras of essentially finite type.
Furthermore, the pure dimension of the algebra $D_{i}$ is $d-u_{i}$ where
$u_{i}=\\#\\{j\,|\,a_{j}\in{\cal N}(s),1\leq j\leq i\\}$.

Proof. $(1\Leftrightarrow 2)$ To prove that the equivalence holds we use
induction on $N$. Suppose that $N=1$. By Theorem 2.2, the algebra $A$ is
regular iff $\sum_{I}\mathfrak{b}_{I}=D$ where the sum is taken over all the
subsets $I$ of the set $\\{1\\}$ such that $|I|\geq 1-d$. There are two cases
to consider: $d=0$ and $d>0$.

Suppose that $d=0$. Then $|I|\geq 1$, i.e. $I=\\{1\\}$. Now,
$D=\sum_{I=\\{1\\}}\mathfrak{b}_{I}=\mathfrak{a}_{r}a_{1}$ iff
$\mathfrak{a}_{r}=D$ and $a_{1}\in D^{\times}$.

Suppose that $d>0$. Then $|I|\geq 1-1=0$, and so $I=\\{1\\}$ or $I=\emptyset$.
Now,

$D=\sum_{I\in\\{\emptyset,\\{1\\}\\}}\mathfrak{b}_{I}=\mathfrak{a}_{\emptyset}a_{1}+\mathfrak{a}_{\\{1\\}}a_{\emptyset}=\mathfrak{a}_{r}a_{1}+\mathfrak{a}_{\\{1\\}}\subseteq\mathfrak{a}_{r}\;\;({\rm
since}\;\;a_{\emptyset}=1)$

iff $\mathfrak{a}_{r}=D$ and $D=(a_{1},\mathfrak{a}_{\\{1\\}})$ iff the
algebra $D$ is regular and the algebra $D/(a_{1})$is regular since
$(a_{1},\mathfrak{a}_{\\{1\\}})$ is the Jacobian ideal of the $D/(a_{1})$ of
essentially finite type of pure dimension $d-1$ (by Krull’s Ptincipal Ideal
Theorem). The proof of the equivalence when $N=1$ is complete.

Suppose that $N>1$, and the equivalence holds for all nartural numbers
$N^{\prime}<N$. Notice that $A=A_{N-1}[X_{1},Y_{1};a_{1}]$ is a GWA of rank 1
where $A_{N-1}=D[X^{\prime},Y^{\prime};a^{\prime}]$ is a GWA of degree $N-1$
where $X^{\prime}=(X_{2},\ldots,X_{N})$, $Y^{\prime}=(Y_{2},\ldots,Y_{N})$ and
$a^{\prime}=(a_{2},\ldots,a_{N})$. Recall that the algebra $D$ is an algebra
of essentially finite type of pure dimension $d$. By Proposition 2.1.(5), the
algebra $A_{N-1}$ is an algebra of essentially finite type of pure dimension
$d+N-1$. By the case $N=1$, $A$ is regular iff the algebra $A_{N-1}$ is
regular, $a_{1}\in{\cal C}_{A_{N-1}}$ $(\Leftrightarrow a_{1}\in{\cal C}_{D})$
and if $a_{1}\not\in A_{N-1}^{\times}$ $(\Leftrightarrow a_{1}\not\in
D^{\rtimes})$ then the algebra

$A_{N-1}/(a_{1})\simeq D/(a_{1})[X^{\prime},Y^{\prime};\overline{a}^{\prime}]$

is regular where $\overline{a}^{\prime}=(a_{2}+(a_{1}),\ldots,a_{N}+(a_{1}))$
iff $a_{1}$ is a homological sequence in $D$ and
$\overline{a}_{2},\ldots,\overline{a}_{N}$ is a homological sequence in
$D_{1}$. This means that the sequence $a_{1},\ldots,a_{N}$ is homological
sequence in $D$.

$(2\Leftrightarrow 3)$ For any permutation $\sigma\in S_{N}$, $A\simeq
D[X^{\sigma},Y^{\sigma};a^{\sigma}]$ where
$X^{\sigma}=(X_{\sigma(1)},\ldots,X_{\sigma(N)})$ and
$a^{\sigma}=(a_{\sigma(1)},\ldots,a_{\sigma(N)})$, and the equivalence follows
from the fact that $(1\Leftrightarrow 2)$. $\Box$

We say that ideals $\mathfrak{a}$ and $\mathfrak{b}$ of an algebra $R$ are co-
prime if $\mathfrak{a}+\mathfrak{b}=R$.

###### Corollary 2.4

Let $D$ be a ring of essentially finite type of pure dimension $d$ over a
perfect field $K$ and $A=D[X,Y;a]$ be a commutative GWA of degree $N$. The
following statements are equivalent.

  1. 1.

The algebra $A$ is a regular algebra.

  2. 2.

$A\simeq\prod_{P\in\min(A)}A/P$ and the algebras $A/P$ are regular.

  3. 3.

The algebra $D$ is regular, elements in $\min(A)$ are pairwise co-prime, and
for each $P=P_{\mathfrak{p},\varepsilon}$, the GWA

$D/\mathfrak{p}[(X_{j_{1}},\ldots,X_{j_{|{\rm
out}(\mathfrak{p})|}}),(Y_{j_{1}},\ldots,Y_{j_{|{\rm
out}(\mathfrak{p})|}});a=(a_{j_{1}},\ldots,a_{j_{|{\rm
out}(\mathfrak{p})|}})]$

is a regular algebra.

Proof. 1. $(1\Leftrightarrow 2)$ The equivalence is true for all algebras of
essentially finite type.

$(2\Leftrightarrow 3)$ The algebra $D$ is reduced ($\mathfrak{n}_{D}=0$) iff
the algebra $A$ is so, by Proposition 2.1.(2), i.e.
$0=\mathfrak{n}_{A}=\bigcap_{P\in\min(A)}P$. So,
$A\simeq\prod_{P\in\min(A)}A/P$ iff the minimal primes of the algebra $A$ are
pairwise co-prime. Now, the equivalence $(2\Leftrightarrow 3)$ follows from
Proposition 2.1.(4). $\Box$

###### Proposition 2.5

Let $D$ be a commutative regular algebra of essentially finite type of pure
dimension $d$ over a perfect field $K$ and $a_{1},\ldots,a_{N}$ be a
homological sequence in $D$ of length $N$. Then

  1. 1.

Every permutation of the sequence $a_{1},\ldots,a_{N}$ is a homological
sequence in $D$ of length $N$.

  2. 2.

There is a permutation, say $a_{i_{1}},\ldots,a_{i_{N}}$, of the sequence
$a_{1},\ldots,a_{N}$ such that

    1. (a)

the algebras $D_{0}:=D,D_{1},\ldots,D_{\nu}$ are regular where
$D_{i}=D/(a_{1},\ldots,a_{i})$ for $i=1,\ldots,\nu$,

    2. (b)

$\overline{a}_{i}\in{\cal C}_{D_{i}}\backslash D_{i}^{\times}$ for
$i=1,\ldots,\nu$, and

    3. (c)

$\overline{a}_{\nu+1},\ldots,\overline{a}_{N}\in D_{s}^{\times}$ where
$\overline{a}_{j}=a_{j}+(a_{1},\ldots,a_{\nu})$.

Proof. 1. Statement 1 follows from Theorem 2.3.

2\. Let $D_{1},\ldots,D_{\nu}$ be as in (4). Then
$D_{\nu}=D/(a_{i_{1}},\ldots,a_{i_{\nu}})$ where $1\leq
i_{1}<\cdots<i_{\nu}\leq N$ and $a_{j}+(a_{i_{1}},\ldots,a_{i_{\nu}})\in
D_{\nu}^{\times}$ for all
$j\in\\{j_{1},\ldots,j_{n-\nu}\\}=\\{1,\ldots,N\\}\backslash\\{i_{1},\ldots,i_{\nu}\\}$,
by Theorem 2.3. Then the sequence
$a_{i_{1}},\ldots,a_{i_{\nu}},a_{j_{1}},\ldots,a_{j_{n-\nu}}$ is required
permutation of the original sequence. $\Box$

##  3  Generators and defining relations of the Poisson enveloping algebra of
a generalized Weyl Poisson algebra

The aim of this section is for an arbitrary generalized Weyl Poisson algebra
to give explicit sets of generators and defining relations for its Poisson
enveloping algebra (Theorem 3.2). We also consider some interesting examples
(Proposition 3.3 and Corollary 3.4). At the begining of the section, examples
of generalized Weyl Poisson algebras are considered.

Examples of GWPAs. 1\. If $D$ is a algebra with trivial Poisson bracket
$(\\{\cdot,\cdot\\}=0)$ then any choice of elements $a=(a_{1},\ldots,a_{n})$
and $\partial=(\partial_{1},\ldots,\partial_{n})\in{\rm Der}_{K}(D)^{n}$ such
that $\partial_{i}(a_{j})=0$ for all $i\neq j$ determines a GWPA
$D[X,Y;a,\partial\\}$ of rank $n$ where ${\rm Der}_{K}(D)$ is the set of
$K$-derivations of the algebra $D$. If, in addition, $n=1$ then there is no
restriction on $a_{1}$ and $\partial_{1}$.

2\. The classical Poisson polynomial algebra
$P_{2n}=K[X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n}]$
($\\{Y_{i},X_{j}\\}=\delta_{ij}$ (the Kronecker delta) and
$\\{X_{i},X_{j}\\}=\\{X_{i},Y_{j}\\}=\\{Y_{i},Y_{j}\\}=0$ for all $i\neq j$)
is a GWPA

$P_{2n}=K[H_{1},\ldots,H_{n}][X,Y;a,\partial\\}$ (5)

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

$K[H_{1}^{\pm 1}][X_{1},Y_{1};a_{1},-H_{1}\frac{d}{dH_{1}}\\}$

where $a_{1}\in K[H_{1}^{\pm 1}]$ is, in fact, isomorphic to a Poisson algebra
in the paper of Cho and Oh [8] which is obtained as a quantization of a
certain GWA with respect to the quantum parameter $q$. In [8, Theorem 3.7], a
Poisson simplicity criterion is given for this Poisson algebra.

Poisson ${\cal P}$-modules. Let $({\cal P},\\{\cdot,\cdot\\})$ be a Poisson
algebra. A left ${\cal P}$-module $M$ is called a Poisson ${\cal P}$-module if
there is a bilinear map

${\cal P}\times M\rightarrow M,\;\;(a,m)\mapsto\delta_{a}m$

which is called a Poisson action of ${\cal P}$ on $M$ such that for all
elements $a,b\in{\cal P}$ and $m\in M$,

(PM1) $\delta_{\\{a,b\\}}=[\delta_{a},\delta_{b}]$,

(PM2) $[\delta_{a},b]=\\{a,b\\}$, and

(PM3) $\delta_{ab}=a\delta_{b}+b\delta_{a}$.

Example. Every Poisson algebra ${\cal P}$ is a Poisson ${\cal P}$-module where
$\delta_{a}={\rm pad}_{a}$.

The Poisson enveloping algebra ${\cal U}({\cal P})$ of a Poisson algebra
${\cal P}$.

For each Poisson algebra ${\cal P}$ there is a unique (up to isomorphism)
associative algebra ${\cal U}({\cal P})$ such that

$\bullet$ every Poisson ${\cal P}$-module is a ${\cal U}({\cal P})$-module,
and vice versa.

The algebra ${\cal U}({\cal P})$ is called the Poisson enveloping algebra
(PEA) of the Poisson algebra ${\cal P}$.

Generators and defining relations of the Poisson enveloping algebra of a
generalized Weyl Poisson algebra. For a Poisson algebra ${\cal P}$ which is
defined by generators and defining relations (as an associative algebra),
Theorem 3.1 gives explicit sets of generators and defining relations for the
Poisson enveloping algebra ${\cal U}({\cal P})$.

###### Theorem 3.1

([7, Theorem 2.2.(2)]) Let ${\cal P}$ be a Poisson algebra, $U({\cal P})$ be
its enveloping algebra (as a Lie algebra) and ${\cal U}({\cal P})$ be its
Poisson enveloping algebra. Then

  1. 1.

${\cal U}({\cal P})\simeq{\cal P}\rtimes_{\rm pad}U({\cal P})/{\cal I}({\cal
P})$ where ${\cal I}({\cal
P})=(\delta_{ab}-a\delta_{b}-b\delta_{a})_{a,b\in{\cal P}}$ is the ideal of
the algebra ${\cal P}\rtimes_{\rm pad}U({\cal P})$ generated by the set
$\\{\delta_{ab}-a\delta_{b}-b\delta_{a}\,|\,a,b\in{\cal P}\\}$.

  2. 2.

If ${\cal P}=S^{-1}K[x_{i}]_{i\in\Lambda}/(f_{s})_{s\in\Gamma}$ where $S$ is a
multiplicative subset of the polynomial algebra $K[x_{i}]_{i\in\Lambda}$
($\Lambda$ and $\Gamma$ are index sets). Then the algebra ${\cal U}({\cal P})$
is generated by the algebra ${\cal P}$ and the elements
$\\{\delta_{i}:=\delta_{x_{i}}\,|\,i\in\Lambda\\}$ subject to the defining
relations (a)–(c): For all elements $i,j\in\Lambda$ such that $i\neq j$ and
$s\in\Gamma$,

    1. (a)

$[\delta_{i},\delta_{j}]=\sum_{k\in\Lambda}\frac{\partial\\{x_{i},x_{j}\\}}{\partial
x_{k}}\delta_{k}$,

    2. (b)

$[\delta_{i},x_{j}]=\\{x_{i},x_{j}\\}$, and

    3. (c)

$\sum_{i\in\Lambda}\frac{\partial f_{s}}{\partial x_{i}}\delta_{i}=0$.

So, the algebra ${\cal U}({\cal P})$ is generated by the algebra ${\cal P}$
and the set $\delta_{\cal P}=\\{\delta_{a}\,|\,a\in{\cal P}\\}$ subject to the
defining relations: For all elenents $a,b\in{\cal P}$ and $\lambda,\mu\in K$,

    1. (a)

$[\delta_{a},\delta_{b}]=\delta_{\\{a,b\\}}$,

    2. (b)

$[\delta_{a},b]=\\{a,b\\}$,

    3. (c)

$\delta_{ab}=a\delta_{b}+b\delta_{a}$,

    4. (d)

$\delta_{\lambda a+\mu b}=\lambda\delta_{a}+\mu\delta_{b}$ and $\delta_{1}=0$.

  3. 3.

The map $\pi_{\cal P}:{\cal U}({\cal P})\rightarrow{\cal D}({\cal P})$,
$a\mapsto a$, $\delta_{b}\mapsto{\rm pad}_{b}=\\{b,\cdot\\}$ is an algebra
homomorphism where $a,b\in{\cal P}$ and its image is the algebra $P{\cal
D}({\cal P})$ of Poisson differential operators of the Poisson algebra ${\cal
P}$.

  4. 4.

The algebra ${\cal P}$ is a subalgebra of ${\cal U}({\cal P})$. Furthermore,
${\cal U}({\cal P})={\cal P}\oplus{\rm ann}_{{\cal U}({\cal P})}(1)$ is a
direct sum of left ${\cal P}$-modules where ${\rm ann}_{{\cal U}({\cal
P})}(1)=\sum_{i\in\Lambda}{\cal U}({\cal P})\delta_{i}$ is the annihilator of
the identity element of the Poisson ${\cal P}$-module ${\cal P}$. The Poisson
${\cal P}$-module structure on the Poisson algebra ${\cal P}$ is obtained from
the ${\cal D}({\cal P})$-module structure on ${\cal P}$ by restriction of
scalars via $\pi_{\cal P}$.

###### Theorem 3.2

Let $A=D[X,Y;a,\partial\\}$ be a GWPA of degree $n$ and
$D=S^{-1}K[t_{\nu}]_{\nu\in I}/(f_{s})_{s\in S}$. Then the Poisson enveloping
algebra ${\cal U}(A)$ of the Poisson algebra $A$ is generated by the algebra
$A$ and elements
$\delta_{X_{1}},\ldots,\delta_{X_{n}},\delta_{Y_{1}},\ldots,\delta_{Y_{n}},\delta_{t_{\nu}}$
(where $\nu\in I$) subject to the defining relations given in statements 1-3:
For all elements $i,j=1,\ldots,n$ and $\nu,\mu\in I$ (where $\delta_{ij}$ is
the Kronecker delta):

  1. 1.

$[\delta_{X_{i}},\delta_{X_{j}}]=0$, $[\delta_{Y_{i}},\delta_{Y_{j}}]=0$,
$[\delta_{Y_{i}},\delta_{X_{j}}]=\delta_{ij}\sum_{\nu\in
I}\frac{\partial}{\partial t_{\nu}}(\partial_{i}(a_{i}))\delta_{t_{\nu}},$
$[\delta_{X_{i}},\delta_{t_{\nu}}]=-\partial_{i}(t_{\nu})\delta_{X_{i}}-X_{i}\sum_{\mu\in
I}\frac{\partial}{\partial t_{\mu}}(\partial_{i}(t_{\nu}))\delta_{t_{\mu}},$
$[\delta_{Y_{i}},\delta_{t_{\nu}}]=\partial_{i}(t_{\nu})\delta_{Y_{i}}+Y_{i}\sum_{\mu\in
I}\frac{\partial}{\partial t_{\mu}}(\partial_{i}(t_{\nu}))\delta_{t_{\mu}},$

  2. 2.

$[\delta_{X_{i}},t_{\nu}]=-\partial_{i}(t_{\nu})X_{i}$,
$[\delta_{Y_{i}},t_{\nu}]=\partial_{i}(t_{\nu})Y_{i}$,
$[\delta_{X_{i}},Y_{j}]=-\delta_{ij}\partial_{i}(a_{i})$,
$[\delta_{Y_{i}},X_{j}]=\delta_{ij}\partial_{i}(a_{i})$,
$[\delta_{X_{i}},X_{j}]=0$, $[\delta_{Y_{i}},Y_{j}]=0$,
$[\delta_{t_{\nu}},t_{\mu}]=\\{t_{\nu},t_{\mu}\\}$,
$[\delta_{t_{\nu}},X_{i}]=\partial_{i}(t_{\nu})X_{i}$,
$[\delta_{t_{\nu}},Y_{i}]=-\partial_{i}(t_{\nu})Y_{i}$, and

  3. 3.

for all elements $s\in S$, $\sum_{\nu\in I}\frac{\partial f_{s}}{\partial
t_{\nu}}\delta_{t_{\nu}}=0$ and for all elements $i=1,\ldots,n$,
$Y_{i}\delta_{X_{i}}+X_{i}\delta_{Y_{i}}-\sum_{\nu\in I}\frac{\partial
a_{i}}{\partial t_{\nu}}\delta_{t_{\nu}}=0$.

Proof. The theorem follows from Theorem 3.1.(2), where the relations 1–3 are
the relations (a)–(c) of Theorem 3.1.(2), respectively, in the case of the
GWPA $A$. $\Box$

###### Proposition 3.3

Let $A=K[H][X,Y;a,\partial=b\partial_{H}\\}$ be a GWPA of degree $1$ where
$K[H]$ is a polynomial algebra in a single variable $H$ with trivial Poisson
bracket, $a,b\in K[H]$ and $\partial_{H}=\frac{d}{dH}=()^{\prime}$. Then the
algebra ${\cal U}(A)$ is generated by the algebra $A=K[H,X,Y]/(XY-a)$ and the
elements $\Delta_{H}$, $\delta_{X}$ and $\delta_{Y}$ subject to the defining
relations given in statements 1-3:

  1. 1.

$[\delta_{Y},\delta_{X}]=(ba^{\prime})^{\prime}\delta_{H}$,
$[\delta_{X},\delta_{H}]=-b\delta_{X}-Xb^{\prime}\delta_{H}$,
$[\delta_{Y},\delta_{H}]=b\delta_{Y}+Yb^{\prime}\delta_{H}$,

  2. 2.

$[\delta_{X},X]=0$, $[\delta_{Y},Y]=0$, $[\delta_{X},H]=-bX$,
$[\delta_{Y},H]=bY$, $[\delta_{X},Y]=-ba^{\prime}$,
$[\delta_{Y},X]=ba^{\prime}$,
$[\delta_{H},H]=0$, $[\delta_{H},X]=bX$, $[\delta_{H},Y]=bY$, and

  3. 3.

$Y\delta_{X}+X\delta_{Y}=a^{\prime}\delta_{H}$.

Proof. The proposition follows from Theorem 3.2. The relations in statements
1–3 of the proposition are precisely the relations 1–3 of Theorem 3.2 for the
GWPA $A$. $\Box$

###### Corollary 3.4

Let the Poisson algebra $A=\bigotimes_{i=1}^{n}A_{i}$ be a tensor product of
the GWPAs
$A_{i}=K[H_{i}][X_{i},Y_{i};a_{i},\partial_{i}=b_{i}\partial_{H_{i}}\\}$ from
Proposition 3.3. Then ${\cal U}(A)=\bigotimes_{i=1}^{n}{\cal U}(A_{i})$ and
the defining relations for the algebra ${\cal U}(A)$ are the union of the
defining relations of all the algebra ${\cal U}(A_{i})$ from Proposition 3.3
and obvious commutation relations that come from the tensor product of
algebras (i.e. $st=ts$).

Proof. [7, Proposition 2.7] states that the Poisson enveloping algebra of the
tensor product of finitely many Poisson algebras is isomorphic to the tensor
product of the corresponding Poisson enveloping algebras. Now, the corollary
follows from Proposition 3.2. $\Box$

###### Corollary 3.5

Let $A=K[H][X,Y;a,\partial=b\partial_{H}\\}$ be a GWPA of degree $1$ as in
Proposition 3.3 such that $a\not\in K\cup\\{0\\}$, $A_{a^{\prime}}$ be the
localization of the algebra $A$ at the powers of the elements
$a^{\prime}=\frac{da}{dH}$. Then the algebra ${\cal U}(A_{a^{\prime}})$ is
generated by the algebra $A_{a^{\prime}}$ and the elements $\delta_{X}$ and
$\delta_{Y}$ subject to the defining relations:

  1. 1.

$[\delta_{Y},\delta_{X}]=\frac{(ba^{\prime})^{\prime}}{a^{\prime}}(Y\delta_{X}+X\delta_{Y})$,

  2. 2.

$[\delta_{X},X]=0$, $[\delta_{Y},Y]=0$, $[\delta_{X},H]=-bX$,
$[\delta_{Y},H]=bY$, $[\delta_{X},Y]=-ba^{\prime}$,
$[\delta_{Y},X]=ba^{\prime}$.

Proof. Proposition 3.3 gives generators and defining relations of the algebra
${\cal U}(A)$. By [7, Theorem 2.10], ${\cal U}(A_{a^{\prime}})\simeq{\cal
U}(A)_{a^{\prime}}$. Since
$\delta_{H}=\frac{1}{a^{\prime}}(Y\delta_{X}+X\delta_{Y})$, the algebra ${\cal
U}(A_{a^{\prime}})$ is generated by the algebra $A_{a^{\prime}}$ and the
elements $\delta_{X}$ and $\delta_{Y}$ subject to the defining relations in
statements 1 and 2 of the corollary which are precisely the relations 1 and 2
of Proposition 3.3, respectively (by direct computations, all the relations
where $\delta_{H}$ is involved in Proposition 3.3 are redundant). $\Box$

##  4 A simplicity criterion for the Poisson enveloping algebra of a
generalized Weyl Poisson algebra

Simplicity criterion for the algebra $P{\cal D}({\cal P})$ of Poisson
differential operators on ${\cal P}$. An ideal $I$ of a Poisson algebra ${\cal
P}$ is called a Poisson ideal if $\\{{\cal P},I\\}\subseteq I$. A Poisson
algebra ${\cal P}$ is called Poisson simple if the ideals $0$ and ${\cal P}$
are the only Poisson ideals of the Poisson algebra ${\cal P}$. Let ${\rm
Der}_{K}({\cal P})$ be the Lie algebra of $K$-derivations of the (associative)
algebra ${\cal P}$. For each element $a\in{\cal P}$, the derivation ${\rm
pad}_{a}:=\\{a,\cdot\\}\in{\rm Der}_{K}({\cal P})$ is called called the
Hamiltonian vector field associated with the element $a$. Then the set of
Hamiltonian vector fields

${\cal H}_{\cal P}:=\\{{\rm pad}_{a}\,|\,a\in{\cal P}\\}$

is a Lie subalgebra of the Lie algebra ${\rm Der}_{K}({\cal P})$.

Definition. Let ${\cal P}$ be a Poisson algebra and ${\cal D}(A)$ be the
algebra of differential operators on ${\cal P}$. The subalgebra of ${\cal
D}({\cal P})$,

$P{\cal D}({\cal P}):=\langle{\cal P},{\cal H}_{\cal P}\rangle,$

is called the algebra of Poisson differential operators of the Poisson algebra
${\cal P}$.

In general, $P{\cal D}({\cal P})\neq{\cal D}({\cal P})$. Theorem 4.1 is a
simplicity criterion for the algebra $P{\cal D}({\cal P})$ of Poisson
differential operators on ${\cal P}$.

###### Theorem 4.1

([7, Theorem 1.1]) Let ${\cal P}$ be a Poisson algebra over an arbitrary field
$K$. Then the following statements are equivalent:

  1. 1.

The algebra $P{\cal D}({\cal P})$ is a simple algebra.

  2. 2.

The Poisson algebra ${\cal P}$ is a Poisson simple algebra.

Poisson simplicity criterion for generalized Weyl Poisson algebras. Let $D$ be
a Poisson algebra and $\partial=(\partial_{1},\ldots,\partial_{n})\in{\rm
PDer}_{K}(D)^{n}$. An ideal $I$ of $D$ is called $\partial$-invariant 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 algebra of $\partial$-constants of $D$, it is a subalgebra of
$D$. Theorem 4.2 is Poisson simplicity criterion for generalized Weyl Poisson
algebras.

###### Theorem 4.2

([6, 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. 1.

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

  2. 2.

for all $i=1,\ldots,n$, $Da_{i}+D\partial_{i}(a_{i})=D$, and

  3. 3.

the algebra ${\rm PZ}(A)$ is a field $(\Leftrightarrow$ char$(K)=0$, ${\rm
PZ}(D)^{\partial}$ is a field and $D_{\alpha}=0$ for all
$\alpha\in{\mathbb{Z}}^{n}\backslash\\{0\\}$ where $D_{\alpha}:=\\{\lambda\in
D^{\partial}\,|\,{\rm
pad}_{\lambda}=\lambda\sum_{i=1}^{n}\alpha_{i}\partial_{i},\;\lambda\alpha_{i}\partial_{i}(a_{i})=0$
for $i=1,\ldots,n\\}$, [6, Proposition 1.2]).

The next corollary follows from Theorem 4.1 and Theorem 4.2.

###### Corollary 4.3

Let $A=D[X,Y;a,\partial\\}$ be a GWPA of rank $n$. Then the following
statements are equivalent:

  1. 1.

The algebra $P{\cal D}(A)$ is a simple algebra.

  2. 2.

    1. (a)

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

    2. (b)

for all $i=1,\ldots,n$, $Da_{i}+D\partial_{i}(a_{i})=D$, and

    3. (c)

the algebra ${\rm PZ}(A)$ is a field $(\Leftrightarrow$ char$(K)=0$, ${\rm
PZ}(D)^{\partial}$ is a field and $D_{\alpha}=0$ for all
$\alpha\in{\mathbb{Z}}^{n}\backslash\\{0\\}$ where $D_{\alpha}:=\\{\lambda\in
D^{\partial}\,|\,{\rm
pad}_{\lambda}=\lambda\sum_{i=1}^{n}\alpha_{i}\partial_{i},\;\lambda\alpha_{i}\partial_{i}(a_{i})=0$
for $i=1,\ldots,n\\}$, [6, Proposition 1.2]).

Simplicity criteria for the Poisson enveloping algebra ${\cal U}({\cal P})$.
Let ${\cal P}$ be a Poisson algebra. The algebra ${\cal P}$ is a ${\cal
D}({\cal P})$-module and hence $P{\cal D}({\cal P})$\- and ${\cal U}({\cal
P})$-module, and so there is a natural algebra epimorphism (Theorem 3.1.(3)):

$\pi_{\cal P}:{\cal U}({\cal P})\rightarrow P{\cal D}({\cal P}),\;\;a\mapsto
a,\;\;\delta_{b}\mapsto{\rm pad}_{b}=\\{b,\cdot\\}$ (6)

where $a,b\in{\cal P}$. Theorem 4.4 is a simplicity criteria for the Poisson
enveloping algebra ${\cal U}({\cal P})$.

###### Theorem 4.4

([7, Theorem 1.2]) Let ${\cal P}$ be a Poisson algebra over an arbitrary field
$K$. Then the following statements are equivalent:

  1. 1.

The algebra ${\cal U}({\cal P})$ is a simple algebra.

  2. 2.

The algebra $P{\cal D}({\cal P})$ is a simple algebra and ${\cal U}({\cal
P})\simeq P{\cal D}({\cal P})$.

  3. 3.

The Poisson algebra ${\cal P}$ is a Poisson simple algebra and ${\cal P}$ is a
faithful left ${\cal U}({\cal P})$-module.

If one of the equivalent conditions holds then ${\cal U}({\cal P})\simeq
P{\cal D}({\cal P})$.

Corollary 4.5 is a simplicity criteria for the Poisson enveloping algebra
${\cal U}(A)$ where $A$ is a generalized Weyl Poisson algebra. It follows from
Theorem 4.2 and Theorem 4.4.

###### Corollary 4.5

Let $A=D[X,Y;a,\partial\\}$ be a GWPA of rank $n$. Then the following
statements are equivalent:

  1. 1.

The algebra ${\cal U}(A)$ is a simple algebra.

  2. 2.

    1. (a)

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

    2. (b)

for all $i=1,\ldots,n$, $Da_{i}+D\partial_{i}(a_{i})=D$,

    3. (c)

the algebra ${\rm PZ}(A)$ is a field, and

    4. (d)

the algebra $A$ is a faithful left ${\cal U}(A)$-module.

Generators and defining relations for the ${\cal A}$-module ${\rm
Der}_{K}({\cal A})$ where ${\cal A}$ is a regular domain of essentially finite
type. A localization of an affine commutative algebra is called an algebra of
essentially finite type. The following notation is fixed:
$P_{n}=K[x_{1},\ldots,x_{n}]$ is a polynomial algebra over perfect field $K$,
$I=(f_{1},\ldots,f_{m})$ is a prime but not a maximal ideal of $P_{n}$, ${\cal
A}=S^{-1}(P_{n}/I)$ is a domain of essentially finite type and $Q=Q({\cal A})$
is its field of fractions, $r=r\Big{(}\frac{\partial f_{i}}{\partial
x_{j}}\Big{)}$ is the rank (over $Q$) of the Jacobian matrix
$\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$ of ${\cal A}$,
$\mathfrak{a}_{r}$ is the Jacobian ideal of the algebra ${\cal A}$ which is
(by definition) generated by all the $r\times r$ minors of the Jacobian matrix
of ${\cal A}$ (the algebra ${\cal A}$ is regular iff $\mathfrak{a}_{r}={\cal
A}$, it is the Jacobian criterion of regularity, [10, Corollary 16.20]),
$\Omega_{\cal A}$ is the module of Kähler differentials for the algebra ${\cal
A}$.

For ${\bf i}=(i_{1},\ldots,i_{r})$ such that $1\leq i_{1}<\cdots<i_{r}\leq m$
and ${\bf j}=(j_{1},\ldots,j_{r})$ such that $1\leq j_{1}<\cdots<j_{r}\leq n$,
$\Delta({\bf i},{\bf j})$ denotes the corresponding minor of the Jacobian
matrix of ${\cal A}$, and the ${\bf i}$ (resp., ${\bf j}$) is called non-
singular if $\Delta({\bf i},{\bf j}^{\prime})\neq 0$ (resp., $\Delta({\bf
i}^{\prime},{\bf j})\neq 0$) for some ${\bf j}^{\prime}$ (resp., ${\bf
i}^{\prime}$). We denote by $\mathbb{I}_{r}$ (resp., $\mathbb{J}_{r}$) the set
of all the non-singular $r$-tuples ${\bf i}$ (resp., ${\bf j}$).

Since $r$ is the rank of the Jacobian matrix of ${\cal A}$, it is easy to show
that $\Delta({\bf i},{\bf j})\neq 0$ iff ${\bf i}\in\mathbb{I}_{r}$ and ${\bf
j}\in\mathbb{J}_{r}$, [4, Lemma 2.1]. We denote by $\mathbb{J}_{r+1}$ the set
of all $(r+1)$-tuples ${\bf j}=(j_{1},\ldots,j_{r+1})$ such that $1\leq
j_{1}<\cdots<j_{r+1}\leq n$ and when deleting some element, say $j_{\nu}$, we
have a non-singular $r$-tuple
$(j_{1},\ldots,\widehat{j_{\nu}},\ldots,j_{r+1})\in\mathbb{J}_{r}$ where the
hat over a symbol means that the symbol is omitted from the list. The set
$\mathbb{J}_{r+1}$ is called the critical set and any element of it is called
a critical singular $(r+1)$-tuple. ${\rm Der}_{K}({\cal A})$ is the ${\cal
A}$-module of $K$-derivations of the algebra ${\cal A}$.

The next theorem gives a finite set of generators and a finite set of defining
relations for the left ${\cal A}$-module ${\rm Der}_{K}({\cal A})$ when ${\cal
A}$ is a regular algebra.

###### Theorem 4.6

([4, Theorem 4.2] if char$(K)=0$; [5, Theorem 1.1] if char$(K)=p>0$;) Let the
algebra ${\cal A}$ be a regular domain of essentially finite type over the
perfect field $K$. Then the left ${\cal A}$-module ${\rm Der}_{K}({\cal A})$
is generated by the derivations $\partial_{{\bf i},{\bf j}}$, ${\bf
i}\in\mathbb{I}_{r}$, ${\bf j}\in\mathbb{J}_{r+1}$ where

$\displaystyle\partial_{{\bf i},{\bf
j}}=\partial_{i_{1},\ldots,i_{r};j_{1},\ldots,j_{r+1}}:={\rm
det}\begin{pmatrix}\frac{\partial f_{i_{1}}}{\partial
x_{j_{1}}}&\cdots&\frac{\partial f_{i_{1}}}{\partial x_{j_{r+1}}}\\\
\vdots&\vdots&\vdots\\\ \frac{\partial f_{i_{r}}}{\partial
x_{j_{1}}}&\cdots&\frac{\partial f_{i_{r}}}{\partial x_{j_{r+1}}}\\\
\partial_{j_{1}}&\cdots&\partial_{j_{r+1}}\\\ \end{pmatrix}$

that satisfy the following defining relations (as a left ${\cal A}$-module):

$\Delta({\bf i},{\bf j})\partial_{{\bf i}^{\prime},{\bf
j}^{\prime}}=\sum_{l=1}^{s}(-1)^{r+1+\nu_{l}}\Delta({\bf
i}^{\prime};j_{1}^{\prime},\ldots,\widehat{j_{\nu_{l}}^{\prime}},\ldots,j_{r+1}^{\prime})\partial_{{\bf
i};{\bf j},j_{\nu_{l}}^{\prime}}$ (7)

for all ${\bf i},{\bf i}^{\prime}\in\mathbb{I}_{r}$, ${\bf
j}=(j_{1},\ldots,j_{r})\in\mathbb{J}_{r}$, and ${\bf
j}^{\prime}=(j_{1}^{\prime},\ldots,j_{r+1}^{\prime})\in{\bf J}_{r+1}$ where
$\\{j_{\nu_{1}}^{\prime},\ldots,j_{\nu_{s}}^{\prime}\\}=\\{j_{1}^{\prime},\ldots,j_{r+1}^{\prime}\\}\backslash\\{j_{1},\ldots,j_{r}\\}$.

Suppose that the algebra ${\cal A}=S^{-1}(P_{n}/I)$ is a Poisson domain of
essentially finite type. Let $d=d_{\cal A}=r({\cal C}_{\cal A})$ be the rank
of the $n\times n$ matrix

${\cal C}_{\cal A}=(\\{x_{i},x_{j}\\})\in M_{n}({\cal A})$

over the field $Q$. For each $l=1,\ldots,n$, let

${\rm ind}_{n}(l)=\\{{\bf i}=(i_{1},\ldots,i_{l})\,|\,1\leq
i_{1}<\cdots<i_{l}\leq n\\}.$

For elements ${\bf i}=(i_{1},\ldots,i_{l})$ and ${\bf j}=(j_{1},\ldots,j_{l})$
of ${\rm ind}_{n}(l)$, let ${\cal C}_{\cal A}({\bf i},{\bf
j})=(\\{x_{i_{\nu}},x_{j_{\mu}}\\})$ be the $l\times l$ submatrix of the
matrix ${\cal C}_{\cal A}$. So, the rows (resp., the columns) of the matrix
${\cal C}_{\cal A}({\bf i},{\bf j})$ are indexed by $i_{1},\ldots,i_{l}$
(resp., $j_{1},\ldots,j_{l}$). The $(i_{\nu},j_{\mu})^{th}$ element of the
matrix ${\cal C}_{\cal A}({\bf i},{\bf j})$ is
$\\{x_{i_{\nu}},x_{j_{\mu}}\\}$. Let $M_{{\cal A},l}=\\{{\cal C}_{\cal A}({\bf
i},{\bf j})\,|\,{\bf i},{\bf j}\in{\rm ind}_{n}(l)\\}$ be the set of all
$l\times l$ submatrices of ${\cal C}_{\cal A}$ and

${\cal C}_{{\cal A},l}=\\{\mu({\bf i},{\bf j}):={\rm det}({\cal C}_{\cal
A}({\bf i},{\bf j}))\,|\,{\bf i},{\bf j}\in{\rm ind}_{n}(l)\\}$

be the set of all $l\times l$ minors of ${\cal C}_{\cal A}$. Let

$\displaystyle\mathbb{I}(l)$ $\displaystyle=$ $\displaystyle\mathbb{I}_{\cal
A}(l)=\\{{\bf i}\in{\rm ind}_{n}(l)\,|\,\mu({\bf i},{\bf j})\neq 0\;\;{\rm
for\;some}\;\;{\bf j}\in{\rm ind}_{n}(l)\\}$ $\displaystyle\mathbb{J}(l)$
$\displaystyle=$ $\displaystyle\mathbb{J}_{\cal A}(l)=\\{{\bf j}\in{\rm
ind}_{n}(l)\,|\,\mu({\bf i},{\bf j})\neq 0\;\;{\rm for\;some}\;\;{\bf
i}\in{\rm ind}_{n}(l)\\}$

The ideal $\kappa_{\cal A}$ of the algebra ${\cal U}({\cal A})$ and its
generators $\delta_{{\bf i},i_{\nu};{\bf j}}$. For each pair of elements ${\bf
i}=(i_{1},\ldots,i_{d})$ and ${\bf j}=(j_{1},\ldots,j_{d})$ of
$\mathbb{I}_{\cal A}(d)=\mathbb{J}_{\cal A}(d)$ and each element
$i_{\nu}\in\\{i_{d+1},\ldots,i_{n}\\}=\\{1,\ldots,n\\}\backslash\\{i_{1},\ldots,i_{d}\\}$,
let us consider the following elements of the algebra ${\cal U}({\cal A})$,
[7],

$\displaystyle\delta_{{\bf i},i_{\nu};{\bf j}}:={\rm
det}\begin{pmatrix}\\{x_{i_{1}},x_{j_{1}}\\}&\ldots&\\{x_{i_{1}},x_{j_{d}}\\}&\delta_{i_{1}}\\\
\vdots&\vdots&\vdots&\vdots\\\
\\{x_{i_{d}},x_{j_{1}}\\}&\ldots&\\{x_{i_{d}},x_{j_{d}}\\}&\delta_{i_{d}}\\\
\\{x_{i_{\nu}},x_{j_{1}}\\}&\ldots&\\{x_{i_{\nu}},x_{j_{d}}\\}&\delta_{i_{\nu}}\\\
\end{pmatrix}=\mu({\bf i},{\bf
j})\delta_{i_{\nu}}+\sum_{s=1}^{d}(-1)^{s+d+1}\mu(i_{1},\ldots,\widehat{i_{s}},\ldots,i_{d},i_{\nu};{\bf
j})\delta_{i_{s}}$ (8)

where $\delta_{i}:=dx_{i}\in\Omega_{\cal A}\subseteq{\cal U}({\cal A})$.

Definition, [7]. Let $\kappa_{\cal A}$ be an ideal of the algebra ${\cal
U}({\cal A})$ which is generated by the finite set of elements $\delta_{{\bf
i},i_{\nu};{\bf j}}\in\Omega_{\cal A}$ where ${\bf i},{\bf
j}\in\mathbb{I}_{\cal A}(d)$, see (8). Then $\kappa_{\cal A}\subseteq{\rm
ker}(\pi_{\cal A})$, [7].

Criteria for ${\rm ker}(\pi_{\cal A})=0$. In the case when the Poisson algebra
${\cal A}$ is a regular domain of essentially finite type, Theorem 4.7 is an
efficient explicit criterion for ${\rm ker}(\pi_{\cal A})=0$, i.e. for the
epimorphism $\pi_{\cal A}:{\cal U}({\cal A})\rightarrow P{\cal D}({\cal A})$
to be an isomorphism.

Since ${\rm Der}_{K}({\cal A})\simeq{\rm Hom}_{\cal A}(\Omega_{\cal A},{\cal
A})$, there is a pairing of left ${\cal A}$-modules (which is an ${\cal
A}$-bilinear map):

${\rm Der}_{K}({\cal A})\times\Omega_{\cal A}\rightarrow{\cal
A},\;\;(\partial,\omega)\mapsto(\partial,\omega):=\partial(\omega).$

###### Theorem 4.7

([7, Theorem 1.9]) Let a Poisson algebra ${\cal A}=S^{-1}(P_{n}/I)$ be a
regular domain of essentially finite type over the field $K$ of characteristic
zero, $d=r({\cal C}_{\cal A})$ and $r=r\Big{(}\frac{\partial f_{i}}{\partial
x_{j}}\Big{)}$. Then the following statements are equivalent (the derivations
$\partial_{{\bf i};{\bf j},j_{\nu}}$ of ${\cal A}$ are defined in Theorem
4.6):

  1. 1.

${\rm ker}(\pi_{\cal A})=0$ $(\Leftrightarrow\pi_{\cal A}:{\cal U}({\cal
A})\simeq P{\cal D}({\cal A}))$.

  2. 2.

$\kappa_{\cal A}=0$.

  3. 3.

$d=n-r$ and $(\partial_{{\bf i};{\bf j},j_{\nu}},\delta_{{\bf i}^{\prime};{\bf
j}^{\prime},j_{\mu}^{\prime}})=0$ for all elements ${\bf i}\in\mathbb{I}_{r}$,
${\bf j}\in\mathbb{J}_{r}$, ${\bf i}^{\prime}\in\mathbb{I}_{\cal A}(d)$, ${\bf
j}^{\prime}\in\mathbb{J}_{\cal A}(d)$, $\nu=r+1,\ldots,n$ and
$\mu=d+1,\ldots,n$ where for ${\bf j}=(j_{1},\ldots,j_{r})$ and ${\bf
i}^{\prime}=(i_{1}^{\prime},\ldots,i^{\prime}_{d})$,
$\\{j_{r+1},\ldots,j_{n}\\}=\\{1,\ldots,n\\}\backslash\\{j_{1},\ldots,j_{r}\\}$
and
$\\{i^{\prime}_{d+1},\ldots,i^{\prime}_{n}\\}=\\{1,\ldots,n\\}\backslash\\{i_{1}^{\prime},\ldots,i_{d}^{\prime}\\}$.

${\rm GK}\,$ stands for the Gelfand-Kirillov dimension and ${\rm Sym}_{\cal
A}(\Omega_{\cal A})$ is the symmetric algebra of the ${\cal A}$-module
$\Omega_{\cal A}$ of Kähler differentials of the algebra ${\cal A}$.

In the case when the Poisson algebra ${\cal P}={\cal A}$ is an algebra of
essentially finite type over a field of characteristic zero, Theorem 4.4 can
be strengthen, see Theorem 4.8.

###### Theorem 4.8

([7, Theorem 1.3]) Let a Poisson algebra ${\cal A}$ be an algebra of
essentially finite type over the field $K$ of characteristic zero. Then the
following statements are equivalent:

  1. 1.

The algebra ${\cal U}({\cal A})$ is a simple algebra.

  2. 2.

The algebra $P{\cal D}({\cal A})$ is a simple algebra and one of the
equivalent conditions of Theorem 4.7 holds.

  3. 3.

The algebra ${\cal A}$ is Poisson simple and one of the equivalent conditions
of Theorem 4.7 holds.

If one of the equivalent conditions holds then the algebra ${\cal
A}=S^{-1}(P_{n}/I)$ is a regular, Poisson simple domain of essentially finite
type over the field $K$ of characteristic zero, the algebra epimorphism
$\pi_{\cal A}:{\cal U}({\cal A})\rightarrow P{\cal D}({\cal A})$ is an
isomorphism, $d=n-r$ where $d=r({\cal C}_{\cal A})$ and
$r=r\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$, and the algebra
${\cal U}({\cal A})$ is a simple Noetherian domain with

${\rm GK}\,\,{\cal U}({\cal A})={\rm GK}\,\,P{\cal D}({\cal A})={\rm
GK}\,\,{\rm gr}\,{\cal U}({\cal A})={\rm GK}\,\,{\rm Sym}_{\cal
A}(\Omega_{\cal A})=2{\rm GK}\,({\cal A})=2(n-r).$

A simplicity criterion for the PEA ${\cal U}(A)$ of a GWPA $A$. Let
$A=D[X,Y;\partial,a\\}$ be a GWPA of degree $N$; a Poisson algebra $D={\cal
A}=S^{-1}(P_{n}/I)$ be a regular domain of essentially finite type;
$I=(f_{1},\ldots,f_{m})$; $r=r\Big{(}\frac{\partial f_{i}}{\partial
x_{j}}\Big{)}$ be the rank of the Jacobian matrix $\Big{(}\frac{\partial
f_{i}}{\partial x_{j}}\Big{)}$ of the algebra $D$; $d=r({\cal C}_{A})$ be the
rank of the $(n+2N)\times(n+2N)$ matrix

${\cal C}_{A}=(\\{s_{i},s_{j}\\})\;\;{\rm
where}\;\;s_{i},s_{j}\in\\{x_{1},\ldots,x_{n},X_{1},\ldots,X_{N},Y_{1},\ldots,Y_{N}\\}.$

Theorem 4.9 is a simplicity criterion for the Poisson enveloping algebra
${\cal U}(A)$ of a GWPA $A$.

###### Theorem 4.9

Let a Poisson algebra $D=S^{-1}(P_{n}/I)$ be a regular domain of essentially
finite type over the field $K$ of characteristic zero and
$r=r\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$ be the rank of the
Jacobian matrix of $D$, $A=D[X,Y;\partial,a\\}$ be a GWPA of degree $N$ and
$d=r({\cal C}_{A})$. Then the following statements are equivalent:

  1. 1.

The algebra ${\cal U}(A)$ is a simple algebra.

  2. 2.

The algebra $P{\cal D}(A)$ is a simple algebra and one of the equivalent
conditions of Theorem 4.7 holds.

  3. 3.

The Poisson algebra $A$ is a Poisson simple algebra and one of the equivalent
conditions of Theorem 4.7 holds.

  4. 4.

    1. (a)

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

    2. (b)

for all $i=1,\ldots,n$, $Da_{i}+D\partial_{i}(a_{i})=D$, and

    3. (c)

the algebra ${\rm PZ}(A)$ is a field $(\Leftrightarrow$ char$(K)=0$, ${\rm
PZ}(D)^{\partial}$ is a field and $D_{\alpha}=0$ for all
$\alpha\in{\mathbb{Z}}^{n}\backslash\\{0\\}$ where $D_{\alpha}:=\\{\lambda\in
D^{\partial}\,|\,{\rm
pad}_{\lambda}=\lambda\sum_{i=1}^{n}\alpha_{i}\partial_{i},\;\lambda\alpha_{i}\partial_{i}(a_{i})=0$
for $i=1,\ldots,n\\}$, [6, Proposition 1.2]).

    4. (d)

$d=n-r+N$ and $(\partial_{{\bf i};{\bf j},j_{\nu}},\delta_{{\bf
i}^{\prime};{\bf j}^{\prime},j_{\mu}^{\prime}})=0$ for all elements ${\bf
i}\in\mathbb{I}_{r}$, ${\bf j}\in\mathbb{J}_{r}$, ${\bf
i}^{\prime}\in\mathbb{I}_{\cal A}(d)$, ${\bf j}^{\prime}\in\mathbb{J}_{\cal
A}(d)$, $\nu=r+1,\ldots,n$ and $\mu=d+1,\ldots,n$ where for ${\bf
j}=(j_{1},\ldots,j_{r})$ and ${\bf
i}^{\prime}=(i_{1}^{\prime},\ldots,i^{\prime}_{d})$,
$\\{j_{r+1},\ldots,j_{n}\\}=\\{1,\ldots,n\\}\backslash\\{j_{1},\ldots,j_{r}\\}$
and
$\\{i^{\prime}_{d+1},\ldots,i^{\prime}_{n}\\}=\\{1,\ldots,n\\}\backslash\\{i_{1}^{\prime},\ldots,i_{d}^{\prime}\\}$.

If one of the equivalent conditions holds then ${\cal U}(A)\simeq P{\cal
D}(A)$, the algebra $A$ is a regular, Poisson simple domain (in particular,
all $a_{i}\neq 0$), $d=n-r+N$ and the algebra ${\cal U}(A)$ is a simple
Noetherian domain with

${\rm GK}\,\,{\cal U}(A)={\rm GK}\,\,P{\cal D}(A)=2{\rm GK}\,(A)=2(n-r+N).$

Proof. The GWPA $A$ is a domain iff the algebra $D$ and all the elements
$a_{i}$ are not equal to zero. Notice that ${\rm GK}\,(A)={\rm
GK}\,(D)+N=n-r+N$. Now, the theorem follows from Theorem 4.8, Theorem 4.2 and
Theorem 4.7. $\Box$

So, Theorem 4.9.(4) is an efficient tool in proving or disproving simplicity
of the algebra ${\cal U}(A)$.

The Gelfand-Kirillov dimension of the algebras ${\cal U}(A)$, ${\rm gr}\,{\cal
U}(A)$ and ${\rm Sym}_{A}(\Omega_{A})$.

###### Theorem 4.10

([7, Theorem 1.4]) Let a Poisson algebra ${\cal A}=S^{-1}(P_{n}/I)$ be a
domain of essentially finite type over a perfect field $K$ where
$I=(f_{1},\ldots,f_{m})$ is a prime ideal of $P_{n}$, and
$r=r\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$ be the rank of the
Jacobian matrix $\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$ . Then
the algebra ${\cal U}({\cal A})$ is a Noetherian algebra with

${\rm GK}\,\,{\cal U}({\cal A})={\rm GK}\,\,{\rm gr}\,{\cal U}({\cal A})={\rm
GK}\,\,{\rm Sym}_{\cal A}(\Omega_{\cal A})=2{\rm GK}\,({\cal A})=2(n-r).$

By Theorem 3.2, the Poisson enveloping algebra ${\cal U}(A)$ of the GWPA
$A=D[X,Y;a,\partial\\}$, where $D=S^{-1}(P_{n}/I)$, admits a filtration by the
total degree of the elements $\delta_{x_{1}},\ldots,\delta_{x_{n}}$,
$\delta_{X_{1}},\ldots,\delta_{X_{n}}$, $\delta_{Y_{1}},\ldots,\delta_{Y_{n}}$
(nonzero elements of the algebra $A$ have degree zero). The associated graded
algebra of ${\cal U}(A)$ is denoted by ${\rm gr}\,{\cal U}(A)$. In the case
when ${\cal U}(A)=P{\cal D}(A)$ and char$(K)=0$, the filtration coincides with
the order filtration on the algebra of Poisson differential operators $P{\cal
D}(A)$ on $A$.

###### Corollary 4.11

Let a Poisson algebra $D=S^{-1}(P_{n}/I)$ be a domain of essentially finite
type over a perfect field $K$, $I=(f_{1},\ldots,f_{m})$,
$r=r\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$ be the rank of the
Jacobian matrix $\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$ of $D$,
$A=D[X,Y;\partial,a\\}$ be a GWPA of degree $N$ with all $a_{i}\neq 0$ and
$d=r({\cal C}_{A})$. Then the algebra ${\cal U}(A)$ is a Noetherian algebra
with

${\rm GK}\,\,{\cal U}(A)={\rm GK}\,\,{\rm gr}\,{\cal U}(A)={\rm GK}\,\,{\rm
Sym}_{A}(\Omega_{A})=2{\rm GK}\,(A)=2(n-r+N).$

Proof. Since ${\rm GK}\,(A)={\rm GK}\,(D)+N=n-r+N$, the corollary follows from
Theorem 4.10. $\Box$

The Gelfand-Kirillov dimension of the algebra $P{\cal D}(A)$ of Poisson
differential operators on $A$. Proposition 4.12 gives the exact figure for the
Gelfand-Kirillov dimension of the algebra $P{\cal D}({\cal A})$ where the
Poisson algebra ${\cal A}$ is a domain of essentially finite type over the
field $K$ of characteristic zero.

###### Proposition 4.12

([7, Theorem 1.5]) Let a Poisson algebra ${\cal A}$ be a domain of essentially
finite type over the field $K$ of characteristic zero and $r$ be the rank of
Jacobian matrix of ${\cal A}$ and $d=r({\cal C}_{\cal A})$. Then

${\rm GK}\,(P{\cal D}({\cal A}))={\rm GK}\,({\cal A})+d=n-r+d.$

###### Corollary 4.13

Let a Poisson algebra $D=S^{-1}(P_{n}/I)$ be a domain of essentially finite
type over the field $K$ of characteristic zero, $I=(f_{1},\ldots,f_{m})$,
$r=r\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$ be the rank of the
Jacobian matrix $\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$ of $D$,
$A=D[X,Y;\partial,a\\}$ be a GWPA of degree $N$ with all $a_{i}\neq 0$ and
$d=r({\cal C}_{A})$. Then

${\rm GK}\,(P{\cal D}(A))={\rm GK}\,(A)+d=n-r+N+d.$

Proof. The GWPA $A$ is a domain since the algebra $D$ is so and all the
elements $a_{i}$ are not equal to zero. Since ${\rm GK}\,(A)={\rm
GK}\,(D)+N=n-r+N$, the corollary follows from Theorem 4.12. $\Box$

The algebra ${\cal U}(A)$ is a domain when $A$ is a regular domain of
essentially finite type. Theorem 4.14 states that the algebra ${\cal U}({\cal
A})$ is a domain provided the algebra ${\cal A}$ is a regular domain of
essentially finite type.

###### Theorem 4.14

([7, Theorem 4.4]) Let a Poisson algebra ${\cal A}=S^{-1}(P_{n}/I)$ be a
regular domain of essentially finite type over the perfect field $K$ where
$I=(f_{1},\ldots,f_{m})$ is a prime but not maximal ideal of $P_{n}$ and
$r=r\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$ is the rank of the
Jacobian matrix $\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$ over the
field of fractions of the domain $P_{n}/I$. Then the algebra ${\cal U}({\cal
A})$ is a Noetherian domain with ${\rm GK}\,\,{\cal U}({\cal A})={\rm
GK}\,\,{\rm gr}\,{\cal U}({\cal A})={\rm GK}\,\,{\rm Sym}_{\cal
A}(\Omega_{\cal A})=2{\rm GK}\,({\cal A})=2(n-r)$.

###### Corollary 4.15

Let a Poisson algebra $D=S^{-1}(P_{n}/I)$ be a domain of essentially finite
type over the perfect field $K$, $I=(f_{1},\ldots,f_{m})$,
$r=r\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$ be the rank of the
Jacobian matrix $\Big{(}\frac{\partial f_{i}}{\partial x_{j}}\Big{)}$ of $D$,
$A=D[X,Y;\partial,a\\}$ be a regular GWPA of degree $N$ with all $a_{i}\neq 0$
and $d=r({\cal C}_{A})$. Then the algebra ${\cal U}(A)$ is a Noetherian domain
with ${\rm GK}\,\,{\cal U}(A)={\rm GK}\,\,{\rm gr}\,{\cal U}(A)={\rm
GK}\,\,{\rm Sym}_{A}(\Omega_{A})=2{\rm GK}\,(A)=2(n-r+N)$.

Proof. Since ${\rm GK}\,(A)={\rm GK}\,(D)+N=n-r+N$ and the algebra $A$ is a
domain (since $D$ is so and all the elements $a_{i}\neq 0$), the corollary
follows from Theorem 4.14. $\Box$

Data Availability. Data sharing is not applicable to this article as no new
data were created or analyzed in this study.

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Department of Pure Mathematics

University of Sheffield

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What is the mechanism of the $T_{4c}(6900)$ tetraquark production ?

A. Szczurek 1,2, R. Maciula 1 and W. Schäfer 1$\star$

1 Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego
152, PL-31-342 Kraków, Poland

2 College of Natural Sciences, Institute of Physics, University of Rzeszów,
ul. Pigonia 1, PL-35-959 Rzeszów, Poland

*<EMAIL_ADDRESS>
Proceedings for the XXVIII International Workshop
on Deep-Inelastic Scattering and Related Subjects,
Stony Brook University, New York, USA, 12-16 April 2021
10.21468/SciPostPhysProc.?

## Abstract

We discuss the production mechanism of fully charm tetraquark, observed
recently by the LHCb at M = 6.9 GeV in the $J/\psi J/\psi$ channel. Both
single parton scattering (SPS) and double parton scattering (DPS) mechanisms
are considered. We calculate the distribution in the invariant mass of the
four-quark system $M_{4c}$ for SPS and DPS production of $cc\bar{c}\bar{c}$ in
the $k_{t}$-factorization approach. We present transverse momentum
distribution in $p_{t,4c}$ for $c\bar{c}c\bar{c}$ system in a mass window
around tetraquark mass. The calculation of the SPS $g^{*}g^{*}\to
T_{4c}(6900)$ fusion mechanism is performed in the $k_{T}$-factorization
approach assuming different spin scenarios ($0^{+}$ and $0^{-}$).

##  1 Introduction

The recent observation by the LHCb collaboration [1] of a sharp peak in the
di-$J/\psi$ channel at $M$ = 6.9 GeV seems to strongly suggest the presence of
a fully charm tetraquark, consisting of $cc\bar{c}\bar{c}$. A number of
theoretical models for the spectroscopy of tetraquarks were developed. The
most popular approach treats the fully heavy ($cc\bar{c}\bar{c}$,
$bb\bar{b}\bar{b}$ or $c\bar{c}b\bar{b}$) tetraquarks as a bound system of a
color antitriplet diquark and color triplet antidiquark.

Assuming the peak observed by LHCb is indeed caused by a new state, models
suggest that it is rather an excited state. However, the spin and parity of
the state are not known at present.

Different $J^{PC}$ combinations are possible in general [2, 3, 4, 5], What is
the mechanism of the fusion of four charm quarks/antiquarks is not clear in
the moment. A large cross section for $c\bar{c}c\bar{c}$ production at the LHC
due to double-parton scattering (DPS) mechanism was predicted in [6]. The
contribution of the single parton scattering to the $c\bar{c}c\bar{c}$
production is much smaller [7, 8, 9, 10]. The production of two $J/\psi$
states was also studied at the LHC [11, 12, 13, 14].

Recently, our group studied production of pseudoscalar [15] and scalar [16]
charmonia via gluon-gluon fusion. We shall use the same formalism also in the
context of the tetraquark production.

##  2 Basic formalism

In Fig.1 we show the dominant reaction mechanisms of $cc\bar{c}\bar{c}$
production: SPS type (left diagram) and DPS type (right diagram).

Figure 1: Two dominant reaction mechanisms of production of $c\bar{c}c\bar{c}$
nonresonant continuum. The left diagram represent the SPS mechanism (box type)
and the left diagram the DPS mechanism.

In our recent study both the SPS and the DPS contributions are calculated in
the framework of $k_{T}$-factorization [17].

In this approach the SPS cross section for $pp\to c\bar{c}c\bar{c}\,X$
reaction can be written as

$d\sigma_{pp\to c\bar{c}c\bar{c}\;X}=\int
dx_{1}\frac{d^{2}k_{1t}}{\pi}dx_{2}\frac{d^{2}k_{2t}}{\pi}{\cal
F}_{g}(x_{1},k_{1t}^{2},\mu^{2}){\cal
F}_{g}(x_{2},k_{2t}^{2},\mu^{2})d{\hat{\sigma}}_{g^{*}g^{*}\to
c\bar{c}c\bar{c}}\;.$ (1)

Above ${\cal F}_{g}(x,k_{t}^{2},\mu^{2})$ is the unintegrated gluon
distribution function (gluon uPDF).

The elementary cross section in Eq. (1) can be written as:

$d{\hat{\sigma}}_{g^{*}g^{*}\to
c\bar{c}c\bar{c}}={1\over(2!)^{2}}\prod_{l=1}^{4}\frac{d^{3}\vec{p}_{l}}{(2\pi)^{3}2E_{l}}(2\pi)^{4}\delta^{4}(\sum_{l=1}^{4}p_{l}-k_{1}-k_{2})\frac{1}{\mathrm{flux}}\overline{|{\cal
M}_{g^{*}g^{*}\to c\bar{c}c\bar{c}}(k_{1},k_{2},\\{p_{l}\\})|^{2}}\;.$ (2)

We consider also the calculation of the SPS-type signal as a fusion of two
(off-shell) gluons for two different spin-parity assignments: $0^{+}$ and
$0^{-}$ of the tetraquark. The corresponding diagram is shown in Fig.2.

Figure 2: The mechanism of gluon-gluon fusion leading to the production of the
$T_{4c}(6900)$ tetraquark.

In this context we use the formalism worked out recently for the inclusive
production of pseudoscalar [15] and scalar [16] quarkonia. The off-shell gluon
fusion cross sections is proportional to a form-factor, which depends on the
virtualities of gluons, $Q_{i}^{2}=-k_{i}^{2}$:

$\displaystyle d\sigma_{g^{*}g^{*}\to 0^{-}}$ $\displaystyle\propto$
$\displaystyle{1\over
k_{1t}^{2}k_{2t}^{2}}\,(\vec{k}_{1t}\times\vec{k}_{2t})^{2}\,F^{2}(Q_{1}^{2},Q_{2}^{2})\,,$
$\displaystyle d\sigma_{g^{*}g^{*}\to 0^{+}}$ $\displaystyle\propto$
$\displaystyle{1\over
k_{1t}^{2}k_{2t}^{2}}\Big{(}(\vec{k}_{1t}\cdot\vec{k}_{2t})(M^{2}+Q_{1}^{2}+Q_{2}^{2})+2Q_{1}^{2}Q_{2}^{2}\Big{)}^{2}\,{F^{2}(Q_{1}^{2},Q_{2}^{2})\over
4X^{2}}\,,$ (3)

with $X=(M^{4}+2(Q_{1}^{2}+Q_{2}^{2})M^{2}+(Q_{1}^{2}-Q_{2}^{2})^{2})/4$.
Note, that for the $0^{+}$ assignment we use only the TT coupling, as in
analogy with [16] we expect the LL contribution to be smaller.

The mechanisms of the background productions are shown in Fig.3.

Figure 3: Two dominant reaction mechanisms of production of $J/\psi J/\psi$
nonresonant continuum. The left diagram represent the SPS mechanism (box type)
and the right diagram the DPS mechanism.

##  3 Selected results

In Fig.4 we present distribution in $p_{t,4c}$ of four quark-antiquark system
within invariant mass window $(M_{R}-\rm{0.1GeV},M_{R}+\rm{0.1GeV})$. In a
naive coalescence model this corresponds to the distribution of the
tetraquark.

Figure 4: Distribution of $p_{t,4c}$ of four quark-antiquark system within
invariant mass window $(M_{R}-\rm{0.1GeV},M_{R}+\rm{0.1GeV})$. Here
$\sqrt{s}=$ 13 TeV and average rapidity of quarks and antiquarks in the
interval (2,4.5). The solid line is for SPS, the dashed line for $gggg\to
c\bar{c}c\bar{c}$ DPS and the dotted line is for $q\bar{q}gg\to
c\bar{c}c\bar{c}$ DPS contribution.

In Fig.5 we show distribution in $M_{J/\psi J/\psi}$ for the two background
mechanisms shown in Fig.3. We see that in the vicinity of the tetraquark mass
the SPS contribution is similar to the DPS one so both of them must be
included in the evaluation of the background.

Figure 5: Distribution in invariant mass of the $J/\psi J/\psi$ system for SPS
(solid line) and DPS (dashed line). In this calculation $\sqrt{s}$ = 13 TeV
and we assumed that both $J/\psi$ mesons have rapidity in the (2,4.5)
interval.

The transverse momentum distribution of the $T_{4c}(6900)$ tetraquark produced
in a single particle mechanism is shown in Fig.6.

Figure 6: Transverse momentum distribution of the $T_{4c}(6900)$ tetraquark
for the $0^{-}$ (left panel) and $0^{+}$ (right panel) assignments. Here
$\sqrt{s}$ = 13 TeV. We show results for the KMR UGDF and $\Lambda$ = 6 GeV
(solid line) and $\Lambda$ = 4 GeV (dashed line).

Since the ratio of signal-to-background improves with transverse momentum of
the tetraquark [1] and knowing relatively well the behaviour of the SPS and
DPS background [17] we can conclude that the $0^{-}$ assignment is disfavoured
by the LHCb experimental results.

##  4 Conclusion

In our recent paper [17] we have considered several aspects related to the
production of $T_{4c}(6900)$ tetraquark observed recently by the LHCb
collaboration in the $J/\psi J/\psi$ channel and the $J/\psi J/\psi$
background.

While the background distributions can in our opinion be reliably calculated,
it is not the case for the signal. In the discussed naive coalescence model we
have adjusted a normalization factor $C$ responsible for the formation
probability $P_{T_{4c}}$ and decay branching fraction $Br(T_{4c}(6900)\to
J/\psi J/\psi)$. We have obtained $C$ = 10-4 for the DPS and C = 10-2 for the
SPS production of $c\bar{c}c\bar{c}$.

We have considered also more explicitly the SPS mechanism of the resonance
production via gluon-gluon fusion in the $k_{T}$-factorization approach. Also
in this case the normalization, related to the underlying formation process
and/or wave function of the tetraquark and the decay branching fraction
$T_{4c}\to J/\psi J/\psi$ must be adjusted to the experimental signal-to-
background ratio. In this study we have considered two examples of the $0^{+}$
and $0^{-}$ assignment. The current data seem to exclude the $0^{-}$
assignment as the final result contradicts qualitatively to the transverse
momentum dependence of the signal-to-background ratio as observed by the LHCb
collaboration [1].

## Acknowledgements

This study was partially supported by the Polish National Science Center grant
UMO-2018/31/ /B/ST2/03537 and by the Center for Innovation and Transfer of
Natural Sciences and Engineering Knowledge in Rzeszów.

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