PBW-basis for universal enveloping algebras of differential graded Poisson algebras

TL;DR

This paper develops a PBW-basis for the universal enveloping algebra of differential graded Poisson algebras, providing explicit formulas and applications to DG symplectic ideals and modules.

Contribution

It introduces a formula for the universal enveloping algebra of DG Poisson algebras and proves a PBW-basis exists for graded commutative polynomial cases.

Findings

Established a PBW-basis for $A^e$ when $A$ is a graded commutative polynomial algebra.

Provided a formula for computing the universal enveloping algebra of DG Poisson algebras.

Applied the PBW-basis to characterize DG symplectic ideals as annihilators of simple modules.

Abstract

For any differential graded (DG for short) Poisson algebra $A$ given by generators and relations, we give a "formula" for computing the universal enveloping algebra $A^e$ of $A$. Moreover, we prove that $A^e$ has a Poincar\'e-Birkhoff-Witt basis provided that $A$ is a graded commutative polynomial algebra. As an application of the PBW-basis, we show that a DG symplectic ideal of a DG Poisson algebra $A$ is the annihilator of a simple DG Poisson $A$-module, where $A$ is the DG Poisson homomorphic image of a DG Poisson algebra $R$ whose underlying algebra structure is a graded commutative polynomial algebra.

Equations

$$\alpha (\{a,b\})$$

$$\beta (ab)$$

$$f(\{a,b\})$$

$$g(ab)$$

$$[\delta (a),\delta (b)](c)$$

$$=\{a,\{b,c\}\}-(-1{)}^{|a||b|}\{b,\{a,c\}\},$$

$$\delta (\{a,b\})(c)$$

$$(\delta d)(a)(b)$$

$$\begin{array}{rl}\delta (a)\gamma (b)(c)-(-1{)}^{|a||b|}\gamma (b)\delta (a)(c)& =\delta (a)(bc)-(-1{)}^{|a||b|}\gamma (b)\{a,c\}\\ & =\{a,bc\}-(-1{)}^{|a||b|}b\{a,c\}\\ & =\gamma (\{a,b\})(c)\end{array}$$

$$\begin{array}{rl}\gamma (a)\delta (b)(c)+(-1{)}^{|a||b|}\gamma (b)\delta (a)(c)& =\gamma (a)\{b,c\}+(-1{)}^{|a||b|}\gamma (b)\{a,c\}\\ & =a\{b,c\}+(-1{)}^{|a||b|}b\{a,c\}\\ & =\delta (ab)(c),\end{array}$$

$$R=\frac{T(V)}{⟨x_{\alpha }\otimes x_{\beta }-(-1{)}^{|x_{\alpha }||x_{\beta }|}{x}_{\beta }\otimes x_{\alpha }|\forall \alpha ,\beta \in \Lambda ⟩}$$

$$\begin{array}{rl}{\psi }_{\alpha }(ab-(-1{)}^{|a||b|}ba)& ={\psi }_{\alpha }(ab)-(-1{)}^{|a||b|}{\psi }_{\alpha }(ba)\\ & =a{\psi }_{\alpha }(b)+(-1{)}^{|a||b|}b{\psi }_{\alpha }(a)-(-1{)}^{|a||b|}(b{\psi }_{\alpha }(a)+(-1{)}^{|a||b|}a{\psi }_{\alpha }(b))\\ & =0.\end{array}$$

$${\psi }_{\alpha }(d(a))-d{\psi }_{\alpha }(a)-\sum _{\beta \in \Lambda }(-1{)}^{|{\psi }_{\beta }(a)|}{\psi }_{\beta }(a){\psi }_{\alpha }(d(x_{\beta }))=0.$$

$$LHSof(2.1)$$

$${\psi }_{\alpha }(d(a))-d{\psi }_{\alpha }(a)-\sum _{\beta \in \Lambda }(-1{)}^{|{\psi }_{\beta }(a)|}{\psi }_{\beta }(a){\psi }_{\alpha }(d(x_{\beta }))=0,$$

$${\psi }_{\alpha }(d(b))-d{\psi }_{\alpha }(b)-\sum _{\beta \in \Lambda }(-1{)}^{|{\psi }_{\beta }(b)|}{\psi }_{\beta }(b){\psi }_{\alpha }(d(x_{\beta }))=0,$$

$$\begin{array}{rl}& {\psi }_{\alpha }(d(ab))-d{\psi }_{\alpha }(ab)-\sum _{\beta \in \Lambda }(-1{)}^{|{\psi }_{\beta }(ab)|}{\psi }_{\beta }(ab){\psi }_{\alpha }(d(x_{\beta }))\\ =& {\psi }_{\alpha }(d(a)b+(-1{)}^{|a|}ad(b))-d(a{\psi }_{\alpha }(b)+(-1{)}^{|a||b|}b{\psi }_{\alpha }(a))\\ & -\sum _{\beta \in \Lambda }(-1{)}^{|{\psi }_{\beta }(ab)|}(a{\psi }_{\beta }(b)+(-1{)}^{|a||b|}b{\psi }_{\beta }(a)){\psi }_{\alpha }(d(x_{\beta }))\\ =& d(a){\psi }_{\alpha }(b)+(-1{)}^{(|a|+1)|b|}b{\psi }_{\alpha }(d(a))+(-1{)}^{|a|}a{\psi }_{\alpha }(d(b))+(-1{)}^{|a|+|a|(|b|+1)}d(b){\psi }_{\alpha }(a)\\ & -d(a){\psi }_{\alpha }(b)-(-1{)}^{|a|}ad{\psi }_{\alpha }(b)-(-1{)}^{|a||b|}d(b){\psi }_{\alpha }(a)-(-1{)}^{(|a|+1)|b|}bd{\psi }_{\alpha }(a)\\ & -\sum _{\beta \in \Lambda }(-1{)}^{|{\psi }_{\beta }(ab)|}a{\psi }_{\beta }(b){\psi }_{\alpha }(d(x_{\beta }))-\sum _{\beta \in \Lambda }(-1{)}^{|{\psi }_{\beta }(ab)|+|a||b|}b{\psi }_{\beta }(a){\psi }_{\alpha }(d(x_{\beta }))\\ =& (-1{)}^{(|a|+1)|b|}b[{\psi }_{\alpha }(d(a))-d{\psi }_{\alpha }(a)-\sum _{\beta \in \Lambda }(-1{)}^{|{\psi }_{\beta }(a)|}{\psi }_{\beta }(a){\psi }_{\alpha }(d(x_{\beta }))]\\ & +(-1{)}^{|a|}a[{\psi }_{\alpha }(d(b))-d{\psi }_{\alpha }(b)-\sum _{\beta \in \Lambda }(-1{)}^{|{\psi }_{\beta }(b)|}{\psi }_{\beta }(b){\psi }_{\alpha }(d(x_{\beta }))]\\ =& 0,\end{array}$$

$$\{a,b\}=\sum _{\alpha \in \Lambda }{\psi }_{\alpha }(a)\{x_{\alpha },b\},$$

$$\{a,b\}$$

$$\{a^{\prime },b^{\prime }\}$$

$$\begin{array}{rl}\{a{a}^{\prime },b{b}^{\prime }\}=& \{a{a}^{\prime },b\}{b}^{\prime }+(-1{)}^{|b|(|a|+|a^{\prime }|)}b\{a{a}^{\prime },b^{\prime }\}\\ =& a\{a^{\prime },b\}{b}^{\prime }+(-1{)}^{|a||a^{\prime }|}{a}^{\prime }\{a,b\}{b}^{\prime }+(-1{)}^{|b|(|a|+|a^{\prime }|)}ba\{a^{\prime },b^{\prime }\}+(-1{)}^{|b|(|a|+|a^{\prime }|)+|a||a^{\prime }|}b{a}^{\prime }\{a,b^{\prime }\}\\ =& \sum _{\alpha \in \Lambda }a{\psi }_{\alpha }(a^{\prime })\{x_{\alpha },b\}{b}^{\prime }+(-1{)}^{|a||a^{\prime }|}\sum _{\alpha \in \Lambda }{a}^{\prime }{\psi }_{\alpha }(a)\{x_{\alpha },b\}{b}^{\prime }\\ & +(-1{)}^{|b|(|a|+|a^{\prime }|)}\sum _{\alpha \in \Lambda }ba{\psi }_{\alpha }(a^{\prime })\{x_{\alpha },b^{\prime }\}+(-1{)}^{|b|(|a|+|a^{\prime }|)+|a||a^{\prime }|}\sum _{\alpha \in \Lambda }b{a}^{\prime }{\psi }_{\alpha }(a)\{x_{\alpha },b^{\prime }\}\\ =& \sum _{\alpha \in \Lambda }a{\psi }_{\alpha }(a^{\prime })\{x_{\alpha },b\}{b}^{\prime }+(-1{)}^{|b||x_{\alpha }|}\sum _{\alpha \in \Lambda }a{\psi }_{\alpha }(a^{\prime })b\{x_{\alpha },b^{\prime }\}\\ & +(-1{)}^{|a||a^{\prime }|}\sum _{\alpha \in \Lambda }{a}^{\prime }{\psi }_{\alpha }(a)\{x_{\alpha },b\}{b}^{\prime }+(-1{)}^{|b||x_{\alpha }|+|a||a^{\prime }|}\sum _{\alpha \in \Lambda }{a}^{\prime }{\psi }_{\alpha }(a)b\{x_{\alpha },b^{\prime }\}\\ =& \sum _{\alpha \in \Lambda }[(a{\psi }_{\alpha }(a^{\prime })+(-1{)}^{|a||a^{\prime }|}{a}^{\prime }{\psi }_{\alpha }(a))(\{x_{\alpha },b\}{b}^{\prime }+(-1{)}^{|b||x_{\alpha }|}b\{x_{\alpha },b^{\prime }\})]\\ =& \sum _{\alpha \in \Lambda }{\psi }_{\alpha }(a{a}^{\prime })\{x_{\alpha },b{b}^{\prime }\},\end{array}$$

$$\psi (f):=\sum _{\alpha \in \Lambda }{\psi }_{\alpha }(f)y_{\alpha }$$

$$\psi (f):=\sum _{\alpha \in \Lambda }{\psi }_{\alpha }(f)y_{\alpha },$$

$$\partial x_{\alpha }:=j(d(x_{\alpha }))=d(x_{\alpha }),\partial y_{\alpha }:=\psi (d(x_{\alpha }))=\sum _{\beta \in \Lambda }{\psi }_{\beta }d(x_{\alpha })y_{\beta }$$

$$\begin{array}{rl}{\partial }^2(y_{\alpha })=& \partial (\sum _{\beta \in \Lambda }{\psi }_{\beta }d(x_{\alpha })y_{\beta })\\ =& \sum _{\beta \in \Lambda }[\partial {\psi }_{\beta }d(x_{\alpha })y_{\beta }+(-1{)}^{|{\psi }_{\beta }d(x_{\alpha })|}{\psi }_{\beta }d(x_{\alpha })\partial (y_{\beta })]\\ =& \sum _{\beta \in \Lambda }[d{\psi }_{\beta }d(x_{\alpha })y_{\beta }+(-1{)}^{|{\psi }_{\beta }d(x_{\alpha })|}{\psi }_{\beta }d(x_{\alpha })\sum _{\gamma \in \Lambda }{\psi }_{\gamma }d(x_{\beta })y_{\gamma }]\\ =& \sum _{\beta \in \Lambda }{\psi }_{\beta }{d}^2(x_{\alpha })y_{\beta }-\sum _{\beta ,\gamma \in \Lambda }(-1{)}^{|{\psi }_{\gamma }d(x_{\alpha })|}{\psi }_{\gamma }d(x_{\alpha }){\psi }_{\beta }d(x_{\gamma })y_{\beta }+\sum _{\beta ,\gamma \in \Lambda }(-1{)}^{|{\psi }_{\beta }d(x_{\alpha })|}{\psi }_{\beta }d(x_{\alpha }){\psi }_{\gamma }d(x_{\beta })y_{\gamma }\\ =& -\sum _{\beta ,\gamma \in \Lambda }(-1{)}^{|{\psi }_{\gamma }d(x_{\alpha })|}{\psi }_{\gamma }d(x_{\alpha }){\psi }_{\beta }d(x_{\gamma })y_{\beta }+\sum _{\beta ,\gamma \in \Lambda }(-1{)}^{|{\psi }_{\beta }d(x_{\alpha })|}{\psi }_{\beta }d(x_{\alpha }){\psi }_{\gamma }d(x_{\beta })y_{\gamma }\\ =& 0\end{array}$$

$$m(fg)$$

$$\partial m(f)$$

$$\partial h(f)$$

$$=\sum _{\alpha \in \Lambda }[\partial {\psi }_{\alpha }(f)y_{\alpha }+(-1{)}^{|{\psi }_{\alpha }(f)|}{\psi }_{\alpha }(f)\partial (y_{\alpha })]$$

$$=\sum _{\alpha \in \Lambda }[d{\psi }_{\alpha }(f)y_{\alpha }+(-1{)}^{|{\psi }_{\alpha }(f)|}{\psi }_{\alpha }(f)(\sum _{\beta \in \Lambda }{\psi }_{\beta }d(x_{\alpha })y_{\beta })]$$

$$=\sum _{\alpha \in \Lambda }[d{\psi }_{\alpha }(f)y_{\alpha }]+\sum _{\beta \in \Lambda }[{\psi }_{\beta }d(f)-d{\psi }_{\beta }(f)]y_{\beta }$$

$$=hd(f)$$

$$\begin{array}{rl}h(fg)& =\psi (fg)=\sum _{\alpha \in \Lambda }{\psi }_{\alpha }(fg)y_{\alpha }=\sum _{\alpha \in \Lambda }f{\psi }_{\alpha }(g)y_{\alpha }+\sum _{\alpha \in \Lambda }(-1{)}^{|f||g|}g{\psi }_{\alpha }(f)y_{\alpha }\\ & =f\psi (g)+(-1{)}^{|f||g|}g\psi (f)=m(f)h(g)+(-1{)}^{|f||g|}m(g)h(f).\end{array}$$

$$\begin{array}{rl}h(x_{\alpha })m(x_{\beta })-(-1{)}^{|x_{\alpha }||x_{\beta }|}m(x_{\beta })h(x_{\alpha })& =\sum _{\gamma \in \Lambda }{\psi }_{\gamma }(x_{\alpha })y_{\gamma }{x}_{\beta }-(-1{)}^{|x_{\alpha }||x_{\beta }|}{x}_{\beta }\sum _{\gamma \in \Lambda }{\psi }_{\gamma }(x_{\alpha })y_{\gamma }\\ & =y_{\alpha }{x}_{\beta }-(-1{)}^{|x_{\alpha }||x_{\beta }|}{x}_{\beta }{y}_{\alpha }\\ & =m(\{x_{\alpha },x_{\beta }\}).\end{array}$$

$$m(\{fg,a\})$$

$$=$$

Full text

This work was supported by National Natural Science Foundation of China (Grant No.11571316) and Natural Science Foundation of Zhejiang Province (Grant No. LY16A010003).

*corresponding author

1. Introduction

The notion of Poisson algebras arises naturally in the study of Hamiltonian mechanics and Poisson geometry. Recently, many important generalizations on Poisson algebras have been obtained in both commutative and noncommutative settings: Poisson orders [1], Poisson PI algebras [13], graded Poisson algebras [4], double Poisson algebras [18], Novikov-Poisson algebras [21], Quiver Poisson algebras [23], noncommutative Leibniz-Poisson algebras [2], left-right noncommutative Poisson algebras [3], noncommutative Poisson algebras [20] and differential graded Poisson algebras [12], etc. An interesting and practical idea to develop Poisson algebras is to study Poisson universal enveloping algebras, which was first introduced by Oh in 1999 [14], and later Oh, Park and Shin studied the Poincaré-Birkhoff-Witt basis (PBW-basis for short) for Poisson universal enveloping algebras [16]. Since then, Poisson universal enveloping algebras have been studied in a series of papers [17, 22]. In particular, the second author and the third author of the present paper studied the universal enveloping algebras of Poisson Hopf algebras and Poisson Ore-extensions [10, 11]. Our main aim of this paper is to study the PBW-basis for universal enveloping algebras of DG Poisson algebras.

In [5, 6, 7, 19], Hodges, Levasseur, Joseph and Vancliff proved that symplectic leaves of certain Poisson varieties correspond bijectively to primitive ideals of the respective quantum algebras. After that, Oh defined the symplectic ideal of a Poisson algebra and proved that there is a one to one correspondence between the primitive ideals of quantum $2\times 2$ matrix algebra and the symplectic ideals of a Poisson algebra constructed appropriately [15]. Moreover, he and two other authors showed that the symplectic ideal of a Poisson homomorphic image of a Poisson polynomial algebra $\Bbbk[x_{1},...,x_{n}]$ is the annihilator of a simple Poisson module. Motivated by the DG version for symplectic ideals, there is a natural question: Is a DG symplectic ideal of a DG Poisson algebra $A$ the annihilator of a simple DG Poisson $A$-module? The present paper gives a positive answer.

The paper is organized as follows. In Section 2, we briefly review some basic concepts related to DG Poisson algebras, DG Poisson modules and universal enveloping algebras of DG Poisson algebras. In particular, we construct the universal enveloping algebra of any DG Poisson algebra given by generators and relations. Section 3 is devoted to the study of PBW-basis for universal enveloping algebras. To be specific, for a DG Poisson algebra $R$ whose underlying algebra structure is a graded commutative polynomial algebra, by using Gröbner-Shirshov basis theory developed in [8] and [9], we prove that the universal enveloping algebra $R^{e}$ has a PBW-basis, which is analogous to the PBW-basis for universal enveloping algebras of Lie algebras. In the last section, we focus on the simple DG Poisson module. As an application of the PBW-basis theorem for universal enveloping algebras of DG Poisson algebras, we prove that a DG symplectic ideal of a DG Poisson algebra $A$ is the annihilator of a simple DG Poisson $A$-module, where $A$ is the DG Poisson homomorphic image of $R$.

Throughout the whole paper, $\mathbb{Z}$ denotes the set of integers, $\Bbbk$ denotes a base field and everything is over $\Bbbk$ unless otherwise stated, all (graded) algebras are assumed to have an identity and all (graded) modules are assumed to be unitary.

2. Universal enveloping algebras of differential graded Poisson algebras

In this section, first we briefly review some basic definitions and properties of DG Poisson algebras and universal enveloping algebras, then we construct the universal enveloping algebra of any DG Poisson algebra $A$ given by generators and relations.

2.1. DG Poisson algebras

By a graded algebra we mean a $\mathbb{Z}$-graded algebra. A DG algebra is a graded algebra with a $\Bbbk$-linear homogeneous map $d:A\rightarrow A$ of degree 1, which is also a graded derivation. Any graded algebra can be viewed as a DG algebra with differential $d=0$; in this case it is called a DG algebra with trivial differential. Let $A$, $B$ be two DG algebras and $f:A\rightarrow B$ be a graded algebra map of degree zero. Then $f$ is called a DG algebra map if $f$ commutes with the differentials.

Definition 2.1**.**

Let $A$ be a graded $\Bbbk$-vector space. If there is a $\Bbbk$-linear map

$\{\cdot,\cdot\}:A\otimes A\rightarrow A$

of degree 0 such that:

1. (i)

(graded antisymmetry): $\{a,b\}=-(-1)^{|a||b|}\{b,a\}$; 2. (ii)

(graded Jacobi identity): $\{a,\{b,c\}\}=\{\{a,b\},c\}+(-1)^{|a||b|}\{b,\{a,c\}\}$,

for any homogeneous elements $a,b,c\in A$, then $(A,\{\cdot,\cdot\})$ is called a graded Lie algebra.

Definition 2.2**.**

[12] Let $(A,\cdot)$ be a graded $\Bbbk$-algebra. If there is a $\Bbbk$-linear map

$\{\cdot,\cdot\}:A\otimes A\rightarrow A$

of degree 0 such that

1. (i)

$(A,\{\cdot,\cdot\})$ is a graded Lie algebra; 2. (ii)

(graded commutativity): $a\cdot b=(-1)^{|a||b|}b\cdot a$; 3. (iii)

(biderivation property): $\{a,b\cdot c\}=\{a,b\}\cdot c+(-1)^{|a||b|}b\cdot\{a,c\}$,

for any homogeneous elements $a,b,c\in A$, then $A$ is called a graded Poisson algebra. If in addition, there is a $\Bbbk$-linear homogeneous map $d:A\rightarrow A$ of degree 1 such that $d^{2}=0$ and

1. (iv)

(graded Leibniz rule for bracket): $d(\{a,b\})=\{d(a),b\}+(-1)^{|a|}\{a,d(b)\}$; 2. (v)

(graded Leibniz rule for product): $d(a\cdot b)=d(a)\cdot b+(-1)^{|a|}a\cdot d(b)$,

for any homogeneous elements $a,b\in A$, then $A$ is called a DG Poisson algebra, which is usually denoted by $(A,\cdot,\{\cdot,\cdot\},d)$, or simply by $(A,\{\cdot,\cdot\},d)$ or $A$ if no confusions arise.

Remark 2.3*.*

For a DG algebra $B$, assume throughout the paper that $B_{P}$ is the DG Poisson algebra $B$ with the “standard graded Lie bracket”: $[a,b]=ab-(-1)^{|a||b|}ba$ for any homogeneous elements $a,b\in B$.

Note that an ideal $I$ of DG Poisson algebra $A$ is called a DG Poisson ideal if $d(I)\subseteq I$, $\{I,A\}\subseteq I$. Let $B$ be another DG Poisson algebra. A graded algebra map $\rho:A\rightarrow B$ is said to be a DG Poisson algebra map if $\rho\circ d_{A}=d_{B}\circ\rho$ and $\rho(\{a,b\}_{A})=\{\rho(a),\rho(b)\}_{B}$ for all homogeneous elements $a,b\in A$. For example, if $I$ is a DG Poisson ideal of $A$, then the canonical projection $\pi_{I}:A\rightarrow A/I$ is a DG Poisson algebra map.

We denote by DG(P)A the category of DG (Poisson) algebras whose morphism space consists of DG (Poisson) algebras map.

Now, we recall the definition of DG Poisson modules over DG Poisson algebras.

Definition 2.4**.**

[12] Let $A=(A,\cdot,\{\cdot,\cdot\},d)\in\textbf{DGPA}$, and $M$ be a left graded module over $A$. We call $M$ a left DG Poisson module over $A$ provided that

1. (i)

$(M,\partial)$ is a left DG module over the DG algebra $A$. That is to say, there is a $\Bbbk$-linear map $\partial:M\rightarrow M$ of degree 1 such that $\partial^{2}=0$ and

$\partial(am)=d(a)m+(-1)^{|a|}a\partial(m)$

for all homogeneous elements $a\in A$ and $m\in M$. Here $\partial$ is also called the differential of $M$. 2. (ii)

$M$ is a left graded Poisson module over the graded Poisson algebra $A$. That is to say, there is a $\Bbbk$-linear map $\{\cdot,\cdot\}_{M}:A\otimes M\rightarrow M$ of degree 0 such that

1. (ia)

$\{a,bm\}_{M}=\{a,b\}_{A}m+(-1)^{|a||b|}b\{a,m\}_{M}$; 2. (ib)

$\{ab,m\}_{M}=a\{b,m\}_{M}+(-1)^{|a||b|}b\{a,m\}_{M}$, and 3. (ic)

$\{a,\{b,m\}_{M}\}_{M}=\{\{a,b\}_{A},m\}_{M}+(-1)^{|a||b|}\{b,\{a,m\}_{M}\}_{M}$,

for all homogeneous elements $a,b\in A$ and $m\in M$. 3. (iii)

the $\Bbbk$-linear map $\partial$ is compatible with the bracket $\{\cdot,\cdot\}_{M}$. That is, we have

$\partial(\{a,m\}_{M})=\{d(a),M\}_{M}+(-1)^{|a|}\{a,\partial(m)\}_{M}$,

for all homogeneous elements $a\in A$ and $m\in M$.

Similarly, a left DG Poisson module $M$ over a DG Poisson algebra $A$ is usually denoted by $(M,\{\cdot,\cdot\}_{M},\partial)$, or simply by $M$ if there are no confusions.

Definition 2.5**.**

Let $A=(A,\cdot,\{\cdot,\cdot\},d)\in\textbf{DGPA}$, and $(M,\{\cdot,\cdot\}_{M},\partial)$ be a left DG Poisson module over $A$. A left graded submodule $N\leq M$ is called a left DG Poisson submodule provided that $\partial(N)\subseteq N$ and $\{A,N\}_{M}\subseteq N$, which is usually denoted by $N\leq_{p}M$.

2.2. Universal enveloping algebras of DG Poisson algebras

In this subsection, we recall the definition and some properties of the universal enveloping algebra of a DG Poisson algebra $A=(A,\cdot,\{\cdot,\cdot\},d)$.

Definition 2.6**.**

[12] Let $A=(A,\cdot,\{\cdot,\cdot\},d)\in\textbf{DGPA}$ and $(A^{ue},\partial)\in\textbf{DGA}$. We call $(A^{ue},\partial)$ is a universal enveloping algebra of $A$ if there exist a DG algebra map $\alpha:(A,d)\rightarrow(A^{ue},\partial)$ and a DG Lie algebra map $\beta:(A,\{\cdot,\cdot\},d)\rightarrow(A_{P}^{ue},[\cdot,\cdot],\partial)$ satisfying

[TABLE]

for any homogeneous elements $a,b\in A$, such that for any $(D,\delta)\in\textbf{DGA}$ with a DG algebra map $f:(A,d)\rightarrow(D,\delta)$ and a DG Lie algebra map $g:(A,\{\cdot,\cdot\},d)\rightarrow(D_{P},[\cdot,\cdot],\delta)$ satisfying

[TABLE]

for all $a,b\in A$, then there exists a unique DG algebra map $\phi:(A^{ue},\partial)\rightarrow(D,\delta)$, making the diagram

[TABLE]

“bi-commute”, i.e., $\phi\alpha=f$ and $\phi\beta=g$.

For an $A\in\textbf{DGPA}$, we denote by $(A^{ue},\alpha,\beta)$ the universal enveloping algebra of $A$. Note that universal enveloping algebra $A^{ue}$ of a DG Poisson algebra $A$ was defined in order that a $\Bbbk$-vector space $M$ is a DG Poisson $A$-module if and only if $M$ is a DG $A^{ue}$-module and that universal enveloping algebra of $A$ is unique up to isomorphic (see [12]).

Proposition 2.7**.**

Let $(A^{ue},\alpha,\beta)$ be the universal enveloping algebra of a finitely generated DG Poisson algebra $A$. Then $\alpha$ is injective.

Proof.

For every $a\in A$, define $\gamma(a)$, $\delta(a)\in End_{\Bbbk}(A)$ by

$\gamma(a)(b)=ab,\delta(a)(b)=\{a,b\}$,

for all $b\in A$. As we all know, $End_{\Bbbk}(A)$ is a graded endomorphism ring if $A$ is a finitely generated $\Bbbk$-module, because

$End_{\Bbbk}(A)=H(A,A)=\oplus_{n\in\mathbb{Z}}H(A,A)_{n}$

and

$H(A,A)_{n}=\{\psi\in Hom_{\Bbbk}(A,A)|~{}\psi(A_{i})\subseteq A_{i+n}\}$.

Let

$d^{\prime}\in End(End_{\Bbbk}(A))$, $d^{\prime}(f)(a)=d(f(a))-(-1)^{|f|}f(d(a))$,

for any elements $f\in End_{\Bbbk}(A)$ and $a\in A$, then $End_{\Bbbk}(A)$ is a DG algebra. It is easy to proof that $\gamma$ is a graded algebra map. Moreover, we have

$(\gamma d)(a)(b)=\gamma(d(a))(b)=d(a)b$

and

$(d^{\prime}\gamma)(a)(b)=d^{\prime}(\gamma(a))(b)=d(\gamma(a)(b))-(-1)^{|\gamma(a)|}\gamma(a)(d(b))=d(ab)-(-1)^{|a|}ad(b),$

for all $a,b\in A$, then $\gamma$ is a DG algebra map by using graded Leibniz rule for product. In fact, for any homogeneous elements $a,b,c\in A$, we have

[TABLE]

and

$(d^{\prime}\delta)(a)(b)=d^{\prime}(\delta(a))(b)=d(\delta(a)(b))-(-1)^{|a|}\delta(a)(d(b))=d(\{a,b\})-(-1)^{|a|}\{a,d(b)\}.$

Note that $A$ is a DG Poisson algebra, by the graded Jacobi identity and graded Leibniz rule for bracket, we have

$\{a,\{b,c\}\}-(-1)^{|a||b|}\{b,\{a,c\}\}=\{\{a,b\},c\}$

and

$d(\{a,b\})-(-1)^{|a|}\{a,d(b)\}=\{d(a),b\}$,

which imply that $\delta$ is a DG Lie algebra map. On the other hand, we see that

[TABLE]

and

[TABLE]

for all $a,b,c\in A$. Hence there exists a DG algebra map $\phi$ from $A^{ue}$ into $End_{\Bbbk}(A)$ such that $\phi\alpha=\gamma$ and $\phi\beta=\delta$. If $a\in ker\alpha$ then $0=\phi\alpha(a)=\gamma(a)$, and so $0=\gamma(a)(1)=a$. It completes the proof. ∎

Henceforce, we identify the DG algebra homomorphic image of a finitely generated DG Poisson algebra $A$ under $\alpha$ to $A$ and denote $\alpha(a)$ by $a$ for all $a\in A$.

2.3. Construction of $A^{e}$

For any DG Poisson algebra $A$ given by generators and relations, we give a “formula” for computing the universal enveloping algebra $A^{e}$ of $A$.

2.3.1. “Anti-differential”

Let $V$ be a graded $\Bbbk$-vector space with a homogeneous $\Bbbk$-basis $\{x_{\alpha}:\alpha\in\Lambda\}$ and

[TABLE]

be a DG Poisson algebra with differential $d$ and Poisson bracket $\{\cdot,\cdot\}$. Here $T(V)$ is the tensor algebra of $V$ over $\Bbbk$ and $|x|$ denotes the degree of the homogeneous element $x$ of $R$.

Now for any $\alpha\in\Lambda$, we define a $\Bbbk$-linear map

$\psi_{\alpha}:R\rightarrow R$

such that

$\psi_{\alpha}(x_{\beta})=\delta_{\alpha\beta}$  and  $\psi_{\alpha}(ab)=a\psi_{\alpha}(b)+(-1)^{|a||b|}b\psi_{\alpha}(a)$,

for all homogeneous elements $a,b\in R$ and $\beta\in\Lambda$. The $\Bbbk$-linear map are usually called anti-differential of $R$.

Remark 2.8*.*

We have the following two observations:

• •

It is easy to see that $|\psi_{\alpha}|=-|x_{\alpha}|$ for any $\alpha\in\Lambda$, and that the $\Bbbk$-linear map $\psi_{\alpha}$ for any $\alpha\in\Lambda$ is well-defined on $R$ since

[TABLE]

• •

If we set $\Lambda_{a}:=\{\alpha\in\Lambda|~{}\psi_{\alpha}(a)\neq 0\}$ for any homogeneous element $a\in R$, then $\Lambda_{a}$ is a finite set since $R$ obviously has a PBW-basis.

Now we give some basic properties of these “anti-differentials”.

Lemma 2.9**.**

For any homogeneous element $a\in R$ and for all $\alpha,\beta\in\Lambda$, we have

[TABLE]

Proof.

Observe that formula (2.1) is true on any $x_{\gamma}$ for all $\gamma\in\Lambda$ since

[TABLE]

In order to prove that formula (2.1) is true for any homogeneous element $a\in R$. It suffices to prove the formula (2.1) is true for $ab$ provided that it is true for any homogeneous elements $a,b\in R$. Thus, assume that we have the following two equations:

[TABLE]

for any homogeneous elements $a,b\in R$. We have

[TABLE]

as required. ∎

Lemma 2.10**.**

We have

[TABLE]

for any homogeneous elements $a,b\in R$.

Proof.

It is clear that formula (2.2) is true on the generators $\{x_{\alpha}\}_{\alpha\in\Lambda}$ of R. Thus in order to complete the proof, we only need to show

$\{aa^{\prime},bb^{\prime}\}=\sum_{\alpha\in\Lambda}\psi_{\alpha}(aa^{\prime})\{x_{\alpha},bb^{\prime}\}$

provided that

[TABLE]

and

[TABLE]

for any homogeneous elements $a,a^{\prime},b,b^{\prime}\in R$.

Indeed, we have

[TABLE]

as required. ∎

2.3.2.

Now we introduce another set of indeterminates $\{y_{\alpha}|~{}\alpha\in\Lambda\}$ such that $|x_{\alpha}|=|y_{\alpha}|$ for all $\alpha\in\Lambda$. Define a graded $R$-free algebra $F(R)$ by

$F(R):=R\langle y_{\alpha}|~{}\alpha\in\Lambda\rangle$

and a $\Bbbk$-linear map

$\psi:R\rightarrow F(R)$

by

[TABLE]

for any $f\in R$. Note that such $\psi$ is well-defined since for any $f\in R$, there are only finite many $\alpha\in\Lambda$ such that $\psi_{\alpha}(f)$ is not zero. Also, let $j:R\rightarrow F(R)$ be the canonical inclusion map.

Now suppose that $(R,d,\{\cdot,\cdot\})$ is a DG Poisson algebra and $I$ is a DG Poisson ideal of $R$. Put $A:=R/I$, then $A$ has a natural DG Poisson algebra structure induced from $R$. Let $\pi^{\prime}$ denote the canonical projection $\pi^{\prime}:R\rightarrow A$, then $\pi^{\prime}$ is a DG Poisson algebra map.

As for $F(R)$, let $J$ be the graded ideal of $F(R)$ generated by

1. (1)

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}I,~{}\psi(I)$, 2. (2)

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}y_{\alpha}y_{\beta}-(-1)^{|x_{\alpha}||x_{\beta}|}y_{\beta}y_{\alpha}-\psi(\{x_{\alpha},x_{\beta}\})$, 3. (3)

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}y_{\alpha}x_{\beta}-(-1)^{|x_{\alpha}||x_{\beta}|}x_{\beta}y_{\alpha}-\{x_{\alpha},x_{\beta}\}$,

where $|x_{\alpha}|=|y_{\alpha}|$ for any $\alpha\in\Lambda$, and

[TABLE]

for any $f\in R$. This induces a canonical projection $\pi:F(R)\rightarrow\mathcal{A}:=F(R)/J$. For the sake of simplicity, we omit the canonical projections $\pi$ and $\pi^{\prime}$ here, and also denote by $d$ the differential of $A$.

Lemma 2.11**.**

$\mathcal{A}$* is a DG algebra.*

Proof.

Note that $|\psi|=0$, it is easy to see that $\mathcal{A}$ is a graded algebra from its construction. Define a $\Bbbk$-linear map $\partial:\mathcal{A}\rightarrow\mathcal{A}$ by

[TABLE]

for all $\alpha\in\Lambda$, and the graded Leibniz rule, i.e.,

$\partial(ab)=\partial(a)b+(-1)^{|a|}a\partial(b)$

for all homogeneous elements $a,b\in\mathcal{A}$. It is obvious that $|\partial|=1$. In order to prove $\mathcal{A}$ is a DG algebra, it suffices to prove that $\partial^{2}=0$.

Indeed, it is easy to see that $\partial(f)=d(f)$ for all $f\in A$ by using induction. We have

$\partial^{2}(x_{\alpha})=\partial(d(x_{\alpha}))=d^{2}(x_{\alpha})=0$,

for all $\alpha\in\Lambda$. Moreover, we have

[TABLE]

by Lemma 2.9. ∎

Note that there are two $\Bbbk$-linear maps $m,h:A\rightarrow\mathcal{A}$ given by

$m(f)=f,~{}h(f)=\psi(f)$,

for all $f\in A$. We can obtain our main result in this subsection.

Theorem 2.12**.**

$(\mathcal{A},m,h)$* is the universal enveloping algebra of a DG Poisson algebra $A$, where $A=R/I$ is defined as above.*

Proof.

By Lemma 2.11, $(\mathcal{A},\partial)$ is a DG algebra. From the construction of $\mathcal{A}$, there are two $\Bbbk$-linear maps $m:A\rightarrow\mathcal{A}$ sending each element $f\in A$ to $f$ and $h:A\rightarrow\mathcal{A}$ sending each element $f\in A$ to $\psi(f)$. In fact, $m$ is the DG algebra map and $h$ is DG $\Bbbk$-linear map since for all homogeneous $f,g\in A$, we have

[TABLE]

and

[TABLE]

by Lemma 2.9.

Notice that for any homogeneous elements $f,g\in A$, we have

[TABLE]

For a monomial $f\in A$, we proceed the proof using induction on the length of $f$. We already know that

[TABLE]

For any monomials $f,g,a\in A$, assume that

$m(\{f,a\})=h(f)m(a)-(-1)^{|f||a|}m(a)h(f)$

and

$m(\{g,a\})=h(g)m(a)-(-1)^{|g||a|}m(a)h(g)$.

Now we have

[TABLE]

Applying the induction, we have

$m(\{f,g\})=h(f)m(g)-(-1)^{|f||g|}m(g)h(f)$,

for all $f,g\in A$. Similarly, we have $h(\{x_{\alpha},x_{\beta}\})=[h(x_{\alpha}),h(x_{\beta})]$. For any monomials $f,g,a\in A$, assume that

$h(\{a,f\})=[h(a),h(f)]$,  $h(\{a,g\})=[h(a),h(g)]$.

Then we have

[TABLE]

Applying the induction, we have

$h(\{f,g\})=[h(f),h(g)]$,

for all $f,g\in A$.

Let $(D,\partial^{\prime})$ be a DG algebra. Let $\gamma:(A,d)\rightarrow(D,\partial^{\prime})$ be a DG algebra map and let $\delta:(A,\{\cdot,\cdot\},d)\rightarrow(D_{P},[\cdot,\cdot],\partial^{\prime})$ be a DG Lie algebra map such that

[TABLE]

for any homogeneous elements $a,b\in A$. Since $F(R)$ is a graded free algebra, there exists a graded algebra map $\phi^{\prime}:F(R)\rightarrow D$ defined by

$\phi^{\prime}(x_{\alpha})=\gamma(x_{\alpha})$, $~{}~{}\phi^{\prime}(y_{\alpha})=\delta(x_{\alpha})$,

for all $\alpha\in\Lambda$.

It is shown to be $\phi^{\prime}\psi=\delta$ using induction on the length of monomials in $R$. Therefore we have

[TABLE]

Thus there exists a graded algebra map $\phi$ from $\mathcal{A}$ into $D$ such that $\phi m=\gamma,\phi h=\delta$.

Further, A graded algebra map $\phi:\mathcal{A}\rightarrow D$ is unique such that $\phi m=\gamma$, $\phi h=\delta$ by its construction and it is also a DG algebra map since for any $\alpha\in\Lambda$, we have

$\phi\partial(x_{\alpha})=\phi d(x_{\alpha})=\phi m(d(x_{\alpha}))=\gamma d(x_{\alpha})=\partial^{\prime}\gamma(x_{\alpha})=\partial^{\prime}\phi(x_{\alpha})$

and

$\phi\partial(y_{\alpha})=\phi\psi(d(x_{\alpha}))=\phi h(d(x_{\alpha}))=\delta d(x_{\alpha})=\partial^{\prime}\delta(x_{\alpha})=\partial^{\prime}\phi(y_{\alpha})$.

Therefore, $(\mathcal{A},m,h)$ is the universal enveloping algebra of a DG Poisson algebra $A$, as required. ∎

Remark 2.13*.*

From now on, we let $A^{e}:=\mathcal{A}$ denote the universal enveloping algebra of a DG Poisson algebra $A$ given by generators and relations.

3. Poincaré-Birkhoff-Witt theorem for universal

enveloping algebras

In this section, we develop an algorithm to find a $\Bbbk$-linear basis of the universal enveloping algebra $A^{e}$.

3.1. Gröbner-Shirshov basis theory

In this subsection, we recall the Gröbner-Shirshov basis theory developed in [8] and [9].

Let $X=\{x_{1},x_{2},\cdots\}$ be a set of alphabets indexed by positive integers. Define a linear ordering on $X$ $\prec$ by setting $x_{i}\prec x_{j}$ if and only if $i<j$. Let $X^{*}$ be the free monoid of associative monomials on $X$. We denote the empty monomial by 1 and the length of a monomial $u$ by $l(u)$ with $l(1)=0$. We consider two linear ordering $<$ and $\ll$ on $X^{*}$ defined as follows [9]:

1. (i)

$1<u$ for any nonempty monomial $u$; and inductively, $u<v$ whenever $u=x_{i}u^{\prime},v=x_{j}v^{\prime}$ and $x_{i}\prec x_{j}$ or $x_{i}=x_{j}$ and $u^{\prime}<v^{\prime}$ with $x_{i},x_{j}\in X$, 2. (ii)

$u\ll v$ if $l(u)<l(v)$ or $l(u)=l(v)$ and $u<v$.

The ordering $<$ (resp. $\ll$) is called the lexicographic ordering (resp. degree-lexicographic ordering). It is easy to see $\ll$ is a monomial order on $X^{*}$, that is, $x\ll y$ implies $axb\ll ayb$ for all $a,b\in X^{*}$.

Let $T_{X}$ be the DG free $k$-algebra generated by $X$, let $I$ be a DG ideal of $T_{X}$ and let $T_{0}=T_{X}/I$, then $T_{0}$ is a DG algebra. The image of $p\in T_{X}$ in $T_{0}$ under the canonical DG quotient map will also be denoted by $p$. Given a nonzero element $p\in T_{0}$, we denote by $\bar{p}$ the maximal monomial appearing in $p$ under the ordering $\ll$. Thus $p=\alpha\bar{p}+\Sigma\beta_{i}w_{i}$ with $\alpha,\beta_{i}\in\Bbbk$, $w_{i}\in X^{*}$, $\alpha\neq 0$ and $w_{i}\ll\bar{p}$. The $\alpha$ is called the leading coefficient of $p$ and if $\alpha=1$, then $p$ is said to be monic. Recall that the composition of $p$ and $q$ as follows.

Definition 3.1**.**

[8] Let $p$ and $q$ be monic elements of $T_{0}$.

1. (a)

If there exist $a$ and $b$ in $X^{*}$ such that $\bar{p}a=b\bar{q}=w$ with $l(\bar{p})>l(b)$, then the composition of intersection is defined to be $(p,q)_{w}=pa-bq$. 2. (b)

If there exist $a$ and $b$ in $X^{*}$ such that $a\neq 1$, $a\bar{p}b=\bar{q}=w$, then the composition of inclusion is defined to be $(p,q)_{w}=apb-q$.

Let $S$ be a subset of monic elements of $T_{0}$ and let $J$ be the DG ideal of $T_{0}$ generated by $S$. Then we say the DG algebra $T_{0}/J$ is defined by $S$. The image of $p\in T_{0}$ in $T_{0}/J$ under the canonical quotient maps will also be denoted by $p$ as long as there is no peril of confusion. Next, let $p,q\in T_{0}$ and $w\in X^{*}$. We define a congruence relation on $T_{0}$ as follows: $p\equiv q$ mod $(J;w)$ if and only if $p-q=\Sigma\alpha_{i}a_{i}s_{i}b_{i}$, where $\alpha_{i}\in\Bbbk$, $a_{i},b_{i}\in X^{*}$, $s_{i}\in S$ and $a_{i}\bar{s}_{i}b_{i}\ll w$. A set $S$ of monic elements of $T_{0}$ is said to be closed under the composition if for any $p,q\in S$ and $w\in X^{*}$ such that $(p,q)_{w}$ is defined, we have $(p,q)_{w}\equiv 0$ mod $(J;w)$.

Now we introduce another definition of monomials and then prove the following generalization of Shirshov’s Composition Lemma to the representations of DG associative algebras.

Definition 3.2**.**

[8] A monomial $u\in X^{*}$ is said to be $S$-standard in $T_{0}$ if $u\neq a\bar{s}b$ for any $s\in S$ and $a,b\in X^{*}$. Otherwise, the monomial $u$ is said to be $S$-reducible in $T_{0}$.

Theorem 3.3**.**

Let $S$ be a subset of monic elements in $T_{0}$ and let $T_{0}/J$ be the DG algebra is defined by $S$. If $S$ is closed under composition in $T_{0}$ and the image of $p\in T_{0}$ is zero in $T_{0}/J$, then the monomial $\bar{p}$ is $S$-reducible in $T_{0}$.

Proof.

Since the image of $p\in T_{0}$ is zero in $T_{0}/J$, we have $p=\Sigma\alpha_{i}a_{i}s_{i}b_{i}$, where $\alpha_{i}\in\Bbbk$, $a_{i},b_{i}\in X^{*}$ and $s_{i}\in S$. Choose the maximal monomial $w$ in the degree-lexicographic ordering $\ll$ among the monomials $\{a_{i}\overline{s_{i}}b_{i}\}$ in the expression of $p$. If $\bar{p}=w$, then we are done. Suppose this is not the case, then $\bar{p}\ll w$ and without loss of generality, we may assume that the following case hold: $w=a_{1}\overline{s_{1}}b_{1}=a_{2}\overline{s_{2}}b_{2}$.

If $w=a_{1}\overline{s_{1}}b_{1}=a_{2}\overline{s_{2}}b_{2}$, then we should show that $a_{1}s_{1}b_{1}\equiv a_{2}s_{2}b_{2}$ mod $(J;w)$. There are three possibilities:

(i) If the monomials $\overline{s_{1}}$ and $\overline{s_{2}}$ have empty intersection in $w$, then we may assume that $a_{1}s_{1}b_{1}=as_{1}b\overline{s_{2}}c$ and $a_{2}s_{2}b_{2}=a\overline{s_{1}}bs_{2}c$, where $a,b,c\in X^{*}$. Thus

$a_{2}s_{2}b_{2}-a_{1}s_{1}b_{1}=-a(s_{1}-\overline{s_{1}})bs_{2}c+as_{1}b(s_{2}-\overline{s_{2}})c$,

which implies $a_{1}s_{1}b_{1}\equiv a_{2}s_{2}b_{2}$ mod $(J;w)$.

(ii) If $\overline{s_{1}}=u_{1}u_{2}$ and $\overline{s_{2}}=u_{2}u_{3}$ for some $u_{2}\neq 1$, then $a_{2}=a_{1}u_{1}$, $b_{1}=u_{3}b_{2}$ and

[TABLE]

Since $\overline{(s_{1},s_{2})_{u_{1}u_{2}u_{3}}}\ll u_{1}u_{2}u_{3}$ and $S$ is closed under composition in $T_{0}$, we obtain $a_{1}s_{1}b_{1}\equiv a_{2}s_{2}b_{2}$ mod $(J;w)$.

(iii) If $\overline{s_{1}}=u_{1}\overline{s_{2}}u_{2}$, then $a_{2}=a_{1}u_{1}$, $b_{2}=u_{2}b_{1}$ and

[TABLE]

Since $\overline{(s_{2},s_{1})_{\overline{s_{1}}}}\ll\overline{s_{1}}$ and $S$ is closed under composition in $T_{0}$, we get $a_{1}s_{1}b_{1}\equiv a_{2}s_{2}b_{2}$ mod $(J;w)$.

Therefore, $p$ can be written as $p=\Sigma\alpha^{\prime}_{i}a^{\prime}_{i}s^{\prime}_{i}b^{\prime}_{i}$, where $a^{\prime}_{i}\overline{s^{\prime}_{i}}b^{\prime}_{i}\ll w$ for all $i$. Choose the maximal monomial $w_{1}$ in the ordering $\ll$ among $\{a^{\prime}_{i}\overline{s^{\prime}_{i}}b^{\prime}_{i}\}$. If $\bar{p}=w_{1}$, then we are done. If this is not the case, repeat the above process. Since $X$ is indexed by the set of positive integers, this process must terminate in finite steps, which completes the proof. ∎

As a corollary, we obtain:

Proposition 3.4**.**

Let $\mathcal{A}\subseteq X^{*}$ form a $\Bbbk$-linear basis of DG algebra $T_{0}=T_{X}/I$, let $S$ be a subset of monic elements of $T_{0}$ and let $T_{0}/J$ be the DG algebra is defined by $S$. Then the following are equivalent:

1. (i)

$S$ is closed under composition in $T_{0}$. 2. (ii)

the subset of $\mathcal{A}$ consisting of $S$-standard monomials in $T_{0}$ forms a $\Bbbk$-linear basis of the DG algebra $T_{0}/J$.

Proof.

Copy the proof of proposition 1.9 in [8]. ∎

3.2. PBW-basis for the universal enveloping algebra

Let $A$ be a DG Poisson homomorphic image of a DG Poisson algebra $R$ with an arbitrary differential $d$ and Poisson bracket $\{\cdot,\cdot\}$, where

[TABLE]

and $\Lambda$ is a finite index set. For the convenience of the narrative, assume that $\Lambda=\{1,2,\cdots,n\}$. In this subsection, we will find a $\Bbbk$-linear basis of $R^{e}$ and develop an algorithm to find a $\Bbbk$-linear basis of the universal enveloping algebra $A^{e}$. From now on, assume that,

$\bullet~{}~{}(R,\{\cdot,\cdot\},d)$ is a DG Poisson algebra, where $R$ is defined as above and $\Lambda$ is a finite index set. Assume that $\Lambda=\{1,2,\cdots,n\}$.

$\bullet~{}~{}A=R/I$, where $I$ is a DG Poisson ideal of $R$.

$\bullet~{}~{}F(R)=R\langle y_{i}|~{}i\in\Lambda\rangle$.

$\bullet~{}~{}\psi:R\rightarrow F(R),~{}~{}\psi(f)=\sum_{r=1}^{n}\psi_{r}(f)y_{r}$.

$\bullet~{}~{}J$ is the graded ideal of $F(R)$ generated

$I$, $~{}\psi(I)$, $~{}y_{i}x_{j}-(-1)^{|x_{i}||x_{j}|}x_{j}y_{i}-\{x_{i},x_{j}\}$, $~{}y_{i}y_{j}-(-1)^{|x_{i}||x_{j}|}y_{j}y_{i}-\psi(\{x_{i},x_{j}\})$

for all $i,j=1,2,\cdots,n$.

$\bullet~{}~{}A^{e}=F(R)/J$ is the universal enveloping algebra of $A$ and let $\pi:F(R)\rightarrow A^{e}$ denote the canonical projection.

Now define a linear ordering on the index set $\Lambda\times\Lambda$ by

$(l,m)<(r,s)\Longleftrightarrow m<s~{}~{}or~{}~{}m=s~{}~{}and~{}~{}l<r$

and a grading on $F(R)$ by

$deg(x_{i})=(1,0)$,   $deg(y_{i})=(0,1)$

for all $i\in\Lambda$, hence the grading of a monomial $u=u_{1}\cdots u_{l}\in F(R)$, where $u_{j}=x_{i}$ or $u_{j}=y_{i}$, is defined by

$deg(u)=deg(u_{1})+\cdots+deg(u_{l})\in\Lambda\times\Lambda$.

We give an ordering $<$ on the set of generators of $F(R)$ by

$x_{1}<x_{2}<\cdots<x_{n}<y_{1}<y_{2}<\cdots<y_{n}$.

Therefore there is a well-ordering $\prec$ on the set of all monomials in $F(R)$. That is, for monomials $u=u_{1}\cdots u_{l}$ and $v=v_{1}\cdots v_{m}$, we denote $u\prec v$ if one of the following conditions holds:

1. (i)

$deg(u)<deg(v)$. 2. (ii)

$deg(u)=deg(v),u_{1}=v_{1},\cdots,u_{r}=v_{r}~{}and~{}u_{r+1}<v_{r+1}$ for some $r\in\Lambda$.

Note that the ordering $\prec$ is a monomial order.

Example 3.5**.**

For $u=x_{2}x_{3}y_{3}^{2}y_{1}$, $v=x_{3}y_{2}x_{1}^{2}$ and $w=x_{2}y_{3}x_{3}y_{2}y_{1}$, we have $deg(u)=deg(w)=(2,3)$ and $deg(v)=(3,1)$, hence $v\prec u\prec w$.

Lemma 3.6**.**

For a monomial $f=x_{i_{1}}\cdots x_{i_{r}}\in F(R)$, $y_{i}f\equiv(-1)^{|x_{i}||f|}fy_{i}+\{x_{i},f\}~{}mod(J;y_{i}f)$.

Proof.

We proceed the proof using induction on $l(f)=r$. If $r=$0 or 1, then it is trivial. Assume that the statement is true for monomials with length less than $r$. Then we have that

[TABLE]

by the induction hypothesis and biderivation property. ∎

Lemma 3.7**.**

For a monomial $f=x_{i_{1}}\cdots x_{i_{r}}\in F(R)$,

$\psi(f)x_{i}\equiv(-1)^{|x_{i}||f|}x_{i}\psi(f)+\{f,x_{i}\}~{}mod(J;\overline{\psi(f)}x_{i})$.

Proof.

By Lemma 2.10 and the definition of $\psi$, we have $\{f,x_{i}\}=\sum_{s}\psi_{s}(f)\{x_{s},x_{i}\}$ and $\psi(f)=\sum_{s}\psi_{s}(f)y_{s}$. Thus

[TABLE]

by Lemma 3.6. ∎

Lemma 3.8**.**

For a monomial $f=x_{i_{1}}\cdots x_{i_{r}}\in F(R)$,

[TABLE]

where $f_{ij}=\{x_{i},x_{j}\}$.

Proof.

Note that $\psi(f)=\sum_{s}\psi_{s}(f)y_{s}$ and $\{x_{i},f\}=\sum_{s}(-1)^{|x_{i}||\psi_{s}(f)|}\psi_{s}(f)f_{is}$, we have

[TABLE]

by Lemma 3.6. ∎

Lemma 3.9**.**

Retain the above notions, we have that

[TABLE]

where $f_{ij}=\{x_{i},x_{j}\}$.

Proof.

Since $\{f,x_{i}\}=\sum_{s}\psi_{s}(f)\{x_{s},x_{i}\}$, we have

[TABLE]

Hence

[TABLE]

∎

Lemma 3.10**.**

For a monomial $f=x_{i_{1}}\cdots x_{i_{r}}\in F(R)$,

[TABLE]

where $f_{ij}=\{x_{i},x_{j}\}$.

Proof.

We proceed the proof using induction on $l(f)=r$. If $r=1$ or $2$, then it is trivial. Assume that the statement is true for monomials with length less than $r$. Set $f_{ij}=\{x_{i},x_{j}\}$, then we have that

[TABLE]

and

[TABLE]

Thus

[TABLE]

by the induction hypothesis. Therefore, we finish the proof. ∎

Lemma 3.11**.**

In $(A^{e},m,h)$, we have

1. (a)

$y_{i}f=(-1)^{|x_{i}||f|}fy_{i}+\{x_{i},f\}$* for $f\in A$,* 2. (b)

$h(f)x_{i}=(-1)^{|x_{i}||f|}x_{i}h(f)+\{f,x_{i}\}$* for $f\in A$,* 3. (c)

$y_{i}h(f)=(-1)^{|x_{i}||f|}h(f)y_{i}+h(\{x_{i},f\})$* for $f\in A$.*

Proof.

From Lemmas 3.6 and 3.7, it is easy to see (a) and (b). Since $\psi(f)=\sum_{t}\psi_{t}(f)y_{t}$ and $\{x_{i},f\}=\sum_{s}(-1)^{|x_{i}||\psi_{s}(f)|}\psi_{s}(f)\{x_{i},x_{s}\}$ for any $f\in R$, we have that

[TABLE]

by Lemma 3.10, where $f_{ij}=\{x_{i},x_{j}\}$. Hence (c) follows from Lemma 3.8. ∎

Theorem 3.12**.**

Let $R=T(V)/\langle x_{\alpha}\otimes x_{\beta}-(-1)^{|x_{\alpha}||x_{\beta}|}x_{\beta}\otimes x_{\alpha}|~{}\forall\alpha,~{}\beta\in\Lambda\rangle$ be a DG Poisson algebra with an arbitrary differential $d$ and Poisson structure $\{\cdot,\cdot\}$. Then the universal enveloping algebra $R^{e}$ has a $\Bbbk$-linear basis

$\mathfrak{B}=\{x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n}^{j_{n}}|~{}i_{r},j_{r}=0,1,2\cdots\}$.

Proof.

Since

[TABLE]

and $\psi(x_{i}x_{j}-(-1)^{|x_{i}||x_{j}|}x_{j}x_{i})=0$, the universal enveloping algebra $R^{e}$ is $R^{e}=F(R)/J^{\prime}$, where $J^{\prime}$ is the DG ideal generated by

1. (1)

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}x_{ij}:x_{i}x_{j}-(-1)^{|x_{i}||x_{j}|}x_{j}x_{i}$, 2. (2)

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}y_{ij}:y_{i}y_{j}-(-1)^{|x_{i}||x_{j}|}y_{j}y_{i}-\psi(\{x_{i},x_{j}\})$, 3. (3)

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}z_{ij}:y_{i}x_{j}-(-1)^{|x_{i}||x_{j}|}x_{j}y_{i}-\{x_{i},x_{j}\}$,

for all $i$, $j$. By Proposition 3.4, it is enough to show that the generators of $J^{\prime}$ is closed under composition in $F(R)$. There are only four possible compositions among the generators of $J^{\prime}$:

• •

$(x_{ij},x_{jk})_{x_{i}x_{j}x_{k}}(i>j>k)$

• •

$(y_{ij},y_{jk})_{y_{i}y_{j}y_{k}}(i>j>k)$

• •

$(y_{ij},z_{jk})_{y_{i}y_{j}x_{k}}(i>j)$

• •

$(z_{ij},x_{jk})_{y_{i}x_{j}x_{k}}(j>k)$

Case1. $(x_{ij},x_{jk})_{x_{i}x_{j}x_{k}}(i>j>k)$

[TABLE]

Case2. $(y_{ij},y_{jk})_{y_{i}y_{j}y_{k}}(i>j>k)$

Set $\{x_{i},x_{j}\}=f_{ij}$. Since $\psi(f)=\sum_{s}\psi_{s}(f)y_{s}$ and $\{x_{i},f\}=\sum_{s}(-1)^{|x_{i}||\psi_{s}(f)|}\psi_{s}(f)f_{is}$, we have

[TABLE]

by Lemma 3.8. It is easy to see that

[TABLE]

by Lemma 3.10.

Hence

[TABLE]

by Lemma 3.9.

Case3. $(y_{ij},z_{jk})_{y_{i}y_{j}x_{k}}(i>j)$

[TABLE]

by the graded Jacobi identity, Lemmas 3.6 and 3.7 .

Case4. $(z_{ij},x_{jk})_{y_{i}x_{j}x_{k}}(j>k)$

[TABLE]

∎

Proposition 3.13**.**

Let $R=T(V)/\langle x_{\alpha}\otimes x_{\beta}-(-1)^{|x_{\alpha}||x_{\beta}|}x_{\beta}\otimes x_{\alpha}|~{}\forall\alpha,~{}\beta\in\Lambda\rangle$ be a DG Poisson algebra with an arbitrary differential $d$ and Poisson structure $\{\cdot,\cdot\}$. Suppose that $A=R/I$ is a DG Poisson homomorphic image of $R$, where $I$ is a subset of monic elements of $R$. If $I\cup\psi(I)$ is closed under composition in $R^{e}$, then

$\{u\in\mathfrak{B}|u$ is $I\cup\psi(I)$-standard$\}$

forms a $\Bbbk$-linear basis of $A^{e}$, where $\mathfrak{B}$ is the one given in Theorem 3.12.

Proof.

It follows immediately from Proposition 3.4 and Theorem 3.12. ∎

Remark 3.14*.*

Although the degree of the Poisson bracket is zero in our definition of DG Poisson algebra, the result obtained in this paper are also true for DG Poisson algebras of degree $n$ with some expected signs, where $n\in\mathbb{Z}$ is the degree of the Poisson bracket.

Example 3.15**.**

Let

[TABLE]

where $|x_{1}=2,|x_{2}|=3$. Let $d:A\rightarrow A$ be a $\Bbbk$-linear map of degree 1 by

$d(x_{1})=x_{2},~{}~{}~{}d(x_{2})=0$.

Moreover, we can define the Poisson bracket by

$\{x_{1},x_{2}\}=-\{x_{2},x_{1}\}=x_{2}^{2},~{}~{}~{}\{x_{1},x_{1}\}=\{x_{2},x_{2}\}=0$.

Note that $0=x_{2}^{2}\in A$, it is easy to see that $A$ is a DG Poisson algebra of degree 1, where the degree of the Poisson bracket is 1. Here, we can suppose that

$R=\Bbbk<x_{1},x_{2}>/(x_{1}x_{2}-x_{2}x_{1},~{}x_{2}^{2})$

and $I=<x_{1}x_{2}>$ is a DG Poisson ideal of $R$. Given an ordering on the set of generators of $R^{e}$ by

$x_{1}<x_{2}<y_{1}<y_{2}$.

Since the set consisting of $x_{1}x_{2},\psi(x_{1}x_{2})=x_{1}y_{2}+x_{2}y_{1}$ is closed under composition in $R^{e}$, the universal enveloping algebra $A^{e}$ has a $\Bbbk$-linear basis

$\{x_{1}^{i_{1}}x_{2}^{i_{2}}y_{1}^{j_{1}}y_{2}^{j_{2}}|~{}i_{1}i_{2}=0,i_{2}j_{1}=0\}$

by Proposition 3.13.

Corollary 3.16**.**

Let $R=T(V)/\langle x_{\alpha}\otimes x_{\beta}-(-1)^{|x_{\alpha}||x_{\beta}|}x_{\beta}\otimes x_{\alpha}|~{}\forall\alpha,~{}\beta\in\Lambda\rangle$ be a DG Poisson algebra with an arbitrary differential $d$ and Poisson structure $\{\cdot,\cdot\}$. Then the universal enveloping algebra $R^{e}$ is a DG free left and right $R$-module with basis

$\mathfrak{C}=\{y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n}^{j_{n}}|~{}j_{r}=0,1,2\cdots\}$.

Proof.

By Theorem 3.12, $R^{e}$ is a DG free left $R$-module. To show that $R^{e}$ is a DG free right $R$-module, it is enough to prove that

$\mathfrak{B^{\prime}}=\{y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n}^{j_{n}}x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots\cdot x_{n}^{i_{n}}|~{}j_{r},i_{r}=0,1,2\cdots\}$

forms a $\Bbbk$-linear basis for $R^{e}$.

We show that $\mathfrak{B^{\prime}}$ is $\Bbbk$- linearly independent. Let

[TABLE]

where $\alpha_{i_{1}\cdots i_{n}},\beta_{j_{1}\cdots j_{n}l_{1}\cdots l_{n}},\gamma_{m_{1}\cdots m_{n}}$ and $\delta\in\Bbbk$. If there exist nonzero $\beta_{j_{1}\cdots j_{n}l_{1}\cdots l_{n}}$’s in the second term of formula (3.1), assume that

$z=y_{1}^{j_{1}}\cdots y_{n}^{j_{n}}x_{1}^{l_{1}}\cdots x_{n}^{l_{n}}$

is maximal among the monomials with nonzero coefficients in the second term of formula (3.1). Using (a) of Lemma 3.11, all monomials in the second term of formula (3.1) can be expressed as $\Bbbk$-linear combinations of monomials in $\mathfrak{B}$ of Theorem 3.12. Then the coefficient of $z$ is the coefficient of $x_{1}^{l_{1}}\cdots x_{n}^{l_{n}}y_{1}^{j_{1}}\cdots y_{n}^{j_{n}}$. Thus all $\beta_{j_{1}\cdots j_{n}l_{1}\cdots l_{n}}$’s in the second term of formula (3.1) are zero and hence all $\alpha_{i_{1}\cdots i_{n}},\gamma_{m_{1}\cdots m_{n}}$ and $\delta$ in formula (3.1) are also zero by Theorem 3.12. By Lemma 3.11, it is easy to see that all monomials in $\mathfrak{B}$ of Theorem 3.12 are spanned by $\mathfrak{B^{\prime}}$. Therefore $\mathfrak{B^{\prime}}$ forms a $\Bbbk$-linear basis of $R^{e}$, as required. ∎

4. Simple DG Poisson modules

In this section, for a given DG Poisson algebra $A=R/I$, where $R$ and $A$ are defined as in Section 3, we prove that a DG symplectic ideal $P$ of $A$ is the annihilator of a simple DG Poisson $A$-module.

Lemma 4.1**.**

Let $\mathcal{A}$ be the set of all DG ideals in a DG Poisson algebra $R$ and let $\mathcal{B}$ be the set of all left DG ideals of $R^{e}$. For $I\in\mathcal{A}$, let $\overline{I}$ be the left DG ideal of $R^{e}$ generated by $I$. Then the map $I\rightarrow\overline{I}$ is an injective map from $\mathcal{A}$ into $\mathcal{B}$ and preserves inclusion, that is, $\overline{I}\cap R=I$. In particular, $I$ is a DG Poisson ideal of $R$ if and only if $\overline{I}$ is an DG ideal of $R^{e}$.

Proof.

Let $I\in\mathcal{A}$. Since $R^{e}$ is a DG free right $R$-module with basis $\mathfrak{C}$ given in Corollary 3.16, every element of $\overline{I}=R^{e}I$ is of the form

$1a_{0}+Y_{1}a_{1}+Y_{2}a_{2}+\cdots+Y_{s}a_{s}$

for some $a_{i}\in I,1\neq Y_{i}\in\mathfrak{C}$. Hence, if $I\subseteq J$ then $\overline{I}\subseteq\overline{J}$ and $I=\overline{I}\cap R$ by Corollary 3.16, thus the map $I\rightarrow\overline{I}$ is injective. The second statement follows immediately from Lemma 3.11. ∎

Remark 4.2*.*

The map $I\rightarrow\overline{I}$ from $\mathcal{A}$ into $\mathcal{B}$ is not surjective. In [16], we can see that in the situation of the DG Poisson algebra concentrated in degree 0 with trivial differential, this map is not a surjective map.

Lemma 4.3**.**

Let $(R^{e},m_{R},h_{R})$ be the universal enveloping algebra of a DG Poisson algebra $(R,d,\{\cdot,\cdot\})$. If $I$ is a DG Poisson ideal of $R$ and $Q$ is the DG ideal of $R^{e}$ generated by $I$ and $h_{R}(I)$, then $(R^{e}/Q,m^{\prime},h^{\prime})$ is the universal enveloping algebra of $A=R/I$, where

$m^{\prime}:A\rightarrow R^{e}/Q,~{}~{}~{}~{}~{}~{}~{}~{}m^{\prime}(r+I)=m_{R}(r)+Q$,

$h^{\prime}:A\rightarrow R^{e}/Q,~{}~{}~{}~{}~{}~{}~{}~{}h^{\prime}(r+I)=h_{R}(r)+Q$.

Proof.

Since $Q$ is the DG ideal of $R^{e}$ generated by $I$ and $h_{R}(I)$, we have

$Q=R^{e}I+R^{e}h_{R}(I)R^{e}$

by Lemma 4.1. Let $(D,\delta)\in\textbf{DGA}$ with a DG algebra map $f:(A,d)\rightarrow(D,\delta)$ and a DG Lie algebra map $g:(A,\{\cdot,\cdot\}_{A},d)\rightarrow(D_{P},[\cdot,\cdot],\delta)$ satisfying

[TABLE]

for any homogeneous elements $a,b\in A$. Denote by $\pi^{\prime}$ the canonical projection from $R$ onto $A$, then there exists a unique DG algebra map $\phi$ from $R^{e}$ into $D$ such that $\phi m_{R}=f\pi^{\prime}$ and $\phi h_{R}=g\pi^{\prime}$ since $(R^{e},m_{R},h_{R})$ is the universal enveloping algebra of a DG Poisson algebra $(R,\{\cdot,\cdot\},d)$. Hence

$\phi(I)=\phi m_{R}(I)=f\pi^{\prime}(I)=0$  and  $\phi h_{R}(I)=g\pi^{\prime}(I)=0$,

so the DG ideal $Q$ is contained in the kernel of $\phi$. Thus $\phi$ induces the DG algebra map $\phi^{\prime}$ from $R^{e}/Q$ into $D$ such that $\phi^{\prime}(u+Q)=\phi(u)$ for all $u\in R^{e}$. Clearly, for all $r+I\in A$, we have that

[TABLE]

If $\phi_{1}$ is an DG algebra map from $R^{e}/Q$ into $D$ such that $\phi_{1}m^{\prime}=f$ and $\phi_{1}h^{\prime}=g$, then we have $\phi_{1}=\phi^{\prime}$ since $R^{e}/Q$ is generated by $m^{\prime}(A)$ and $h^{\prime}(A)$. It completes the proof. ∎

Remark 4.4*.*

From the Lemma 4.3, we have the following two observations:

• •

Since $R^{e}$ is the universal enveloping algebra of a DG Poisson algebra $R$, there is an DG algebra homomorphism $\eta$ from $R^{e}$ into $A^{e}$ such that $\eta m_{R}=m_{A}\pi^{\prime}$ and $\eta h_{R}=h_{A}\pi^{\prime}$.

[TABLE]

• •

$\eta$ is an epimorphism, since $A^{e}$ is generated by $A=m_{A}(A)$ and $h_{A}(A)$. Set

$I=ker\pi^{\prime},~{}~{}~{}~{}Q=R^{e}I+R^{e}h_{R}(I)R^{e}$.

We have $ker(\eta)=Q$.

Lemma 4.5**.**

If $f$ and $g$ are DG $\Bbbk$-linear maps from a DG Poisson algebra $A$ into a DG algebra $B$ such that

[TABLE]

for any homogeneous elements $a,b\in A$, then

[TABLE]

Proof.

For any homogeneous elements $a,b\in A$, we have

[TABLE]

Note that

$\{a,b\}=-(-1)^{|a||b|}\{b,a\},~{}~{}ab=(-1)^{|a||b|}ba$.

Hence

[TABLE]

and

[TABLE]

Therefore, we have the conclusion. ∎

Lemma 4.6**.**

Let $(R^{e},m_{R},h_{R})$ be the universal enveloping algebra of a DG Poisson algebra $R$. Then

[TABLE]

Proof.

Since

$h_{R}(ab)=h_{R}(a)m_{R}(b)+(-1)^{|a||b|}h_{R}(b)m_{R}(a)=h_{R}(a)b+(-1)^{|a||b|}h_{R}(b)a$

for all elements $a,b\in R$ by Lemma 4.5. Observe that formula (4.1) is true on $n=1$ and $n=2$ since

[TABLE]

Set

[TABLE]

Thus in order to complete the proof, we only need to show

$h_{R}(x_{i}^{n+1})=\bigtriangleup(n+1)y_{i}x_{i}^{n}$

provided that $h_{R}(x_{i}^{n})=\bigtriangleup(n)y_{i}x_{i}^{n-1}$.

Indeed if $n=2k$, then

[TABLE]

On the other hand, if $n=2k+1$, then

[TABLE]

as required. ∎

Using the above notation, we have the following lemma.

Lemma 4.7**.**

Let $(R^{e},m_{R},h_{R})$ be the universal enveloping algebra of a DG Poisson algebra $R$. Then

[TABLE]

Proof.

For each monomial $x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}$ of $R$, we proceed the proof using induction on the number of indeterminates. By Lemma 4.6, it is clear that formula (4.2) is true on the $x_{1}^{i_{1}}$. Thus in order to complete the proof, we only need to show the formula (4.2) on the $x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}$ provided that

[TABLE]

Since

$h_{R}(ab)=h_{R}(a)m_{R}(b)+(-1)^{|a||b|}h_{R}(b)m_{R}(a)=h_{R}(a)b+(-1)^{|a||b|}h_{R}(b)a$

for all elements $a,b\in R$ by Lemma 4.5, we have

[TABLE]

as required. ∎

Proposition 4.8**.**

Let $(R^{e},m_{R},h_{R})$ be the universal enveloping algebra of a DG Poisson algebra $R$. Then, for every monomial $x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}\in R$, the expression of $z=y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n}^{j_{n}}h_{R}(x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}})$ as a right $R$-combination of monomial in $\mathfrak{C}$ of Corollary 3.16 is of the form

$k_{1}Y_{1}a_{1}+k_{2}Y_{2}a_{2}+\cdots+k_{s}Y_{s}a_{s},~{}k_{i}\in\Bbbk,~{}1\neq Y_{i}\in\mathfrak{C}$ and $a_{i}\in R$.

That is, the coefficient of 1 under the expression of $z$ as a right $R$-combination of monomials in $\mathfrak{C}$ is zero.

Proof.

Since

$h_{R}(ab)=m_{R}(a)h_{R}(b)+(-1)^{|a||b|}m_{R}(b)h_{R}(a)=ah_{R}(b)+(-1)^{|a||b|}bh_{R}(a)$,

for all elements $a,b\in R$, we have $h_{R}(1)=0$, that is, $h_{R}(k)=0$ for every $k\in\Bbbk$. Express $h_{R}(x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}})$ as given in Lemma 4.7 and then it suffices to prove the statement for the case $z=y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n}^{j_{n}}y_{i}$.

For a monomial $y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n}^{j_{n}}$, we proceed the proof using induction on the $r=j_{1}+j_{2}+\cdots+j_{n}$. If $r=0$ or $1$, then it is trivial. Assume that the statement is true for monomials with length less than $j_{1}+j_{2}+\cdots+j_{n}$, we should proof the statement is also true for monomials with length equal $j_{1}+j_{2}+\cdots+j_{n}$. There are two possibilities: (I) $j_{n}\neq 0$, (II) $j_{n}=0$ (without loss of generality, we can further assume that $j_{n-1}\neq 0$).

Case I. $j_{n}\neq 0$: Note that

$y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n}^{j_{n}}y_{i}=y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n}^{j_{n}-1}((-1)^{|x_{n}||x_{i}|}y_{i}y_{n}+h_{R}(\{x_{n},x_{i}\}))$

by Lemma 3.11. By the induction hypothesis, we have the expression of $y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n}^{j_{n}-1}y_{i}$ as a right $R$-combination of monomial in $\mathfrak{C}$ of Corollary 3.16 is of the form

$k_{1}Y_{1}a_{1}+k_{2}Y_{2}a_{2}+\cdots+k_{s}Y_{s}a_{s},~{}k_{i}\in\Bbbk,~{}1\neq Y_{i}\in\mathfrak{C}$ and $a_{i}\in R$.

Then the result on the case

$(-1)^{|x_{n}||x_{i}|}y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n}^{j_{n}-1}y_{i}y_{n}$

is true by $(a)$ of Lemma 3.11. On the other hand, since $\{x_{n},x_{i}\}$ can expressed as polynomial in $R$, we can see that the result on the case

$y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n}^{j_{n}-1}h_{R}(\{x_{n},x_{i}\})$

is also true by Lemma 4.7 and the induction hypothesis.

Case II. $j_{n}=0$: Note that

$y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n-1}^{j_{n-1}}y_{i}=y_{1}^{j_{1}}y_{2}^{j_{2}}\cdots y_{n-1}^{j_{n-1}-1}((-1)^{|x_{n-1}||x_{i}|}y_{i}y_{n-1}+h_{R}(\{x_{n-1},x_{i}\}))$.

Similarly, we have

[TABLE]

by Lemma 3.11 and the induction hypothesis, where $~{}k_{i}^{\prime}\in\Bbbk,~{}1\neq Y_{i}^{\prime}\in\mathfrak{C}$ and $a_{i}^{\prime}\in R$. Thus, the result on the case

$\sum_{i}(-1)^{|x_{n-1}||a_{i}^{\prime}|}k_{i}^{\prime}Y_{i}^{\prime}y_{n-1}a_{i}^{\prime}$

is true by the case I. Further, similar to the proof of the case I, it is easy to see that the result on the case II is also true.

Therefore, we finish the proof. ∎

Lemma 4.9**.**

Let $(R^{e},m_{R},h_{R})$ be the universal enveloping algebra of a DG Poisson algebra $R$. If $M\neq R$ is a DG ideal of $R$ and $Q$ is a DG Poisson ideal contained in $M$, then

$R^{e}M+R^{e}h_{R}(Q)R^{e}\neq R^{e}.$**

Proof.

Since $R^{e}$ is a DG free right $R$-module by Corollary 3.16, every element of $R^{e}M+R^{e}h_{R}(Q)R^{e}$ is of the form

$\sum_{i}b_{i}m_{i}+c_{i}h_{R}(q_{i})a_{i},~{}a_{i},b_{i},c_{i}\in R^{e},~{}m_{i}\in M,~{}q_{i}\in Q.$

Express each $a_{i}$ as a right $R$-combination of monomials $Y_{j}$ in $\mathfrak{C}$ of Corollary 3.16. Since

$y_{i}h_{R}(q_{i})=(-1)^{|x_{i}||q_{i}|}h_{R}(q_{i})y_{i}+h_{R}(\{x_{i},q_{i}\})$

for each $i=1,2,\cdots,n$ by $(c)$ of Lemma 3.11 and since $\{x_{i},q_{i}\}\in Q$, we may assume that $a_{i}\in R$ for all $i$. Next express each $c_{i}$ as a right $R$-combination of monomials $Y_{j}$ in $\mathfrak{C}$ of Corollary 3.16. For $r\in R$, since

$h_{R}(q_{i})r=(-1)^{|r||q_{i}|}rh_{R}(q_{i})+\{q_{i},r\}$

by $(b)$ of Lemma 3.11 and $\{q_{i},r\}\in Q\subseteq M$, we may assume that each $c_{i}$ is a $\Bbbk$-linear combination of monomials in $\mathfrak{C}$ of Corollary 3.16. Thus by Proposition 4.8, $\sum_{i}c_{i}h_{R}(q_{i})a_{i}$ can be written as

$\sum_{i}c_{i}h_{R}(q_{i})a_{i}=\sum_{i}k_{i}^{\prime}Y_{i}^{\prime}r_{i}^{\prime},~{}k_{i}^{\prime}\in\Bbbk,~{}1\neq Y_{i}^{\prime}\in\mathfrak{C}$ and $r_{i}^{\prime}\in R.$

Finally, express each $b_{i}$ as a right $R$-combination of monomials $Y_{j}$ in $\mathfrak{C}$ of Corollary 3.16. Then every element of $R^{e}M+R^{e}h_{R}(Q)R^{e}$ is of the form

$1m_{0}+k_{1}Y_{1}r_{1}+k_{2}Y_{2}r_{2}+...+k_{s}Y_{s}r_{s},~{}m_{0}\in M,~{}k_{i}\in\Bbbk,~{}1\neq Y_{i}\in\mathfrak{C}$ and $r_{i}\in R.$

Thus $R^{e}M+R^{e}h_{R}(Q)R^{e}$ does not contain the unity since $M\neq R$ is a DG ideal of $R$. Therefore, we complete the proof. ∎

Now, we study the simple DG Poisson module.

Let $(B,\cdot,\{\cdot,\cdot\},d)\in\textbf{DGPA}$ and let $(B^{e},m_{B},h_{B})$ be the universal enveloping algebra of $B$, the definition of a DG Poisson module $M$ over a DG Poisson algebra $B$ is given in Definition 2.4. Note that the annihilator of a DG Poisson $B$-module $M$ is defined to be

$ann_{B}(M)=\{b\in B|~{}b\cdot M=0\}$,

which is a DG Poisson ideal of $B$. Since $M$ is a left DG $B^{e}$-module, it is easy to prove that

$ann_{B}(M)=m_{B}^{-1}(ann_{B^{e}}(M))=ann_{B^{e}}(M)\cap B$,

where $B$ is a finitely generated DG Poisson algebra.

Now follow the idea of [16], we can define a DG symplectic ideal. A DG Poisson ideal $Q$ of $B$ is said to be DG symplectic ideal if there is a maximal DG ideal $M$ of $(B,\cdot,\{\cdot,\cdot\},d)$ such that $Q$ is the largest DG Poisson ideal contained in $M$. Note that a simple DG Poisson module over a DG Poisson algebra $B$ is the DG Poisson module over $B$ that has no non-zero proper DG Poisson submodule.

Theorem 4.10**.**

Let $R=T(V)/\langle x_{\alpha}\otimes x_{\beta}-(-1)^{|x_{\alpha}||x_{\beta}|}x_{\beta}\otimes x_{\alpha}|~{}\forall\alpha,~{}\beta\in\Lambda\rangle$ be a DG Poisson algebra with an arbitrary differential $d$ and Poisson structure $\{\cdot,\cdot\}$. Suppose that $A=R/I$ is a DG Poisson homomorphic image of $R$. If $P$ is a DG symplectic ideal of $A$, then $P$ is the annihilator of a simple DG Poisson $A$-module.

Proof.

From the above, let $\pi^{\prime}$ be the canonical projection from $R$ onto $A$ and let $M$ be a maximal DG ideal of $A$ such that $P$ is the largest DG Poisson ideal contained in $M$. By Remark 4.4, there is an epimorphism $\eta$ from $R^{e}$ onto $A^{e}$ such that $ker(\eta)=R^{e}I+R^{e}h_{R}(I)R^{e}$.

[TABLE]

Note that $\pi^{\prime-1}(M)$ is a maximal DG ideal of $R$ and $\pi^{\prime-1}(P)$ is the largest DG Poisson ideal contained in $\pi^{\prime-1}(M)$. Since

$R^{e}\pi^{\prime-1}(M)+R^{e}h_{R}(\pi^{\prime-1}(P))R^{e}\neq R^{e}$

by Lemma 4.9, there is a maximal left DG ideal $N$ of $R^{e}$ containing $R^{e}\pi^{\prime-1}(M)+R^{e}h_{R}(\pi^{\prime-1}(P))R^{e}$. Then $X=R^{e}/N$ is a simple left DG $R^{e}$-module and so $X$ is a simple left DG Poisson $R$-module.

Nota that $ann_{R^{e}}(X)$ is the largest DG ideal of $R^{e}$ contained in $N$:

since $ann_{R^{e}}(X)=\{u\in R^{e}|~{}u\cdot X=0\}$, we have

$0=u(u^{\prime}+N)=uu^{\prime}+N$

for all $u^{\prime}+N\in X$, then $uu^{\prime}\in N$. Set $u^{\prime}=1$, we have $u\in N$, that means $ann_{R^{e}}(X)\subseteq N$. If there exist a DG ideal $B$ strictly contained in $N$ such that $ann_{R^{e}}(X)\subsetneq B$, then there exist a $b\in B$ such that $b\notin ann_{R^{e}}(X)$, we have $b(u_{1}+N)\neq 0$ for some $u_{1}+N\in X$, thus $bu_{1}\notin N$. But $b\in B,u_{1}\in R^{e},B$ is a DG ideal, then $bu_{1}\in B\subseteq N$, which contradicts $bu_{1}\notin N$. Hence $ann_{R^{e}}(X)$ is the largest DG ideal of $R^{e}$ contained in $N$.

Since $R^{e}\pi^{\prime-1}(P)$ is a DG ideal by Lemma 4.1 and $R^{e}\pi^{\prime-1}(P)\subseteq R^{e}\pi^{\prime-1}(M)\subseteq N$, we have

$\pi^{\prime-1}(P)=(R^{e}\pi^{\prime-1}(P))\cap R\subseteq ann_{R^{e}}(X)\cap R\subseteq N\cap R$

by Lemma 4.1. Since $\pi^{\prime-1}(M)\subseteq N\cap R\neq R$ and $\pi^{\prime-1}(M)$ is a maximal DG ideal of $R$, we have $\pi^{\prime-1}(M)=N\cap R$. Hence $\pi^{\prime-1}(P)=ann_{R^{e}}(X)\cap R=ann_{R}(X)$ since $\pi^{\prime-1}(P)$ is the largest DG Poisson ideal contained in $\pi^{\prime-1}(M)$.

Note that $ker(\eta)$ is a DG ideal of $R^{e}$ contained in $N$ by Remark 4.4 since $I=ker(\pi^{\prime})\subseteq\pi^{\prime-1}(P)$ and $I\subseteq\pi^{\prime-1}(M)$. Hence $ker(\eta)X=0$ and so $X$ is a simple left DG $A^{e}$-module with module structure induced by $\eta$. Thus $X$ becomes a simple DG Poisson $A$-module.

Moreover, we have

[TABLE]

which completes the proof. ∎