DG Poisson adjoint action and its application


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 1, 2019, 14-24
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

DG Poisson adjoint action and its application

Xiaojie Li1, Xianguo Hu1, Jiafeng Lü1, Xingting Wang2,∗

1 Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004
P.R. China
2 Department of Mathematics, Howard University, Washington DC, 20059, USA

Abstract. In this paper, the differential graded (DG for short) Poisson adjoint action on M is
introduced, whereM is a DG Poisson module over a DG Poisson Hopf algebra A. As an application,
we give a new DG poisson module structure over the DG Poisson Hopf algebra A, which depends
heavily on the structure of A.

2010 Mathematics Subject Classifications: 16E45, 16S10, 17B35, 17B63

Key Words and Phrases: Differential graded Poisson Hopf algebras, differential graded Poisson
modules, the DG Poisson adjoint action

1. Introduction

The Poisson bracket was originally introduced by French Mathematician Siméon Denis
Poisson in search for integrals of motion in Hamiltonian mechanics. Recently, different gen-
eralizations of Poisson algebras have been introduced by several people: Poisson orders [1],
noncommutative Leibniz-Poisson algebras [2], Left-right noncommutative Poisson algebras
[3], graded Poisson algebras [4], Poisson Ore-extensions [8], differential graded Poisson al-
gebra [9], Poisson PI algebras [12], double Poisson algebras [19], Novikov-Poisson algebras
[20] and Quiver Poisson algebras [22], etc. One of the most interesting features in this
area is the Poisson universal enveloping algebra, which was first introduced by Oh [13]
in order to describe the category of Poisson modules. Most recently, many people show
their interests in Poisson universal enveloping algebras [8, 10, 16, 18, 21]. In particular,
the second author of the present paper studied the universal enveloping algebras of DG
Poisson algebras and propose a definition for the DG Poisson module [9].

We know that, Poisson Hopf algebras arise naturally in Poisson geometry and quantum
groups. Recently, Poisson Hopf algebras are studied by many authors from different
perspectives [5, 7, 14, 15]. In [5], the authors developed the theory of Poisson Hopf algebras,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i1.3368

Email addresses: jiafenglv@zjnu.edu.cn (J.-F. Lü), 2015210420@zjnu.edu.cn (X.-G. Hu),
407586291@qq.com (X.-J. Li), xingting.wang@howard.edu (X. Wang)

http://www.ejpam.com 14 c© 2019 EJPAM All rights reserved.



X.-J. Li, X.-G. Hu,J.-F. Lü, X. Wang / Eur. J. Pure Appl. Math, 12 (1) (2019), 14-24 15

given the definition of a DG Poisson Hopf algebra A, and discussed the structures for the
universal enveloping algebra of A.

Considering the importance of DG Poisson Hopf algebras and DG Poisson modules,
our aim in this paper is to study the DG Poisson adjoint action on M , where M is a DG
Poisson modules over a DG Poisson Hopf algebras A. Furthermore, according to the DG
Poisson adjoint action on M , we will construct a new DG Poisson module structure over
A, which relies heavily on the structure of A.

The paper is organized as follows. In Section 2, we briefly review some basic definitions
and results related to DG Poisson Hopf algebras and DG Poisson modules over DG Poisson
algebras. In Section 3, we first propose a definition for the DG Poisson adjoint action,
then discuss some basic properties of the DG Poisson adjoint action. As an application,
we construct a new DG Poisson module structure over a DG Poisson Hopf algebra A.

Throughout the whole paper, Z denotes the set of integers, k denotes a base field of
characteristic zero unless otherwise stated, and all (graded) algebras are assumed to have
an identity and all (graded) modules are assumed to be unitary. We always take the
grading to be Z-graded. In addition, Let V and W be graded vector spaces, the twisting
map T : V ⊗W →W ⊗ V is defined for homogeneous elements v ∈ V and w ∈W by

T (v ⊗ w) = (−1)|v||w|w ⊗ v

and extends to all elements of V and W through linearity.

2. Preliminaries

In this section, we will recall some definitions and properties of DG Poisson Hopf
algebras and DG Poisson modules.

By a graded algebra A we mean a Z-graded algebra (A, u, η), where u : A ⊗ A → A
and η : k → A are called the multiplication and unit of A, respectively. For convenience,
we shall write u(a⊗ b) as ab, ∀a, b ∈ A, whenever this does not cause confusion.

Recall that a graded coalgebra C over k is a Z-graded vector space with the graded linear
maps ∆ : C → C ⊗ C and ε : C → k of degree 0 such that the obvious (usual) diagrams
commute, where ∆ and ε are called the comultiplication and counit of C, respectively.

For any homogeneous element c ∈ C, we shall use the Sweedler’s notation, that is
∆(c) =

∑
(c) c(1)⊗ c(2). In this notation, the comultiplication and counit property may be

expressed as

(∆⊗ I)∆(c) = (I ⊗∆)∆(c) =
∑
(c)

c(1) ⊗ c(2) ⊗ c(3),

c =
∑
(c)

ε(c(1))c(2) =
∑
(c)

c(1)ε(c(2))

for any homogeneous elements c ∈ C, respectively.
Let H be a graded algebra with multiplication u and unit η, and at the same time a

graded coalgebra with comultiplication ∆ and counit ε. If ∆ and ε are graded algebra



X.-J. Li, X.-G. Hu,J.-F. Lü, X. Wang / Eur. J. Pure Appl. Math, 12 (1) (2019), 14-24 16

homomorphisms, then H is called a graded bialgebra. Further, if H admits a graded vector
space homomorphism S : H → H of degree 0 which satisfies the following defining relation:

u ◦ (I ⊗ S) ◦∆ = u ◦ (S ⊗ I) ◦∆ = η ◦ ε.

Then H is called a graded Hopf algebra, and S is called the antipode of H.
Similarly, a graded Poisson algebra can be defined as a graded algebra with a compatible

graded Lie structure.

Definition 1. [4] Let (A, ·) be a graded k-algebra. If there is a k-linear map

{·, ·} : A⊗A→ A

of degree 0 such that

(i) (A, {·, ·}) is a graded Lie algebra. That is to say, we have

(ia) {a, b} = −(−1)|a||b|{b, a};
(ib) {a, {b, c}} = {{a, b}, c}+ (−1)|a||b|{b, {a, c}},

(ii) (graded commutativity): a · b = (−1)|a||b|b · a;

(iii) (biderivation property): {a, b · c} = {a, b} · c+ (−1)|a||b|b · {a, c},

for any homogeneous elements a, b, c ∈ A, then A is called a graded Poisson algebra.

Definition 2. [5] Let A be a graded k-vector space. If there is a k-linear map

{·, ·} : A⊗A→ A

of degree 0 such that:

(i) (A, u, η, {·, ·}) is a graded Poisson algebra;

(ii) (A, u, η,∆, ε) is a graded Hopf algebra;

(iii) ∆({a, b}A) = {∆(a),∆(b)}A⊗A for all a, b ∈ A, where the Poisson bracket {·, ·}A⊗A
on A⊗A is defined by

{a⊗ a′, b⊗ b′}A⊗A = (−1)|a
′||b|({a, b} ⊗ a′b′ + ab⊗ {a′, b′}) (1)

for any homogeneous elements a, b, a′, b′ ∈ A.

Then A is called a graded Poisson Hopf algebra. If in addition, there is a k-linear homo-
geneous map d : A→ A of degree 1 such that d2 = 0 and

(iva) d({a, b}) = {d(a), b}+ (−1)|a|{a, d(b)};

(ivb) d(a · b) = d(a) · b+ (−1)|a|a · d(b);



X.-J. Li, X.-G. Hu,J.-F. Lü, X. Wang / Eur. J. Pure Appl. Math, 12 (1) (2019), 14-24 17

(ivc) ε ◦ d = 0 and ∆d(a) =
∑

(a) d(a(1)) ⊗ a(2) +
∑

(a)(−1)|a(1)|a(1) ⊗ d(a(2)), where
∆(a) =

∑
(a) a(1) ⊗ a(2),

for any homogeneous elements a, b ∈ A, then A is called a DG Poisson Hopf algebra, which
is usually denoted by

(A, u, η,∆, ε, S, {·, ·}, d).

Remark 1. By the formula (2.1) and ∆({a, b}A) = {∆(a),∆(b)}A⊗A, we have

∆({a, b}) =
∑
(a)(b)

(−1)|a(2)||b(1)|({a(1), b(1)} ⊗ a(2)b(2) + a(1)b(1) ⊗ {a(2), b(2)})

for all homogeneous elements a, b of a DG Poisson Hopf algebra A.

If A is just a graded Hopf algebra, then the antipode S has the following properties
[6, 11, 17].

Lemma 1. Let A be a graded Hopf algebra and S its antipode; then

(i) S ◦ u = u ◦ T ◦ (S ⊗ S),

(ii) S ◦ η = η,

(iii) ε ◦ S = ε,

(iv) T ◦ (S ⊗ S) ◦∆ = ∆ ◦ S,

(v) if A is graded commutative or graded cocommutative, then S ◦ S = I,

where I : A→ A is the identity morphism and T : A⊗A→ A⊗A is the twisting morphism.

Lemma 2. [5] If (A, u, η,∆, ε, S, {·, ·}, d) is a DG Poisson Hopf algebra, then dS = Sd,
S({a, b}) = (−1)|a||b|{S(b), S(a)} and ε({a, b}) = 0 for all a, b ∈ A.

Definition 3. [9] Let (A, ·, {·, ·}A, d) be a DG Poisson algebra. We call a Z-graded vector
space

M =
⊕
i∈Z

M i

a left DG Poisson module over A provided that the following conditions are satisfied:

(i) (M, ·, ∂) is a left DG module over the DG algebra A. Equivalently,

(ia) there is a k-bilinear function − · − : A⊗M →M of degree 0 such that M is a
left graded module over A, i.e., Ai ·M j ⊆M i+j for all i, j ∈ Z,

(ib) there is a k-linear map ∂ : M →M of degree 1 such that ∂2 = 0 and ∂(a ·m) =
d(a) ·m+ (−1)|a|a · ∂(m) for all homogeneous elements a ∈ A and m ∈M .



X.-J. Li, X.-G. Hu,J.-F. Lü, X. Wang / Eur. J. Pure Appl. Math, 12 (1) (2019), 14-24 18

(ii) (M, ·, {·, ·}) is a left Z-graded Poisson module over the graded Poisson algebra A.
That is to say, there is another bilinear bracket {·, ·}M : A ⊗M → M of degree 0
such that

(iia) {a, b ·m}M = {a, b}A ·m+ (−1)|a||b|b · {a,m}M ,

(iib) {a · b,m}M = a · {b,m}M + (−1)|a||b|b · {a,m}M and

(iic) {a, {b,m}M}M = {{a, b}A,m}M + (−1)|a||b|{b, {a,m}M}M

for all homogeneous elements a, b ∈ A and m ∈M .

(iii) The linear function ∂ is compatible with the bracket {−,−}M . That is, we have
∂({a,m}M ) = {d(a),m}M + (−1)|a|{a, ∂(m)}M for all homogeneous elements a ∈ A
and m ∈M .

We usually call ∂ the differential of M and use a quadruple (M, ·, {·, ·}M , ∂) to denote
a DG Poisson module.

3. DG Poisson adjoint action and its application

In this section, we define the DG Poisson adjoint action, and construct a new DG
Poisson module over a DG Poisson Hopf algebra A by studying the DG Poisson adjoint
action, which will take advantage of the structure of a DG Poisson Hopf algebra.

We begin with the following definition.

Definition 4. Let M be a DG Poisson module over a DG Poisson Hopf algebra

(A, u, η,∆, ε, S, d)

. Then the DG Poisson adjoint action on M is defined by

ada(z) =
∑
(a)

(−1)|a(1)||a(2)|S(a(2)){a(1), z}, a ∈ A, z ∈M,

where ∆(a) = Σ(a)a(1) ⊗ a(2)
There exists a canonical DG Poisson adjoint action on A given by

ada(z) =
∑
(a)

(−1)|a(1)||a(2)|S(a(2)){a(1), z}, a, z ∈ A,

since A is a DG Poisson A-module with Poisson module structure {a, z}A and az is the
multiplication in A for all a, z ∈ A. Moreover, the canonical DG Poisson adjoint action
on A satisfies the following relation:

ada(zy) = ada(z) · y + (−1)|a||z|z · ada(y), a, z, y ∈ A.

In particular, if |a| = 1, then the canonical DG Poisson adjoint action is a graded derivation
on A.



X.-J. Li, X.-G. Hu,J.-F. Lü, X. Wang / Eur. J. Pure Appl. Math, 12 (1) (2019), 14-24 19

Lemma 3. Let (A, u, η,∆, ε, S, d) be a DG Poisson Hopf algebra and let M be a DG
Poisson A-module. Then

ada(z) = −
∑
(a)

a(1){S(a(2)), z},

where a ∈ A and z ∈M .

Proof. By Definition 3, it is easy to see that {1A, z} = 0, for any homogeneous element
z ∈M . Then for any homogeneous elements a ∈ A, z ∈M , we have

0 = {ε(a)1A, z} = {
∑
(a)

a(1)S(a(2)), z} =
∑
(a)

(a(1){S(a(2)), z}+(−1)|a(1)||a(2)|S(a(2)){a(1), z}).

Therefore
ada(z) = −

∑
(a)

a(1){S(a(2)), z}.

Lemma 4. Let (A, u, η,∆, ε, S, d) be a DG Poisson Hopf algebra and let M be a DG
Poisson A-module. For any homogeneous elements a, b ∈ A, we have

adab = ε(a)adb + (−1)|a||b|ε(b)ada.

Proof. Since ∆(ab) =
∑

(a)(b)(−1)|a(2)||b(1)|a(1)b(1) ⊗ a(2)b(2) and M is a DG Poisson
module over A, then for any homogeneous elements a, b ∈ A, z ∈M , we have

adab(z) =
∑
(a)(b)

(−1)|a(1)b(1)||a(2)b(2)|+|a(2)||b(1)|S(a(2)b(2)){a(1)b(1), z}

=
∑
(a)(b)

(−1)|a(1)b(1)||a(2)b(2)|+|a(2)||b(1)|+|a(2)||b(2)|S(b(2))S(a(2))(a(1){b(1), z}

+ (−1)|a(1)||b(1)|b(1){a(1), z})

=
∑
(a)(b)

(−1)|b(1)||b(2)|a(1)S(a(2))S(b(2)){b(1), z}

+ (−1)|a||b|
∑
(a)(b)

(−1)|a(1)||a(2)|b(1)S(b(2))S(a(2)){a(1), z}

= ε(a)adb(z) + (−1)|a||b|ε(b)ada(z)

by Lemma 1.

Lemma 5. Let (A, u, η,∆, ε, S, d) be a DG Poisson Hopf algebra and let M be a DG
Poisson A-module. For any homogeneous elements a, b ∈ A, we have

ad{a,b} = adaadb − (−1)|a||b|adbada.



X.-J. Li, X.-G. Hu,J.-F. Lü, X. Wang / Eur. J. Pure Appl. Math, 12 (1) (2019), 14-24 20

Proof. Since ∆({a, b}) =
∑

(a)(b)(−1)|a(2)||b(1)|({a(1), b(1)}⊗a(2)b(2)+a(1)b(1)⊗{a(2), b(2)})
and M is a DG Poisson module over A, then for any homogeneous elements a, b ∈ A, z ∈
M , we have

ad{a,b}(z)

=
∑
(a)(b)

(−1)|{a,b}(1)||{a,b}(2)|S({a, b}(2)){{a, b}(1), z}

=
∑
(a)(b)

(−1)(|a(1)|+|b(1)|)(|a(2)|+|b(2)|)+|a(2)||b(1)|(S(a(2)b(2)){{a(1), b(1)}, z}

+ S({a(2), b(2)}){a(1)b(1), z})

=
∑
(a)(b)

(−1)|a(1)||a(2)b(2)|+|b(1)||b(2)|+|a(2)||b(2)|S(b(2))S(a(2))[{a(1), {b(1), z}} − (−1)|a(1)||b(1)|{b(1), {a(1), z}}]

+
∑
(a)(b)

(−1)|a(1)||a(2)b(2)|+|b(1)||b(2)|+|a(2)||b(2)|{S(b(2)), S(a(2))}[a(1){b(1), z}+ (−1)|a(1)||b(1)|b(1){a(1), z}]

by Lemmas 1 and 2.
But∑

(a)(b)

(−1)|a(1)||a(2)|+|a(1)||b(2)|+|b(1)||b(2)|+|a(2)||b(2)|{S(b(2)), S(a(2))}a(1){b(1), z}

=
∑
(a)(b)

−(−1)|a(1)||a(2)|+|a(1)||b(2)|+|b(1)||b(2)|{S(a(2)), S(b(2))}a(1){b(1), z}

=
∑
(a)(b)

−(−1)|b(1)||b(2)|a(1)[{S(a(2)), S(b(2)){b(1), z}} − (−1)|a(2)||b(2)|S(b(2)){S(a(2)), {b(1), z}}]

= adaadb(z) +
∑
(a)(b)

(−1)|b(1)||b(2)|+|a(2)||b(2)|a(1)S(b(2)){S(a(2)), {b(1), z}}

by Lemma 3.
Similarly, we have∑
(a)(b)

(−1)|a(1)||a(2)|+|a(1)||b(2)|+|b(1)||b(2)|+|a(2)||b(2)|+|a(1)||b(1)|{S(b(2)), S(a(2))}b(1){a(1), z}

=− (−1)|a||b|adbada(z)−
∑
(a)(b)

(−1)|a||b|+|a(1)||a(2)|+|a(2)||b(2)|b(1)S(a(2)){S(b(2)), {a(1), z}}.

Since for any homogeneous element z ∈M , we have {1A, z} = 0. Thus

ad{a,b}(z)

= adaadb(z) +
∑
(a)(b)

(−1)|b(1)||b(2)|+|a(2)||b(2)|a(1)S(b(2)){S(a(2)), {b(1), z}}

− (−1)|a||b|adbada(z)−
∑
(a)(b)

(−1)|a||b|+|a(1)||a(2)|+|a(2)||b(2)|b(1)S(a(2)){S(b(2)), {a(1), z}}



X.-J. Li, X.-G. Hu,J.-F. Lü, X. Wang / Eur. J. Pure Appl. Math, 12 (1) (2019), 14-24 21

+
∑
(a)(b)

(−1)|a(1)||a(2)|+|a(1)||b(2)|+|b(1)||b(2)|+|a(2)||b(2)|S(b(2))S(a(2)){a(1), {b(1), z}}

−
∑
(a)(b)

(−1)|a(1)||a(2)|+|a(1)||b(2)|+|b(1)||b(2)|+|a(2)||b(2)|+|a(1)||b(1)|S(b(2))S(a(2)){b(1), {a(1), z}}

= adaadb(z)− (−1)|a||b|adbada(z) +
∑
(b)

(−1)|b(1)||b(2)|+|a||b(2)|S(b(2)){ε(a)1A, {b(1), z}}

−
∑
(a)

(−1)|a||b|+|a(1)||a(2)|+|a(2)||b|S(a(2)){ε(b)1A, {a(1), z}}

= adaadb(z)− (−1)|a||b|adbada(z).

As an application, we are ready to state and prove our main result.

Theorem 1. Let M be a DG Poisson module over a DG Poisson Hopf algebra

(A, u, η,∆, ε, S, d)

. Define

α : A×M →M, (a, z) 7→ a ◦ z = ε(a)z;

β : A×M →M, (a, z) 7→ a ∗ z = ada(z) :=
∑
(a)

(−1)|a(1)||a(2)|S(a(2)){a(1), z}.

Then (M, ◦, ∗, ∂) is a DG Poisson A-module.

Proof. Note that (A, u, η,∆, ε, S, d) is a DG Poisson Hopf algebra, we have that ε is
a graded algebra homomorphism. The fact (M, ◦) is a left Z-graded module over A is
straightforward and follows easily from the Definition of a graded module.

Now, let us prove that (M, ◦, ∗) is a left Z-graded Poisson module over the graded
Poisson algebra A. By Lemma 2, we know that ε({a, b}) = 0, ∀a, b ∈ A. Note that k is a
trivial DG Poisson Hopf algebra concentrated in degree 0 with trivial Poisson bracket and
trivial differential, hence

{a, b}◦m+(−1)|a||b|b◦(a∗m) = ε({a, b})m+(−1)|a||b|ε(b)ada(m) = ε(b)ada(m) = a∗(b◦m).

For any homogeneous elements a, b ∈ A,m ∈M , we have

(ab) ∗m = adab(m) = ε(a)adb(m) + (−1)|a||b|ε(b)ada(m) = a ◦ (b ∗m) + (−1)|a||b|b ◦ (a ∗m)

and

{a, b}∗m = ad{a,b}(m) = adaadb(m)−(−1)|a||b|adbada(m) = a∗(b∗m)−(−1)|a||b|b∗(a∗m)

by Lemmas 4 and 5. Thus (M, ◦, ∗) is a left Z-graded Poisson module over A.



REFERENCES 22

Finally, we show that the differential ∂ satisfies the corresponding conditions. For any
homogeneous elements a ∈ A,m ∈M , we have on one hand:

d(a) ◦m+ (−1)|a|a ◦ ∂(m) = ε(d(a))m+ (−1)|a|ε(a)∂(m) = ε(a)∂(m) = ∂(a ◦m),

since εd = 0 and k is a trivial DG Poisson Hopf algebra concentrated in degree 0; and on
the other hand:

∂(a ∗m) = ∂(ada(m)) = ∂(
∑
(a)

(−1)|a(1)||a(2)|S(a(2)){a(1),m})

=
∑
(a)

(−1)|a(1)||a(2)|(dS(a(2)){a(1),m}+ (−1)|a(2)|S(a(2))∂({a(1),m}))

=
∑
(a)

(−1)|a(1)||a(2)|Sd(a(2)){a(1),m}

+
∑
(a)

(−1)|a(1)||a(2)|+|a(2)|S(a(2))[{d(a(1)),m}+ (−1)|a(1)|{a(1), ∂(m)}],

d(a) ∗m+ (−1)|a|a ∗ ∂(m)

= add(a)(m) + (−1)|a|ada(∂(m))

=
∑
(a)

(−1)|d(a(1))||a(2)|S(a(2)){d(a(1)),m}+
∑
(a)

(−1)|a(1)||d(a(2))|+||a(1)||Sd(a(2)){a(1),m}

+ (−1)|a|(
∑
(a)

(−1)|a(1)||a(2)|S(a(2)){a(1), ∂(m)})

by Lemma 2 and the following identity:

∆d(a) = d(a(1))⊗ a(2) + (−1)|a(1)|a(1) ⊗ d(a(2)).

Hence
∂(a ∗m) = d(a) ∗m+ (−1)|a|a ∗ ∂(m).

Therefore (M, ◦, ∗, ∂) is a DG Poisson A-module.

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