Lie Bialgebroid of Pseudo-differential Operators

Abstract

We associate a Lie bialgebroid structure to the algebra of formal Pseudo-differential operators, as the classical limit of a quantum groupoid. As a consequence, the non-commutative Kadomtsev–Petviashvili hierarchy is naturally obtained by an algebraic procedure.

1 Introduction

Consider the formal pseudo-differential operator L as

$$\begin{aligned} L=\frac{\partial }{\partial x}+u_1\left( \frac{\partial }{\partial x}\right) ^{-1}+u_2\left( \frac{\partial }{\partial x}\right) ^{-2}+ \dots, \end{aligned}$$

where $u_1, u_2, \dots,$ are infinite number of variables which are dependent variables of systems of partial differential equations. The classical Kadomtsev–Petviashvili (KP) hierarchy is the infinite system of equations

$$\begin{aligned} \partial _{x_m}L=[(L^m)_+,L], \ \ \ \ m=1,2,3, \dots, \end{aligned}$$

where $(L^m)_+$ denotes the projection of the pseudo-differential operator $L^m$ into the space of differential operators, and $x_1, x_2, \dots$ denote independent variables.

The non-commutative KP hierarchy usually has been defined (see e.g. [11]) using non-commutativity of the coordinates

$$\begin{aligned}{}[x_k, x_l]=i\theta ^{kl}, \end{aligned}$$

in Pseudo-differential operators, for the independent variables $x_k.$ Here $\theta ^{kl}$ are real constants and called the non-commutative parameters. Hence, the non-commutative KP hierarchy is described by

$$\begin{aligned} \partial _{x_m}L=[(L^m)_+,L]_{\star }, \end{aligned}$$

where the star-product $\star$ is represented by the Moyal product.

In this note, we are going to show that one can obtain naturally the non-commutative structure on KP equations from some algebraic data. To this end we explain the relation between Lie bialgebroids and quantum groupoids from work of Xu in [23], i.e. “ deformation of a bialgebroid ” (known as a quantum groupoid) induces a Lie bialgebroid as a classical limit. For this, during Sects. 2, 3, and 4 we review several algebraic structures such as bialgebroids, Lie bialgebras and Lie bialgebroids with their examples. Gathering together these concepts helps to understand the relation between them (especially for a non-familiar reader to these subjects) and provides the ingredients which we use later in the last sections. Then, in Sect. 5 we explain deformation of bialgebroids based on [23]. In Sect. 6 we use the result of Xu, to associate a Lie bialgebroid to the algebra of formal Pseudo-differential operators. In Sect. 7 we present non-commutative KP equations based on quantum groupoid associated to the algebra of formal Pseudo-differential operators. Solving KP equations, in particular showing existence and uniqueness of solutions of KP equations has been an interesting subject of study. For instance, in a recent work in [18] (also see references there in), the Cauchy problem of the KP hierarchy has been solved and formulated in several non-standard cases such as non-commutative case described by Moyal product. Now, with the algebraic construction in this paper, one can expect that the non-commutative KP equations can be solved more generally.

1.1 Preliminaries

Consider an associative algebra as a triple $(A,\mu,\eta )$ where A is a vector space over a field k,  the map $\mu :A\otimes A \rightarrow A$ denotes the multiplication, and $\eta :k\rightarrow A$ denotes the unit of the algebra. The maps $\mu$ and $\eta$ are linear and satisfy the following associative properties.

• $\mu \circ (\mu \otimes id)=\mu \circ (id\otimes \mu )$

• $\mu \circ (\eta \otimes id)=\mu \circ (id\otimes \eta )=id.$

If A is commutative then we also have $\mu \circ \ \tau _{A,A}=\mu$ where $\tau _{A,A}$ is the flip switching, that is $\tau _{A,A}(a\otimes a')=a'\otimes a.$ A morphism between algebras $(A,\mu,\eta )$ and $(A',{\mu },{\eta }')$ is a linear map $f:A\rightarrow A'$ such that ${\mu }'\circ (f\otimes f)=f\circ \mu$ and $f\circ \eta ={\eta }'$.

The notion of coalgebra is dual to the notion of algebra in the following sense.

Definition 1

A co-associative coalgebra is a triple $(C,\Delta,\varepsilon )$ where C is a vector space over a field k and the maps $\Delta :C\rightarrow C\otimes C$ and $\varepsilon :C\rightarrow k$ are linear satisfying the co-associative properties

• $(id\otimes \Delta )\circ \Delta =(\Delta \otimes id)\circ \Delta$

• $(id\otimes \varepsilon )\circ \Delta =(\varepsilon \otimes id)\circ \Delta =id.$

If C is co-commutative then we also have $\tau _{C,C}\circ \Delta =\Delta.$ A morphism between coalgebras $(C,\Delta,\varepsilon )$ and $(C',{\Delta }',{\varepsilon }')$ is a linear map $f:C\rightarrow C'$ such that $(f\otimes f)\circ \Delta ={\Delta }'\circ f$ and $\varepsilon ={\varepsilon }'\circ f.$

Definition 2

Suppose H is a vector space over the field k equipped with an algebra structure $(H,\mu,\eta )$ and a coalgebra structure $(H,\Delta,\varepsilon )$ then $(H,\mu,\eta,\Delta,\varepsilon )$ is a bialgebra verifying the compatibility conditions between these two structures i.e.,

• 1. The maps $\mu$ and $\eta$ are morphisms of coalgebras, or equivalently

• 2. the maps $\Delta$ and $\varepsilon$ are morphisms of algebras.

Given an algebra $(A,\mu,\eta )$ and a coalgebra $(C,\Delta ,\varepsilon )$ one can define a bilinear map called the convolution on the vector space Hom(C, A),  denoted by $\star :Hom(C,A)\rightarrow Hom(C,A),$ of linear maps from C to A. If f, g are such linear maps then the convolution $f\star g$ is the composition of the following maps

$$\begin{aligned} C{\mathop {\rightarrow }\limits ^{\Delta }}C\otimes C{\mathop {\rightarrow }\limits ^{f\otimes g}}A\otimes A{\mathop {\rightarrow }\limits ^{\mu }}A. \end{aligned}$$

Let $(H,\mu,\eta,\Delta,\varepsilon )$ be a bialgebra. An endomorphism S of H is called an antipode for the bialgebra H if $S\star id_H=id_H\star S =\eta \circ \varepsilon$.

Remark 1

A bialgebra does not necessarily have an antipode, but if it does, it has only one. Because if $S'$ is another antipode then we have

$$\begin{aligned} S=S\star (\eta \varepsilon )=S\star (id_H\star S')=(S\star id_H)\star S'=(\eta \varepsilon )\star S'=S'. \end{aligned}$$

Definition 3

A Hopf algebra is a bialgebra with antipode, and is denoted by $(H,\mu,\eta, \Delta,\varepsilon,S).$ A morphism of Hopf algebras is a morphism between the underlying bialgebras commuting with the antipode.

Example 1

Any algebra A is the quotient of a free algebra $K\{X\}$ where K is a field and X is a generating set. Since, it suffices to take any generating set X and $A=K\{X\}/I$ where I is a two sided ideal of $K\{X\}$.

Let $X=\{x_1,...,x_n\}$ and let I be the two sided ideal of $K\{x_1,...,x_n\}$ generated by elements of the form $x_ix_j-x_jx_i$ where i, j run over all integers from 1 to n. The quotient algebra $K\{x_1,...,x_n\}/I$ is isomorphic to the polynomial algebra $K[x_1,...,x_n]$ in n variables with coefficient in the ground field K.

The structure $(K[x],\mu,\eta,\Delta, \varepsilon,S)$ is a Hopf algebra where the algebra K[x] is called affine line and $\mu$ is the multiplication and $\eta$ is the unit of K[x]. The co-multiplication, co-unit and antipode are defined by

$$\begin{aligned}&\Delta :K[x]\rightarrow K[x',x'']; \ \ \ \ \Delta (x)=x'+x'' \\&\varepsilon :K[x]\rightarrow K; \ \ \ \ \varepsilon (x)=0 \\&S:K[x]\rightarrow K[x]; \ \ \ \ S(x)=-x. \end{aligned}$$

If $X=G$ is a finite group with commutative product, then K[G] is a bialgebra with co-multiplication

$$\begin{aligned}&\Delta :k[G]\rightarrow k[G]\otimes k[G]\\&\Delta (f)(x,y)=f(xy) \end{aligned}$$

and co-unit $\varepsilon (x)=1$. Also, if $X=G$ is a finite group with convolution product $(f\star g)(x)=\sum _{x_1,x_2}f(x_1)g(x_2)$, where $x_1x_2=x,$ then K[G] is a bialgebra with $\Delta (x)=x\otimes x$, and $\varepsilon (x)=1$.

Example 2

Let I be the two sided ideal of $K\{x,y\}$ generated by $xy-yx$, the quotient algebra $K\{x,y\}/I$ is isomorphic to the polynomial algebra K[x, y] with two variables x, y with coefficients in the ground field K. The algebra K[x, y] is called affine plane. In the affine plane if we define the two sided ideal $I_q=<xy-qyx>$ then we obtain a new algebra $K\{x,y\}/I_q=K_q[x,y]$ which is clearly non-commutative and is called quantum plane. In this algebra instead of the relation $xy=yx$ in commutative case we have $xy=qyx$. This is a bialgebra equipped with

$$\begin{aligned}&\Delta (x)=x\otimes x,\ \ \Delta (y)=y\otimes 1+x\otimes y \\&\varepsilon (x)=1,\ \ \varepsilon (y)=0. \end{aligned}$$

Example 3

Let V be a vector space. Define $T^0(V)=k, T^1(V)=V, T^n(V)=V^{\otimes n}$ (The tensor product of n copies of V) if $n>1$.

The canonical isomorphisms $T^n(V)\otimes T^m(V)\cong T^{n+m}(V)$ induce an associative product

$$\begin{aligned} (x_1\otimes...\otimes x_n)(x_{n+1}\otimes... \otimes x_{n+m})=x_1\otimes...\otimes x_n\otimes x_{n+1}\otimes...\otimes x_{n+m} \end{aligned}$$

on the vector space $T(V)=\bigoplus _{n\ge 0}T^n(V)$. The unit for this product is the image of the unit element 1 in $k=T^0(V)$. The vector space T(V) equipped with this algebra structure, is called the tensor algebra of V. If V is a vector space, the symmetric algebra S(V) is the quotient $S(V)=T(V)/I(V)$ of the tensor algebra T(V) by the two sided ideal I(V) generated by all elements $xy-yx$ where x, y run over V.

To any Lie algebra L we can assign an (associative) algebra U(L) called envelo** algebra of L,  with a morphism of Lie algebras $i_L:L\rightarrow Lie(U(L)).$ More precisely, let I(L) be the two sided ideal of the tensor algebra T(L) generated by all elements of the form $xy-yx-[x,y]$ where $x,y\in L$ and [., .] is the Lie bracket. We define $U(L):=T(L)/I(L)$. If L is a Lie algebra then the universal envelo** algebra of L,  denoted by U(L) is a Hopf algebra equipped with co-multiplication

$$\begin{aligned}&\Delta :U(L)\rightarrow U(L)\otimes U(L)\\&\Delta (X)=X\otimes 1+1\otimes X, \end{aligned}$$

and with co-unit and antipode

$$\begin{aligned}&\varepsilon :U(L)\rightarrow k, \ \ \varepsilon (X)=0, \\&S:U(L)\rightarrow U(L), \ \ S(X)=-X, \end{aligned}$$

for $X\in L.$ As an example, for the Lie algebra ${\mathfrak {sl}}(2)$ of traceless $2\times 2$ matrices, the envelo** algebra $U({\mathfrak {sl}}(2))$ is a Hopf algebra, see [12]. Moreover, we can construct a Hopf algebra $U_q({\mathfrak {sl}}(2))$ which is a one-parameter deformation of the envelo** algebra of the Lie algebra ${\mathfrak {sl}}(2)$.

2 Bialgebroids

Definition 4

A bialgebroid is a pair of algebras A and R together with an algebra homomorphism $\alpha :R\rightarrow A$, an algebra anti-homomorphism $\beta :R\rightarrow A$ such that the images of $\alpha$ and $\beta$ commute in A i.e. $\alpha (r_1)\beta (r_2)=\beta (r_2)\alpha (r_1)$, for each $r_1,r_2\in R$. By these two morphisms we can assign an (R, R)-bimodule structure on A in natural way by $r.a=\alpha (r)a$ and $a.r=\beta (r)a$, $r\in R,\ a\in A$. With this bimodule structure on A we can also consider $A\otimes _R A$ as an (R, R)-bimodule which is given as a left module by $r.(a_1\otimes a_2)=\alpha (r)a_1\otimes a_2$ and as a right module by $(a_1\otimes a_2).r=a_1\otimes \beta (r)a_2$, $r\in R,\ \ a_1,a_2\in A$.

We define the co-product $\Delta :A\rightarrow A\otimes _R A$ and the co-unit $\varepsilon :A\rightarrow R$ as (R, R)-bimodule maps satisfying the co-associativity axiom for $\Delta$ and co-unit as below

• $(id_A\otimes _R\Delta )\circ \Delta =(\Delta \otimes _R id_A)\circ \Delta$

• $\varepsilon (1_A)=1_R$, $\varepsilon \beta =\varepsilon \alpha =id_R$, and $(id_A\otimes \varepsilon )\circ \Delta =(\varepsilon \otimes id_A)\circ \Delta =id_A.$

The co-product $\Delta$ and the algebra structure on A should be compatible in the sense that the kernel of the following map

$$\begin{aligned} \Phi :A\otimes A\otimes A\rightarrow & {} A\otimes _RA\\ a_1\otimes a_2\otimes a_3\mapsto & {} (\Delta a_1)(a_1\otimes a_2) \end{aligned}$$

is a left ideal of $A\otimes A^{op}\otimes A^{op}$. Here we are using the fact that $A\otimes A$ acts on $A\otimes _RA$ from the right by right multiplications. Also we need the co-unit map is compatible with the algebra structure on A in the sense that the kernel of $\varepsilon$ is a left ideal of A. We denote a bialgebroid by $(A,R,\alpha,\beta,\Delta,\varepsilon )$ where A is called total algebra, R is called the base algebra, and the maps $\alpha ,\beta$ are called source map and target map respectively.

Recall that a groupoid G over a set $G_0$ is a set together with a pair of structure maps $G{\mathop {\rightarrow }\limits ^{s}}G_0$ and $G{\mathop {\rightarrow }\limits ^{t}}G_0,$ where s is called source and t is called target. Moreover, G is equipped with a product, identity, and an inversion, (for more details see for example [4]). In fact, we can think of an element $g\in G$ as an arrow (morphism) from $x=s(g)$ to $y=t(g)$ in $G_0.$

Example 4

If we consider a group G as a groupoid $\mathcal G$ over a set $X=\{x\}$ then the source map and target map are as constant maps $s,t:G\rightarrow \{x\},\ s(g)=t(g)=x$ for each $g\in G$. The arrows or elements of G are as $G=s^{-1}(x)\rightarrow t^{-1}(x)=G.$ We can consider them as $G{\mathop {\rightarrow }\limits ^{g}}G$ such that $h\mapsto gh$ for each $h,g\in G$. Also the unit for our groupoid is $\varepsilon :X\rightarrow G$ with $\varepsilon (x)=e$ (e is the identity element of the group G). The inverse map is $i:G\rightarrow G$ with $g\mapsto g^{-1}$. The multiplication map

$$\begin{aligned} m:\mathcal G_{(2)}=\mathcal G\times _{\{x\}}\mathcal G\rightarrow \mathcal G \end{aligned}$$

where $\mathcal G_{(2)}=\{(g,h)\in G\times G \mid s(g)=t(h)\}=G\times G$, is in fact the multiplication of the group G.

Now, we consider the space of smooth functions on G,  i.e. $C^\infty (G)$ as the total algebra and R the real numbers as the base algebra. We define $\alpha =s^*$ and $\beta =t^*$ such that

$$\begin{aligned} s^*=t^*:R=C^\infty (\{x\})\rightarrow C^\infty (G). \end{aligned}$$

It is clear that $(s^*f)(g)=f\circ s(g)=f(x)\in R.$ In other words $s^*$ assigns to each $r\in R$ the constant function $r\in C^\infty (G).$ So $s^*(r)=r$ similarly $t^*(r)=r$ for $r\in R.$

Moreover,

$$\begin{aligned} \varepsilon ^*:C^\infty (G)\rightarrow C^\infty (\{x\})=R \end{aligned}$$

with $(\varepsilon ^*f)(x)=f\circ \varepsilon (x)=f(e).$ It is clear that $(\varepsilon ^*id)=id(e)=1\in R$ hence $\varepsilon (1_{C^\infty (G)})=1$. Therefore $\varepsilon ^*$ can be considered as the co-unit. Finally,

$$\begin{aligned} m^*:C^\infty (G)\rightarrow C^\infty (G)\otimes C^\infty (G) \end{aligned}$$

defines the co-product, with $\Delta (f)=f\otimes f.$ Clearly $C^\infty (G)$ is an (R, R)-bimodule. It is easy to check that $\varepsilon ,\Delta$ are (R, R)-bimodule maps and they are compatible with the algebra structure on $C^\infty (G)$. Therefore, we constructed a bialgebroid $(C^\infty (G),R,s^*,t^*,\Delta ,\varepsilon )$ by the groupoid $\mathcal G$ which is the group G.

Recall that a smooth map $f:M\rightarrow N$ between smooth manifolds M and N is a local diffeomerphism (or étale map) if $(df)_x$ is an isomorphism for any $x\in M.$ An étale groupid is a groupoid with the source map s as a local diffeomorphism. For more details and examples see [19].

Example 5

Let $G_1{\mathop {\rightarrow }\limits ^{{\mathop {\rightarrow }\limits ^{s}}}}G_0$ be an étale groupoid. We define

$$\begin{aligned} \alpha :=s^*,\ \beta :=t^*:C^\infty (G_0)\rightarrow C^\infty (G_1) \end{aligned}$$

and we consider the co-unit as $\varepsilon ^*:C^\infty (G_1)\rightarrow C^\infty (G_0)$. The multiplication of groupoid is

$$\begin{aligned} m:G_2=G_1\times _{G_0}G_1\rightarrow G_1 \end{aligned}$$

where $G_2=\{(g,h)\in G_1 | s(g)=t(h)\}$. So, the pull-back of this multiplication, i.e.

$$\begin{aligned} \Delta =m^*:C^\infty (G_1)\rightarrow C^\infty (G_2)=C^\infty (G_1\times _{G_0}G_1)\cong C^\infty (G_1)\otimes _{C^\infty (G_0)}C^\infty (G_1) \end{aligned}$$

defines a co-product. Hence, $C^\infty (G_1)$ over $C^\infty (G_0)$ is a bialgebroid.

Example 6

Again consider the étale groupoid $G{\mathop {\rightarrow }\limits ^{{\mathop {\rightarrow }\limits ^{s}}}}G_0.$ The Connes algebra $C_c^\infty (G)$ of (smooth) complex (or real) functions with compact support on G (see [2, 6, 7]), together with the base algebra $(C_c^\infty (G))_0$ defines a bialgebroid, where $(C_c^\infty (G))_0$ is the subalgebra of $C_c^\infty (G)$ of functions with support in $G_0\subset G.$ This subalgebra is commutative and may be identified with the commutative algebra $C_c^\infty (G_0).$

Considering the inclusion $i:C_c^\infty (G_0)\hookrightarrow C_c^\infty (G)$ we take $\alpha =\beta =i$. The product is the convolution

$$\begin{aligned} (aa')(g'')=\sum _{gg'=g''}a(g)a'(g'),\ \ \forall a,a'\in C_c^\infty (G),\ g''\in G, \end{aligned}$$

where the sum is over all possible decompositions of $g''\in G$. We take the co-unit

$$\begin{aligned} \varepsilon :C_c^\infty (G)\rightarrow C_c^\infty (G_0) \end{aligned}$$

by

$$\begin{aligned} \varepsilon (a)(x)=\sum _{s(g)=x}a(g),\ \ \forall a\in C_c^\infty (G),\ x\in G_0. \end{aligned}$$

This sum is over all the elements g of G which satisfy $s(g)=x$. The coproduct $\Delta :C_c^\infty (G)\rightarrow C_c^\infty (G)\otimes _{C_c^\infty (G_0)}C_c^\infty (G)$ is defined as follows: Let $d:G\rightarrow G^s\times _{G_0}^sG$ be the diagonal open embedding, i.e. $d(g)=(g,g)$. This map gives the inclusion $C_c^\infty (G)\rightarrow C_c^\infty (G\times _{G_0}G)$. We define $\Delta$ as the composition of this inclusion with the inverse of the isomorphism

$$\begin{aligned}&\Omega :C_c^\infty (G)\otimes _{C_c^\infty (G_0)}C_c^\infty (G)\rightarrow C_c^\infty (G\times _{G_0}G) \\&\Omega (a\otimes a')(g,g')=a(g)a'(g'). \end{aligned}$$

(For the proof that $\Omega$ is an isomorphism see [20]).

If $a\in C_c^\infty (G)$ has the support in an open subset U of G which is so small that $s|_U$ is injective, then $\Delta (a)=a\otimes \xi =\xi \otimes a$ where $\xi$ is any smooth function with compact support in U which constantly equals 1 on the support of a. The functions $a\in C_c^\infty (G)$ which satisfy the above condition generate the linear space $C_c^\infty (G)$.

Example 7

1) If A is any algebra then there is a bialgebroid structure on $H=A\otimes A^{op}$ over A with

• 1. $\Delta :H\rightarrow H\otimes _A H$ , $a\otimes b\mapsto (a\otimes 1)\otimes (1\otimes b)$,

• 2. $\varepsilon :H\rightarrow A$, $a\otimes b\mapsto ab$,

• 3. $\alpha :A\rightarrow H$, $a\mapsto a\otimes 1$, $\beta :A\rightarrow H$, $a\mapsto 1\otimes a$.

2) For any finite dimensional algebra over k,  the algebra of k-linear maps from A to itself, i.e. $H=End_k(A),$ has a bialgebroid structure over A,  see [15].

Definition 5

A morphism between two bialgebroids $(A,R,\alpha ,\beta ,\Delta ,\varepsilon )$ and $(A',R',{\alpha }',{\beta }',{\Delta }',{\varepsilon }')$ consists of an algebra morphism $T:A\rightarrow A'$ and an algebra morphism $t:R\rightarrow R'$ which commute with all the structure maps.

As for Hopf algebras we expect that a Hopf algebroid is a bialgebroid with antipode.

Definition 6

The antipode in a bialgebroid $(A,R,\alpha ,\beta ,\Delta ,\varepsilon )$ is a bijective map $\tau :A\rightarrow A$ which has the following properties

• 1. $\tau$ is an algebra anti-isomorphism for A.

• 2. $\tau \beta =\alpha$

• 3. $\mu (\tau \otimes id)\Delta =\beta \varepsilon \tau :A\rightarrow A$, (it is the same as the definition of antipode in section 1.)

• 4. There exists a linear map $\gamma :A {\otimes }_R A \rightarrow A\otimes A$ with:

  • (a) $\gamma$ is a section for the natural projection $p:A\otimes A\rightarrow A{\otimes }_R A$

  • (b) $\mu (id\otimes \tau )\gamma \Delta =\alpha \varepsilon :A\rightarrow A$.

Proposition 1

The maps $\tau :A\rightarrow A$ and $id:R\rightarrow R$ define a bialgebroid morphism.

Proof

In the above definition, condition (1) means that for all $x,y\in A$, $\tau (xy)=\tau (y)\tau (x)$ and condition (2) means that $\tau (\beta (1_R))=\alpha (1_R)$ or $\tau (1)=1.$ On the other hand $(\tau \otimes \tau )\Delta =\Delta ^{op}\tau$, and $\varepsilon \circ \tau =\varepsilon.$ Moreover by condition (2) $\tau$ commutes with structure maps. Clearly id is a morphism of algebra and it commutes with the structure maps. $\square$

Proposition 2

Bialgebras and bialgebroids are equivalent over a field k as a base algebra.

Proof

For any bialgebra $(A,\mu ,\eta ,\Delta ,\varepsilon )$ we can consider $R=k$ , where k is the ground field for the algebra A,  and $\alpha ,\beta :k\rightarrow A$ by $\alpha (r)=r$ and $\beta (r)=r$, $(r\in k)$. Hence, $(A,k,\alpha ,\beta ,\Delta ,\varepsilon )$ is a bialgebroid because for $r,s\in k$ we have $\alpha (rs)=rs=\alpha (r)\alpha (s)$ so $\alpha$ is an algebra homomorphism, also $\beta (rs)=rs=sr=\beta (s)\beta (r)$ so $\beta$ is an anti-homomorphism, and finally $\alpha (r)\beta (s)=rs=sr=\beta (s)\alpha (r)$ so the images of $\alpha$ and $\beta$ commute. Conversely, if R is the field k the definition of a bialgebroid over k is reduced to a bialgebra over k, see [15]. $\square$

Example 8

1. If L is a Lie algebra then the universal envelo** algebra of L,  that is U(L) can be considered as a bialgebroid by the above Proposition 2. Consider Hopf algebras $SL_q(2)$ and $U_q({\mathfrak {sl}}(2))$ as mentioned before (there is a duality between SL(2) and $U({\mathfrak {sl}}(2)),$ see [12]). By the above proposition they are also examples of bialgebroids.

Let $(A,R,\alpha ,\beta ,\Delta ,\tau )$ be a Hopf algebroid over the field k of characteristic zero. And let $End_k R$ be the algebra of linear endomorphisms of R over k. When R acts on $End_k R$ from the left by left multiplication and acts from the right by right multiplication, we have $End_k R$ as an (R, R)-bimodule. Assume that R is a left A-module and moreover the representation $\rho :A\rightarrow End_k R$ is an (R, R)-bimodule map. We define $\phi _{\alpha },\phi _{\beta }:(A{\otimes }_R A)\otimes R\rightarrow A$ by

$$\begin{aligned} \phi _{\alpha }(x\otimes _Ry\otimes a)= & {} \rho (x)(a).y\\ \phi _{\beta }(x\otimes _Ry\otimes a)= & {} x.\rho (y)(a) \end{aligned}$$

where $x,y\in A, a\in R$.

Definition 7

Given a bialgebroid $(A,R,\alpha ,\beta ,\Delta ,\varepsilon )$, an anchor map is a representation $\rho :A\rightarrow End_k R$ which is an (R, R)-bimodule map such that

• 1. $\phi _{\alpha }(\Delta x\otimes a)=x\alpha (a)$ and $\phi _{\beta }(\Delta x\otimes a)=x\beta (a)$, $\forall x\in A, a\in R$,

• 2. $x(1_R)=\varepsilon (x)$, $\forall x\in A$.

For a bialgebra $R=k$ and $End_kR\cong R,$ clearly we can take the co-unit as the anchor. In general, for a bialgebroid, the existence of an anchor map is stronger than the existence of a co-unit.

Example 9

Let P be a smooth manifold and D be the algebra of differential operators on P. Let R be the algebra of smooth functions on P. Then $(D,R,\alpha ,\beta ,\Delta ,\varepsilon )$ is a bialgebroid where $\alpha =\beta$ is the embedding $R\rightarrow D$ and $\Delta :D\rightarrow D\otimes _R D$ is defined by

$$\begin{aligned} \Delta (d)(f,g)=d(fg), \forall d\in D \end{aligned}$$

and $f,g\in R.$ Also, the usual action of differential operators on $C^\infty (P)$ defines an anchor $\mu :D\rightarrow End_kR$. The co-unit $\varepsilon :D\rightarrow R$ is the natural projection to its 0-order part of a differential operator.

Below we see a construction of new bialgebroid from a given bialgebroid, which is called the twist construction. First we need the following proposition.

Proposition 3

(Xu, Proposition 4.2 in [23]) Let $(H, R, \alpha , \beta , m,\Delta , \varepsilon )$ be a Hopf algebroid with anchor $\mu.$ Let $F\in H\otimes _R H$ and define $\alpha _F,\beta _F:R\rightarrow H$ by

$$\begin{aligned} \alpha _F(a)=\phi _{\alpha }(F\otimes a),\ \beta _F(a)=\phi _\beta (F\otimes a),\ \forall a\in R, \end{aligned}$$

and for any $a,b\in R$, set $a*_Fb=\alpha _F(a)(b)$. Now assume that F satisfies:

$$(\Delta \otimes _R id )FF^{12}=(id\otimes _R\Delta )FF^{23} \ \ in\ \ H\otimes _RH\otimes _RH,$$

(1)

and

$$(\varepsilon \otimes _Rid)F=1_H, (id\otimes _R\varepsilon )F=1_H$$

(2)

where $F^{12}=F\otimes 1\in (H\otimes _R H)\otimes H),\ F^{23}=1\otimes F\in H\otimes (H\otimes _RH)$. Then

• $(R,*_F)$ is an associative algebra, and $1_R*_Fa=a*_F1_R=a,\ \forall a\in R$.

• $\alpha _F:R_F\rightarrow H$ is an algebra homomorphism, and $\beta _F:R_F\rightarrow H$ is an algebra anti-homomorphism. Here $R_F$ stands for the algebra $(R,*_F)$.

• $(\alpha _Fa)(\beta _Fb)=(\beta _Fb)(\alpha _Fa),\ \forall a,b\in R$.

Definition 8

Let $M_1$ and $M_2$ be given left H-modules. If the linear map

$$\begin{aligned} F^{\#}:M_1\otimes _{R_F}M_2\rightarrow & {} M_1\otimes _RM_2\\ (m_1\otimes _{R_F}m_2)\rightarrow & {} F.(m_1\otimes m_2) \end{aligned}$$

(for $m_1\in M_1$ and $m_2\in M_2$) is an isomorphism of vector spaces, we say that F is invertible. An element $F\in H\otimes _RH$ is called a twistor if it is invertible and satisfies Eq. (1) and (2) in above proposition.

Theorem 1

(Xu, Theorem 4.14 in [23]) Assume that $(H,R,\alpha ,\beta ,\Delta ,\varepsilon )$ is a bialgebroid with anchor $\mu$, and $F\in H\otimes _R H$ is a twistor. Then $(H,R_F,\alpha _F,\beta _F,\Delta _F,\varepsilon )$ is a bialgebroid which still admits $\mu$ as an anchor.

Example 10

([23]) Consider a smooth manifold P with the algebra D of differential operators on it, and $R=C^{\infty }(P).$ We denote the space of formal power series in $\hbar$ with coefficients in D by $D[[\hbar ]].$ The Hopf algebroid structure on D naturally can be extended to a Hopf algebroid structure on $D[[\hbar ]]$ over the base algebra $R[[\hbar ]],$ and it admits a natural anchor map. One can check that

$$\begin{aligned} F=1\otimes _R1+\hbar B_1+\dots \in D\otimes _R D[[\hbar ]] (\cong D[[\hbar ]]\otimes _{R[[\hbar ]]} D[[\hbar ]]) \end{aligned}$$

as a formal power series of bidifferential operatos is a twistor if and only if the multiplication on $R[[\hbar ]]$ defined by

$$\begin{aligned} f*_{\hbar }g=F(f,g), \ \ \forall f,g\in R[[\hbar ]] \end{aligned}$$

is associative with identity being the constant function 1,  in other words, $*_{\hbar }$ is a star product on P. Hence, the bracket

$$\begin{aligned} \{f,g\}=B_1(f,g)-B_1(g,f), \ \ \forall f,g \in C^{\infty }(P), \end{aligned}$$

defines a Poisson structure on P and $f*_{\hbar }g=F(f,g)$ is a deformation quantization of this Poisson structure.

Now, set $D_{\hbar }=D[[\hbar ]]$ equipped with the usual multiplication, $R_{\hbar }=R[[\hbar ]]$ equipped with the $*$-product defined above. Consider $\alpha _{\hbar }: R_{\hbar }\rightarrow D_{\hbar }$ and $\beta _{\hbar }:R_{\hbar }\rightarrow D_{\hbar }$ given by

$$\begin{aligned} \alpha _{\hbar }(f)g=f*_{\hbar }g, \ \ \ \beta _{\hbar }(f)g=g*_{\hbar }f, \ \ \ \forall f,g \in R. \end{aligned}$$

Moreover, consider the co-product $\Delta _{\hbar }:D_{\hbar }\rightarrow D_{\hbar }\otimes _{R_{\hbar }} D_{\hbar }$ defined by $\Delta _{\hbar }=F^{-1}\Delta F,$ and co-unit as the projection $D_{\hbar }\rightarrow R_{\hbar }.$ By Theorem 1 we obtain twisted Hopf algebroid $(D_{\hbar }, R_{\hbar }, \alpha _{\hbar }, \beta _{\hbar }, m, \Delta _{\hbar }, \varepsilon ).$ This twisted Hopf algebroid is called the quantum groupoid associated to the star product $*_{\hbar }.$

3 Lie Bialgebras

Definition 9

Let G be a Lie group and $\rho :G\rightarrow GL(V)$ be the representation of G on the vector space V. Also, let $d\rho :\mathfrak g\rightarrow End(V)$ be the infinitesimal representation of the group representation. then

• $\varphi :G\rightarrow V$ is a 1-cocycle on G if $\varphi (gh)=\varphi (g)+\rho (g)\varphi (h),$

• $\phi :\mathfrak g\rightarrow V$ is a 1-cocycle on $\mathfrak g$ if $\phi ([u,v])=d\rho (u)\phi (v)-d\rho (v)\phi (u).$

Note that if $\varphi$ is 1-cocycle on G then $\phi =d_e\varphi$ is 1-cocycle on $\mathfrak g$. Also, if G is simply connected then the 1-cocycle on $\mathfrak g$ can be integrated to 1-cocycle on G.

Definition 10

Let $\mathfrak g$ be a Lie algebra. A Lie bialgebra structure on $\mathfrak g$ is a linear map $\delta _{\mathfrak g}:\mathfrak g\rightarrow \mathfrak g\otimes \mathfrak g$, called co-commutator, such that

• 1. $\delta _{\mathfrak g}^*:\mathfrak g^*\otimes \mathfrak g^*\rightarrow \mathfrak g^*$ is a Lie bracket on $\mathfrak g^*$

• 2. $\delta _{\mathfrak g}$ is a 1-cocycle of $\mathfrak g$ with values in $\mathfrak g\otimes \mathfrak g.$

A homomorphism of Lie bialgebras $\varphi :\mathfrak g\rightarrow \mathfrak h$ is a homomorphism of Lie algebras such that $(\varphi \otimes \varphi )\circ \delta _{\mathfrak g}=\delta _{\mathfrak h}\circ \varphi.$

Definition 11

Let G be a Lie group. A Poisson Lie group is $(G,\Pi )$ where $\Pi$ is a Poisson structure such that $m:G\times G\rightarrow G$ is Poisson map. In this case $\Pi$ is called multiplicative.

Remark 2

A Poisson structure $\Pi$ is multiplicative if and only if $\Pi _{gh}=(l_g)_*\Pi _h+(r_h)_*)\Pi _g$, $g,h\in G$, where $l_g$ and $r_h$ are left and right translations, see [5].

Example 11

Any Lie group can be seen as a Poisson Lie group with $\Pi =0$.

Example 12

Let $\mathfrak g$ be a finite dimensional Lie algebra with Lie bracket $[.,.]_{\mathfrak g}$. Each $\mu \in \mathfrak g^*$ is a function on $\mathfrak g$. Given functions $f,g\in C^\infty (\mathfrak g^*)$ the new function $\{f,g\}\in C^\infty (\mathfrak g^*)$ evaluated at $\mu$ is $\{f,g\}(\mu )=\mu ([Df(\mu ),Dg(\mu )]_{\mathfrak g})$. Then $\mathfrak g^*$ as an abelian Lie group (with addition) is a Poisson Lie group with this Poisson structure (Poisson bracket).

The following proposition gives an example of Lie bialgebras.

Proposition 4

Let $(G,\Pi )$ be a Poisson Lie group with the Lie algebra $\mathfrak g$. Also, let $F:\mathfrak g\rightarrow \mathfrak g\wedge \mathfrak g$ be 1-cocycle, and $F^*$ Lie bracket then $(\mathfrak g,\mathfrak g^*)$ is a Lie bialgebra.

Proof

Let $(G,\Pi )$ be a Poisson Lie group, and $F=\Pi ^{(1)}$ the linear part of $\Pi$ at the point e. The map $\Pi ^{(1)}:\mathfrak g\rightarrow \mathfrak g\wedge \mathfrak g$ is linear Poisson structure on $\mathfrak g$ where $\mathfrak g=T_eG$. Then $F^*:\mathfrak g^*\wedge \mathfrak g^*\rightarrow \mathfrak g^*$ is a Lie algebra structure on $\mathfrak g^*$. In fact, for $\Pi :G\rightarrow \bigwedge ^2TG$, if we consider $\tilde{\Pi }:G\rightarrow \bigwedge ^2\mathfrak g$ for which $g\mapsto (r_g)_*\Pi _g$, then We can write $\Pi ^{(1)}=d_e\tilde{\Pi }$. Note that, as we mentioned in the previous remark, $\Pi$ is multiplicative if and only if $\tilde{\Pi }$ satisfies $\tilde{\Pi }(gh)=\tilde{\Pi }(g)+Ad_g\tilde{\Pi }(h).$ By above explanation, F satisfies $F([u,v])=ad_uF(v)-ad_vF(u).$ So F is 1-cocycle on $\mathfrak g$ with values in $\bigwedge ^2\mathfrak g$. That is if $(G,\Pi )$ is a Poisson Lie group then $F=d_e\tilde{\pi }:\mathfrak g\rightarrow \mathfrak g\wedge \mathfrak g$ such that

• 1. $F^*:\mathfrak g^*\wedge \mathfrak g^*\rightarrow \mathfrak g^*$ is Lie bracket. (equivalently, $\Pi$ is Poisson.)

• 2. F is a 1-cocycle. (equivalently, $\Pi$ is multiplicative.)

$\square$

Lemma 1

Let $\Lambda$ be a bivector in the Lie algebra, i.e. $\Lambda \in \bigwedge ^2\mathfrak g$ and $\Pi _g=(l_g)_*\Lambda -(r_g)_*\Lambda$. Then $\Pi$ is multiplicative.

Proof

We have

$$\begin{aligned} (l_g)_*\Pi _h+(r_h)_*\Pi _g= & {} (l_g)_*((l_h)_*\Lambda -(r_h)_*\Lambda )+(r_h)_*((l_g)_*\Lambda -(r_g)_*\Lambda )\\= & {} (l_{gh})_*\Lambda -(l_g)_*(r_h)_*\Lambda +(r_h)_*(l_g)_*\Lambda -(r_{gh})_*\Lambda \\= & {} \Pi _{gh}. \end{aligned}$$

Hence, by Remark 2 in above, $\Pi$ is multiplicative. $\square$

Proposition 5

$\Pi$ is Poisson structure if and only if $[\Lambda ,\Lambda ]\in \bigwedge ^3\mathfrak g$ is Ad-invariant.

Proof

By the above lemma we write $\Pi =\Lambda ^l-\Lambda ^r$. Moreover, a bivector field $\Pi$ is a Poisson bivector field if and only if $[\Pi ,\Pi ]=0$. So, we have

$$\begin{aligned}{}[\Pi ,\Pi ]=0\Leftrightarrow & {} [\Lambda ^l-\Lambda ^r,\Lambda ^l-\Lambda ^r]=[\Lambda ^l,\Lambda ^l]+[\Lambda ^r,\Lambda ^r]=0\\\Leftrightarrow & {} [\Lambda ,\Lambda ]^l=[\Lambda ,\Lambda ]^r\\\Leftrightarrow & {} Ad_g[\Lambda ,\Lambda ]=[\Lambda ,\Lambda ]. \end{aligned}$$

For more details see [5] or [22]. $\square$

Definition 12

If $\Pi =\Lambda ^l-\Lambda ^r$ then we call $(G,\Pi )$ a coboundary Poisson-Lie group.

If $\Lambda$ is such that $[\Lambda ,\Lambda ]$ is Ad-invariant then we call it r-matrix. In the special case, $[\Lambda ,\Lambda ]=0$ is called the classical Yang-Baxter equation and $\Lambda$ is called triangular r-matrix.

4 Lie Bialgebroids

Definition 13

A Lie algebroid over a manifold P is a vector bundle A over the manifold P together with a Lie algebra structure $[.,.]_A$ on the space $\Gamma (A)$ of smooth sections of A,  and a bundle map $\rho :A\rightarrow TP$ (called the anchor), such that

• The induced map $\Gamma (\rho ):\Gamma (A)\rightarrow \chi (P)$ is a Lie algebra homomorphism i.e., $\rho ([X,Y])=[\rho (X),\rho (Y)],\ (X,Y\in \Gamma (A))$ and

• For any $f\in C^\infty (P)$ and for any smooth sections X and Y of A the Leibniz identity holds i.e., $[X,fY]_A=f[X,Y]_A+(\rho (X).f)Y$.

Remark 3

• 1) The map $\Gamma (\rho )$ may be denoted simply by $\rho$ and also called the anchor.

• 2) For every $X\in \Gamma (A)$, we define A-Lie derivative operations on both $\Gamma (A)$ and $C^\infty (P)$ by $L_XY=[X,Y]_A$, and $L_Xf=\rho (X).f.$ Then the Leibniz identity is as a derivation

  $$\begin{aligned} L_X(fY)=f(L_XY)+(L_Xf)Y. \end{aligned}$$

Example 13

Let P be a Poisson manifold with Poisson Tensor $\Pi$. Then $T^*P$ carries a natural Lie algebroid structure, called the cotangent bundle Lie algebroid of the Poisson manifold P. The anchor map $\Pi ^{\#}:T^*P\rightarrow TP$ is defined by

$$\begin{aligned} \Pi ^{\#}_p:T_p^*P\rightarrow T_pP:\Pi ^{\#}(\xi )(\eta )=\Pi (\xi ,\eta ) \end{aligned}$$

for each $\xi ,\eta \in T^*_pP$ and the Lie bracket of 1-forms $\alpha$ and $\beta$ is given by

$$\begin{aligned}{}[\alpha ,\beta ]=L_{\Pi ^{\#}(\alpha )}\beta -L_{\Pi ^{\#}(\beta )}\alpha -d(\Pi (\alpha ,\beta )). \end{aligned}$$

Definition 14

Let $(A,\rho ,[.,.]_A)$ be a Lie algebroid over the manifold P,  and $\bigwedge ^*A^*$ be the exterior algebra of its dual $A^*$. Sections of $\bigwedge ^*A^*$ are called A-differential forms on P, or A-forms on P.

If $\omega \in \Gamma (\bigwedge ^kA^*)$, we say that $\omega$ is homogeneous, and its degree is $|\omega |=k$, and we call it as an A-k-form.

Definition 15

There is a differential operator which takes an A-k-form $\omega$ to an A-$(k+1)$-form $d_A\omega$ as below

$$\begin{aligned} d_A\omega (v_1,...,v_{k+1})= & {} \sum _i(-1)^{i+1}\rho (v_i).\omega (v_1,...,\hat{v_i},...,v_{k+1})\\ & \quad + \sum _{i<j}(-1)^{i+j}\omega ([v_i,v_j]_A,v_1,...,\hat{v_i},...,\hat{v_j},...,v_{k+1}) \end{aligned}$$

where $v_1,...,v_{k+1}\in \Gamma (A)$ are A-vector fields.

Remark 4

([4]) The Lie algebroid axioms for A implies that:

• 1. $d_A$ is $C^\infty (X)$-multilinear,

• 2. $d^2_A=0$,

• 3. $d_A(\omega _1\wedge \omega _2)=d_A\omega _1\wedge \omega _2+(-1)^{|\omega _1|}\omega _1\wedge d_A\omega _2$.

The triple $(\Gamma \bigwedge ^*A^*,\wedge ,d_A)$ forms a differential graded algebra, the same as the usual algebra of differential forms.

Proposition 6

There is a one-to-one correspondence between Lie algebroid structures on A and differential operators on $\Gamma (\bigwedge ^*A^*)$ satisfying properties 1-3 in above.

Proof

The anchor map is obtained from $d_A$ on functions by

$$\begin{aligned} \rho (v).f=(d_Af)(v),\ v\in \Gamma (A),\ f\in C^\infty (P). \end{aligned}$$

The lie bracket $[.,.]_A$ is determined by

$$\begin{aligned} i_{[v,w]_A}\omega= & {} \rho (v).\omega (w)-\rho (w).\omega (v)-d_A\omega (v,w)\\= & {} i_v(d_A(i_w\omega ))-i_w(d_A(i_v\omega ))-i_{(v\wedge w)}d_A\omega , \end{aligned}$$

where $(v,w\in \Gamma (A),\ \omega \in \Gamma (A^*))$. $\square$

Definition 16

The exterior differential algebra $\Gamma (\bigwedge ^*A^*),\wedge ,d_A)$ associated to a Lie algebroid $(A,\rho ,[.,.]_A)$ determines de Rham cohomology groups, called Lie algebroid cohomology of A or A-cohomology.

Example 14

Let $A=\mathfrak g$ be a Lie algebra i.e., a Lie algebroid over a one-point space, the cohomology of the differential complex

$$\begin{aligned} \mathbb {R}\longrightarrow \mathfrak g^*\longrightarrow \mathfrak g^*\wedge \mathfrak g^*\longrightarrow... \end{aligned}$$

is the standard Lie algebra cohomology $(\bigwedge ^*\mathfrak g^*,\wedge ,d\mathfrak g)$.

Example 15

Let $A=TP$ be a tangent bundle of a manifold P,  the cohomology computed by $(\Gamma (\bigwedge ^*A^*),\wedge ,d_A)=(\Omega ^*(P),\wedge ,d_{de Rham})$ is the usual de Rham cohomology.

Definition 17

Lie algebroid multivector fields or A-multivector fields are sections of the exterior algebra $\bigwedge ^*A$ of a Lie algebroid $(A,\rho ,[.,.]_A)$. If $v\in \Gamma (\bigwedge ^kA)$, then v is called homogeneous with degree $|v|=k$.

Remark 5

([4])

• 1) The extension of $[.,.]_A$ to arbitrary A-multivector fields by setting it on homogeneous A-multivector fields v, w is

  $$\begin{aligned} i_{[v,w]_A}\omega =(-1)^{(|v|-1)(|w|-1)}i_v(d_A(i_w\omega ))-i_w(d_A(i_v\omega ))-(-1)^{|v|-1}i_{(v\wedge w)}d_A\omega \end{aligned}$$

• 2) The bracket $[.,.]_A$ on A-multivector fields has the following properties.

• 1. $[.,.]_A$ allows us to extend the A-Lie derivative operation defined for A-vector fields in definition of Lie algebroids to arbitrary elements of $v,w\in \Gamma (\bigwedge ^*A),$ i.e. $L_vw:=[v,w]_A$.

• 2. $[.,.]_A$ is a graded Lie algebra structure i.e.

  $$\begin{aligned} {[}v,w]_A=-(-1)^{(|v|-1)(|w|-1)}[w,v]_A \end{aligned}$$

• 3. $[.,.]_A$ satisfies a super-Jacobi identity:

  $$\begin{aligned} {[}v,[w,y]_A]_A+(-1)^{(|y|-1)(|v|+|w|)}[y,[v,w]_A]_A+(-1)^{(|v|-1)(|w|+|y|)}[w,[y,v]_A]_A=0 \end{aligned}$$

• 4. $[.,.]_A$ satisfies a super-Leibniz identity:

  $$\begin{aligned} {[}v,w\wedge y]_A=[v,w]_A\wedge y+(-1)^{(|v|-1)|w|}w\wedge [v,y]_A. \end{aligned}$$

Definition 18

The triple $(\Gamma (\bigwedge ^*A),\wedge ,[.,.]_A)$ is called the Gerstenhaber algebra of the Lie algebroid $(A,\rho ,[.,.]_A)$, or just the A-Gerstenhaber algebra. We will refer to the bracket $[.,.]_A$ on $\Gamma (\bigwedge ^*A)$ as the A-Gerstenhaber bracket.

In general,

Definition 19

A Gerstenhaber algebra $(\mathfrak a,\wedge ,[.,.])$ is a graded vector space $\mathfrak a=\mathfrak a_0\oplus \mathfrak a_1\oplus...$ together with a super commutative associative multiplication of degree 0, $\mathfrak a_i\wedge \mathfrak a_j\subseteq \mathfrak a_{i+j}$ and a super-Lie algebra structure of degree $-1$, $[\mathfrak a_i,\mathfrak a_j]\subseteq \mathfrak a_{i+j-1}$, satisfying the super-Leibniz identity

$$\begin{aligned}{}[a,b\wedge c]=[a,b]\wedge c+(-1)^{(|a|-1)|b|}b\wedge [a,c]. \end{aligned}$$

Remark 6

From a Lie algebroid structure on A,  i.e. $(A,\rho ,[.,.]_A)$, we obtain a differential algebra structure on $\Gamma (\bigwedge ^*A^*)$, i.e. $(\Gamma (\bigwedge ^*A^*),\wedge ,d_A)$, and from that we get a Gerstenhaber algebra structure on $\Gamma (\bigwedge ^*A)$, i.e. $(\Gamma (\bigwedge ^*A),\wedge ,[.,.]_A)$.

Definition 20

For the tangent bundle Lie algebroid $(A,\rho ,[.,.]_A)=(TP,id,[.,.])$, $d_A$ is the de Rham differential and A-Gerstenhaber bracket is usually called the Schouten-Nijenhuis bracket on multivector fields.

A bivector field $\Pi \in \Gamma (\bigwedge ^2TP)$ is called a Poisson bivector field if and only if $[\Pi ,\Pi ]=0$. This is equivalent to $d^2_{\Pi }=0$ for the differential operator $d_{\Pi }:=[\Pi ,.]$.

The notion of Poisson structure naturally generalizes to arbitrary Lie algebroid as follows:

Let $(A,\rho ,[.,.]_A)$ be a Lie algebroid over P. An element $\Pi \in \Gamma (\bigwedge ^2A)$ is called an A-Poisson bivector field when $[\Pi ,\Pi ]_A=0$, where $[.,.]_A$ is the A-Gerstenhaber bracket.

Example 16

Consider $A=\mathfrak g$ as a Lie algebra, and a $\mathfrak g$-Poisson bivector field $\Pi \in \mathfrak g\wedge \mathfrak g$ to a left invariant Poisson structure on the underlying Lie group G. The equation $[\Pi ,\Pi ]_{\mathfrak g}=0$ is called the classical Yang-Baxter equation.

Remark 7

([4]) The push-forward $\rho _*\Pi$ of an A-Poisson bivector field $\Pi$ by the anchor $\rho :\Gamma (\bigwedge ^2A)\rightarrow \Gamma (\bigwedge ^2TP)$ defines an ordinary Poisson structure on the manifold P. By the Jacobi identity, an arbitrary (not necessarily Poisson) element $\Theta \in \Gamma (\bigwedge ^2A)$ satisfies

$$\begin{aligned} d^2_{\Theta }+\left[\frac{1}{2}[\Theta ,\Theta ]_A,.\right]_A=0 \end{aligned}$$

which is similar to the equation for a flat connection.

In the first section we saw how one defines the universal envelo** algebra for a Lie algebra. Now we want to see the definition for Lie algebroids.

Definition 21

If $(A,\rho ,[.,.]_A)$ is a Lie algebroid over the manifold P,  then the $C^\infty (P)$-module $C^\infty (P)\oplus \Gamma (A)$ is a Lie algebra over $\mathbb R$ with the Lie bracket

$$\begin{aligned} {[}f+X,g+Y]=(\rho (X).g-\rho (Y).f)+[X,Y]. \end{aligned}$$

Considering the universal envelo** algebra of this Lie algebra as $U=U(C^\infty (P)\oplus \Gamma (A))$, we denote $f'$ and $X'$ as the canonical images of $f\in C^\infty (P)$ and $X\in \Gamma (A)$ in U. If I is the two-sided ideal of U generated by all elements of the form $(fg)'-f'g'$ and $(fX)'-f'X'$ then the universal envelo** algebra of the Lie algebroid A is defined by $U(A)=U/I$.

Theorem 2

The universal envelo** algebra U(A) of a Lie algebroid A admits a co-commutative bialgebroid structure.

Proof

Let $R=C^\infty (P)$, and $\alpha =\beta :R\rightarrow U(A)$ be the natural embedding. For the co-product define

$$\begin{aligned}&\Delta (f)=f\otimes _R1,\ \ \forall f\in R \\&\Delta (X)=X\otimes _R1+1\otimes _RX,\ \ \forall X\in \Gamma (A). \end{aligned}$$

This formula extends to a co-product $\Delta :U(A)\rightarrow U(A)\otimes _RU(A)$ by the compatibility condition.

The co-unit map is the projection $\varepsilon :U(A)\rightarrow R$. Also, the map $\mu :U(A)\rightarrow End_kR$ defined by

$$\begin{aligned} (\mu x)(f)=(\rho x)(f),\ \ \forall x\in UA,\ f\in R \end{aligned}$$

is the anchor for our bialgebroid where $\rho : U(A)\rightarrow TP$ is the algebra homomorphism extending the anchor of the Lie algebroid A. Hence, $(U(A),R,\alpha ,\beta ,m,\Delta ,\varepsilon )$ is a co-commutative bialgebroid with anchor $\mu.$ $\square$

The notion of Lie bialgebroids is a natural generalization of Lie bialgebras.

Definition 22

A Lie bialgebroid is a dual pair $(A,A^*)$ of vector bundles equipped with Lie algebroid structures such that the differential d on $\bigoplus _k\Gamma (\bigwedge ^kA^*)$ is defined by

$$\begin{aligned} d:\Gamma \left( \bigwedge ^kA^*\right) \rightarrow \Gamma \left( \bigwedge ^{k+1}A^*\right) \end{aligned}$$

and the differential $d_*$ on $\bigoplus _k\Gamma (\bigwedge ^k(A^*)^*)\cong \bigoplus _k\Gamma (\bigwedge ^kA)$ is defined by

$$\begin{aligned} d_*:\Gamma \left( \bigwedge ^kA\right) \rightarrow \Gamma \left( \bigwedge ^{k+1}A\right) \end{aligned}$$

coming from the structure on $A^*$ is a derivation of the Schouten bracket on $\bigoplus _k\Gamma (\bigwedge ^kA)$, equivalently, $d_*$ is a derivation for sections of A i.e.,

$$\begin{aligned} d_*[X,Y]=[d_*X,Y]+[X,d_*Y],\ \ \forall X,Y\in \Gamma (A). \end{aligned}$$

In other words, $(\bigoplus _k\Gamma (\bigwedge ^kA),\wedge ,[.,.],d_*)$ is a differential Gerstenhaber algebra.

Example 17

For a Poisson manifold $(P,\Pi ),$ the dual pair $(TP,T^*P)$ where TP is the standard tangent bundle and $T^*P$ is the cotangent Lie algebroid, together with

$$\begin{aligned} d_{\Pi }[X,Y]=[d_{\Pi }X,Y]+[X,d_{\Pi }Y] \end{aligned}$$

is a Lie bialgebroid. As before we saw, $d_{\Pi }=[\Pi ,.]$ which is obtained from the graded Jacobi identity, see [16].

Proposition 7

In fact, a Lie bialgebroid is equivalent to a strong differential Gerstenhaber algebra structure on $\bigoplus _k\Gamma (\bigwedge ^k(A)$. (See Proposition 2.3 in [24]).

Remark 8

([23]) In a Lie bialgebroid $(A,A^*)$, the base P has a natural Poisson structure as

$\{f,g\}=<df,d_*g>,\ \ \forall f,g\in C^\infty (P)$ which satisfies $[df,dg]=d\{f,g\}$.

The same as Lie bialgebras, a useful method of constructing Lie bialgebroids is using r-matrices. Recall that an r-matrix is a section $\Lambda \in \Gamma (\bigwedge ^2A)$ satisfying

$$\begin{aligned} L_X[\Lambda ,\Lambda ]=[X,[\Lambda ,\Lambda ]]=0,\ \ \forall X\in \Gamma (A). \end{aligned}$$

An r-matrix $\Lambda$ defines a Lie bialgebroid, where the differential $d_*:\Gamma (\bigwedge ^*A)\rightarrow \Gamma (\bigwedge ^{*+1}A)$ is $d_*=[.,\Lambda ]$. The bracket on $\Gamma (A^*)$ is

$$\begin{aligned} {[}\xi ,\eta ]=L_{\Lambda ^{\#}\xi }\eta -L_{\Lambda ^{\#}\eta }\xi -d[\Lambda (\xi ,\eta )]. \end{aligned}$$

The anchor is $\rho \circ \Lambda ^{\#}:A^*\rightarrow TP$, where $\Lambda ^{\#}$ is the bundle map $A^*\rightarrow A$ with

$$\begin{aligned} \Lambda ^{\#}(\xi )(\eta )=\Lambda (\xi ,\eta ),\ \forall \xi ,\eta \in \Gamma (A^*). \end{aligned}$$

This Lie bialgebroid is called a coboundary Lie bialgebroid, and it is called a triangular Lie bialgebroid if $[\Lambda , \Lambda ]=0$. If $\Lambda$ is also of constant rank then we call it as a regular triangular Lie bialgebroid.

Remark 9

([23]) When P has only one point, that means A is a Lie algebra, then $L_X[\Lambda ,\Lambda ]=[X,[\Lambda ,\Lambda ]]=0$ is equivalent to that $[\Lambda ,\Lambda ]$ is Ad-invariant, so $\Lambda$ is an r-matrix.

If A is the tangent bundle TP with the standard Lie algebroid structure, $L_X[\Lambda ,\Lambda ]=[X,[\Lambda ,\Lambda ]]=0$ is equivalent to $[\Lambda ,\Lambda ]=0$ i.e., $\Lambda$ is a Poisson tensor.

5 Deformation of Bialgebroids

We consider a topological bialgebra $(A,\mu ,\eta ,\Delta ,\varepsilon )$ where A is a module over the ring $k[[\hbar ]]$ and $\mu :A\otimes A\rightarrow A, \ \ \eta :k[[\hbar ]]\rightarrow A, \ \Delta :A\rightarrow A\otimes A$ and $\varepsilon :A\rightarrow k[[\hbar ]]$ are $k[[\hbar ]]$-linear maps. Similarly, a topological bialgebroid is a bialgebroid which is a module over $k[[\hbar ]]$ and all structure maps are $k[[\hbar ]]$-linear maps.

Remember from Sect. 2, using twist construction, in Theorem 1 one obtains a new bialgebroid from a given bialgebroid. Also remember the Example 10, in which using twisted construction we obtained twisted Hopf algebroid $(D_{\hbar }, R_{\hbar }, \alpha _{\hbar }, \beta _{\hbar }, m, \Delta _{\hbar }, \varepsilon )$ which is called the quantum groupoid associated to the star product $*_{\hbar }.$

Definition 23

A deformation of a bialgebroid $(A,R,\alpha ,\beta ,m,\Delta ,\varepsilon )$ over a field k is a topological bialgebroid $(A_{\hbar },R_{\hbar },\alpha _{\hbar },\beta _{\hbar },m_{\hbar },\Delta _{\hbar },\varepsilon _{\hbar })$ over the ring $k[[\hbar ]]$ of formal power series in $\hbar$ such that

• 1. $A_\hbar$ is isomorphic to $A[[\hbar ]]$ as $k[[\hbar ]]$-module with identity $1_A$, and $R_{\hbar }$ is isomorphic to $R[[\hbar ]]$ as $k[[\hbar ]]$-module with identity $1_R$,

• 2. $\alpha _{\hbar }=\alpha (mod \hbar ),\ \beta _{\hbar }=\beta (mod \hbar ),\ m_{\hbar }=m(mod \hbar ),\ \varepsilon _{\hbar }=\varepsilon (mod \hbar )$,

• 3. $\Delta _{\hbar }=\Delta (mod \hbar )$.

In this case, we simply say that the quotient $A_{\hbar }/\hbar A_{\hbar }$ is isomorphic to A as a bialgebroid.

Lemma 2

(Xu, [23]) By the conditions (1), (2) in the above definition, $A_{\hbar }\otimes _{R_{\hbar }}A_{\hbar }/\hbar (A_{\hbar }\otimes _{R_{\hbar }}A_{\hbar })$ is isomorphic to $A\otimes _RA$ as k-module.

Definition 24

A quantum universal envelo** algebroid (or QUE algebroid), also called a quantum groupoid is a deformation of the standard bialgebroid $(UA,R,\alpha ,\beta ,m,\Delta ,\varepsilon )$ of a Lie algebroid A.

Suppose $(U_{\hbar }A,R_{\hbar },\alpha _{\hbar },\beta _{\hbar },m_{\hbar },\Delta _{\hbar },\varepsilon _{\hbar })$ is a quantum groupoid. Note that $U_{\hbar }A=UA[[\hbar ]]$ and $R_{\hbar }=R[[\hbar ]]$. Then $R_{\hbar }$ defines a star product on the base manifold P for the Lie algebroid A,  so that

$$\begin{aligned} \{f,g\}=lim_{\hbar \rightarrow 0}\frac{1}{\hbar }(f*_{\hbar }g-g*_{\hbar }f) \end{aligned}$$

(with $f,g\in R$) is a Poisson structure on the base P. Define

• $\delta f:=lim_{\hbar \rightarrow 0}\frac{1}{\hbar }(\alpha _{\hbar }f-\beta _{\hbar }f)\in UA,\ \forall f\in R$,

• $\Delta ^1X:=lim_{\hbar \rightarrow 0}\frac{1}{\hbar }(\Delta _{\hbar }X-(1\otimes _{R_{\hbar }}X+X\otimes _{R_{\hbar }}1))\in UA\otimes _R UA,\ \forall X\in \Gamma (A)$, and

• $\delta X:=\Delta ^1X-(\Delta ^1X)_{21}\in UA\otimes _RUA.$

For any $f,g\in R,x,y\in UA$ write

$$\begin{aligned}&\alpha _{\hbar }f=f+\hbar \alpha _1f+\hbar ^2\alpha _2f+O(\hbar ^3), \\&\beta _{\hbar }f=f+\hbar \beta _1f+\hbar ^2\beta _2f+O(\hbar ^3), \\&f*_{\hbar }g=fg+\hbar B_1(f,g)+O(\hbar ^2), \\&x*_\hbar y=xy+\hbar m_1(x,y)+O(\hbar ^2) \end{aligned}$$

where $\alpha _1f,\beta _1f,\alpha _2f,\beta _2f,m_1(x,y)$ are elements in U(A). So, $\{f,g\}=B_1(f,g)-B_1(g,f)$ and $\delta f=\alpha _1f-\beta _1f$.

Proposition 8

(Xu, [23]) For any $f,g\in R$ and $X\in \Gamma (A)$ we obtain

• 1. $\delta f\in \Gamma (A)$ and $\delta X\in \Gamma (\bigwedge ^2A)$,

• 2. $\delta (fg)=f\delta g+g\delta f$,

• 3. $\delta (fX)=f\delta X+\delta f\wedge X$,

• 4. $[\delta f,g]=\{f,g\}$,

• 5. $\delta ^2f=0.$

By properties (1)–(3) we can extend $\delta$ to a well-defined degree 1 derivation $\delta :\Gamma (\bigwedge ^*A)\rightarrow \Gamma (\bigwedge ^{*+1}A).$ Then the algebra $(\bigoplus \Gamma (\bigwedge ^*A),\wedge ,[.,.],\delta )$ is a strong differential Gerstenhaber algebra. Since, Considering the above proposition, it suffices to show that $\delta$ is a derivation with respect to [., .], and $\delta ^2=0$. The proof of these two are in Propositions 5.13 and 5.14 in [23].

By above proposition the dual pair $(A,A^*)$ is a Lie bialgebroid, which is called the classical limit of the quantum groupoid $U_{\hbar }A$.

Therefore, the following theorem is obtained.

Theorem 3

(Xu, [23]) A quantum groupoid $(U_{\hbar }A,R_{\hbar },\alpha _{\hbar },\beta _{\hbar },m_{\hbar },\Delta _{\hbar },\varepsilon _{\hbar })$ naturally induces a Lie bialgebroid $(A,A^*)$ as a classical limit. The induced Poisson structure of this Lie bialgebroid on the base manifold P coincides with the one obtained as the classical limit of the base $*$-algebra $R_{\hbar }$.

In [10] it was shown that every Lie bialgebra is quantizable. Now, one can ask if every Lie bialgebroid is quantizable.

Definition 25

A Quantization of a Lie bialgebroid $(A,A^*)$ is a quantum groupoid $(U_{\hbar }A,R_{\hbar },\alpha _{\hbar },\beta _{\hbar },m_{\hbar },\Delta _{\hbar },\varepsilon _{\hbar })$ whose classical limit is $(A,A^*)$.

Remark 10

In [13], Kontsevich has shown that given a Poisson structure $\{.,.\}$ on a Poisson manifold P one can find a $*$-product such that

$$\begin{aligned} f*g=fg+\{f,g\}t+\dots \end{aligned}$$

So, we can say that the Lie bialgebroid $(TP,T^*P)$ associated to a Poisson manifold P is always quantizable.

In a special case, Xu in [23] showed that any regular triangular Lie bialgebroid is quantizable. In general case, in [3] it was shown that every Lie bialgebroid is quantizable.

6 The Lie Bialgebroid of Formal Pseudo-differential Operators

In this section we are going to indicate the Theorem 3 for formal pseudo-differential operators. In other words, we see how naturally the algebra of formal pseudo-differential operators can be considered as a Lie bialgebroid. First we recall some definitions. Here we consider the formal pseudo-differential operators in a general algebraic setting described for instance in [1].

Let A be a commutative k–algebra with unit 1,  where k is a field of the zero-characteristic. Assume that A is equipped with a derivation, i.e. there is a k–linear map $D: A\rightarrow A$ satisfying the Leibnitz rule $D(f.g)=(Df).g+f.(Dg)$ for all $f,g\in A.$ We say that an element is constant if $Df=0.$ Consider $\xi$ as a formal variable not in A. The algebra of symbols over A is the vector space

$$\begin{aligned} \Psi _{\xi }(A)= \left\{ P_{\xi }=\sum _{\nu \in \mathbb {Z}} a_{\nu }\xi ^{\nu } \mid a_{\nu }\in A, a_{\nu }=0 \ \ for \ \ \nu \gg 0\right\} \end{aligned}$$

which is equipped with the associative multiplication $\circ$ defined by

$$\begin{aligned} P_{\xi }\circ Q_{\xi }=\sum _{k\ge 0} \frac{1}{k!}\frac{\partial ^k}{\partial \xi ^k}D^kQ_{\xi }. \end{aligned}$$

Note that the algebra A is included in $\Psi _{\xi }(A),$ and for $f\in A$ and $\nu \in Z$ the multiplication $\circ$ is given by $\xi ^{\nu }\circ f=\sum _{k\ge 0} \frac{\nu (\nu -1)\dots (\nu -k+1)}{k!}(D^kf)\xi ^{\nu -k}.$

Definition 26

The algebra of formal pseudo-differential operators over A is the vector space

$$\begin{aligned} \Psi (A)=\left\{P=\sum _{\nu \in \mathbb {Z}} a_{\nu }D^{\nu } \mid a_{\nu }\in A, a_{\nu }=0 \ \ for \ \ \nu \gg 0\right\} \end{aligned}$$

which is equipped with the unique multiplication which makes the map $\sum _{\nu \in \mathbb {Z}}a_{\nu }\xi ^{\nu }\mapsto \sum _{\nu \in \mathbb {Z}} a_{\nu }D^{\nu }$ an algebra homomorphism. The algebra $\Psi (A)$ is associative but not commutative. It is a Lie algebra over k with the Lie bracket $[P,Q]:=PQ-QP.$

Now, consider $A=\mathbb {R}^n$ as the commutative $\mathbb {R}$–algebra with unit 1,  over the field $\mathbb {R},$ equipped with the Hadamard product (entrywise product) which for $n=1$ is the usual product on $\mathbb {R}.$

Theorem 4

There is a natural Lie bialgebroid structure on the algebra of formal pseudo-differential operators $\Psi (\mathbb {R}^n)$ as the classical limit of the quantum groupoid associated to it.

Proof

Remember from above definition that the algebra A is included in $\Psi (A).$ Moreover, if we consider $\mathbb {R}^n$as a manifold, then we can define the bundle map $\pi : \Psi (\mathbb {R}^n)\rightarrow \mathbb R^n$ as the natural projection. Hence, $\Psi (\mathbb {R}^n)$ is a Lie algebroid with the Lie bracket defined in above definition. By Definition (21), $U(\Psi (\mathbb {R}^n))=U(\Gamma (\Psi (\mathbb {R}^n))\oplus C^\infty (\mathbb R^n))$ is the universal envelo** algebra of our Lie algebroid. Therefore, by Theorem 2, we have the bialgebroid $(U(\Psi (\mathbb {R}^n)),C^\infty (\mathbb R^n),\alpha ,\beta ,\Delta ,\varepsilon )$ in which $\alpha =\beta =i:C^\infty (\mathbb R^n)\hookrightarrow U(\Psi (\mathbb {R}^n))$ are the natural embeddings, and $\varepsilon :U(\Psi (\mathbb {R}^n))\rightarrow C^\infty (\mathbb R^n)$ is the projection. Moreover, we have

$$\begin{aligned} \Delta (f)= & {} f\otimes _{C^\infty (\mathbb R^n)}1, \ \ \ \forall f\in R=C^\infty (\mathbb R^n)\\ \Delta (X)= & {} X\otimes _{C^\infty (\mathbb R^n)}1+1\otimes _{C^\infty (\mathbb R^n)}X,\ \ \ \forall X\in \Gamma (U(\Psi (\mathbb {R}^n))). \end{aligned}$$

Now, consider $(U_{\hbar }(\Psi (\mathbb {R}^n)), C^\infty (\mathbb R^n)_{\hbar },\alpha _{\hbar },\beta _{\hbar },\Delta _{\hbar },\varepsilon _{\hbar })$ as the quantum universal envelo** algebra of the above bialgebroid. Here we define $\alpha _{\hbar },\beta _{\hbar }:C^\infty (\mathbb R^n)_{\hbar }\rightarrow U_{\hbar }(\Psi (\mathbb {R}^n))$ by

$$\begin{aligned} \alpha _{\hbar }f= & {} f+\hbar \sum _{i=1}^m\partial _i f+\hbar ^2\alpha _2f+O\left( \hbar ^3\right) \\ \beta _{\hbar }f= & {} f+\hbar \left( -\sum _{i=m}^n\partial _i f\right) +\hbar ^2\beta _2f+O\left( \hbar ^3\right) , \end{aligned}$$

where $\partial _i$ means differentiation with respect to the i–th variable and m is fixed such that $1\le m\le n.$

Also, if we consider the coordinates $(q_i,p_i)$ on $T\mathbb {R}^n$ we define

$$\begin{aligned} f*_{\hbar } g= & {} fg+\hbar \sum _{i=1}^n \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}+O(\hbar ^2),\\ x*_{\hbar }y= & {} xy+\hbar m_1(x,y)+O(\hbar ^2), \end{aligned}$$

moreover, we consider $\varepsilon _{\hbar }(x) = \varepsilon (x)+\hbar \varepsilon _1(x)+O(\hbar ^2),$ ($\varepsilon _1(x)\in C^\infty (\mathbb R^n)$), and $\Delta _{\hbar } X$ with

$$\begin{aligned} \Delta ^1 X= & {} lim_{\hbar \rightarrow 0}\frac{1}{\hbar }(\Delta _{\hbar } X-(1\otimes _{C^\infty (\mathbb R^n)_{\hbar }}X+X\otimes _{C^\infty (\mathbb R^n)_{\hbar }}1))\\\in & {} U(\Psi (\mathbb {R}^n))\otimes _{C^\infty (\mathbb R^n)}U(\Psi (\mathbb {R}^n)),\ \forall X\in \Gamma (U(\Psi (\mathbb {R}^n))), \end{aligned}$$

for any $f,g\in C^\infty (\mathbb R^n),\ x,y\in U(\Psi (\mathbb {R}^n))$, where $\alpha _2f, \beta _2f,m_1(x,y)$ are elements in $U(\Psi (\mathbb {R}^n))$.

Therefore, we have

$$\begin{aligned} \{f,g\}=\, & {} lim_{\hbar \rightarrow 0}\frac{1}{\hbar }(f*_{\hbar } g-g*_{\hbar }f)\\= \,& {} lim_{\hbar \rightarrow 0}\frac{1}{\hbar }\left( fg+\hbar \sum _{i=1}^n \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}+O\left( \hbar ^2\right) -gf-\hbar \sum _{i=1}^n \frac{\partial g}{\partial q_i}\frac{\partial f}{\partial p_i}-O\left( \hbar ^2\right) \right) \\=\, & {} \sum _{i=1}^n \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}-\frac{\partial g}{\partial q_i}\frac{\partial f}{\partial p_i} \end{aligned}$$

which is the standard Poisson bracket on the base manifold $\mathbb R^n$. Moreover,

$$\begin{aligned} \delta f=\, & {} lim_{\hbar \rightarrow 0}\frac{1}{\hbar }(\alpha _{\hbar }f-\beta _{\hbar }f)\\= \,& {} lim_{\hbar \rightarrow 0}\frac{1}{\hbar }\left( \hbar \sum _{i=1}^m\partial _i f -\hbar \left( -\sum _{i=m}^n\partial _i f\right) +\hbar ^2\alpha _2f-\hbar ^2\beta _2f+O\left( \hbar ^3\right) -O\left( \hbar ^3\right) \right) \\=\, & {} \sum _{i=1}^m\partial _i f-\left( -\sum _{i=m}^n\partial _i f\right) \\= \, & {} df. \end{aligned}$$

Thus we obtain the ordinary derivation which clearly satisfies all the properties of $\delta$ in Proposition 8. Therefore, by Theorem 3 the quantum groupoid $(U_{\hbar }(\Psi (\mathbb {R}^n),C^\infty (\mathbb R^n)_{\hbar },\alpha _{\hbar },\beta _{\hbar },\Delta _{\hbar },\varepsilon _{\hbar })$ naturally induces the Lie bialgebroid $(\Psi (\mathbb {R}^n), (\Psi (\mathbb {R}^n))^*)$ as a classical limit.

With the bracket $[.,.]_{U(\Psi (\mathbb {R}^n)}$ and the exterior product $\wedge$ we have $(\bigoplus \Gamma (\bigwedge ^*U(\Psi (\mathbb {R}^n)),\wedge ,[.,.],\delta )$ as a differential Gerstenhaber algebra, and

$$\begin{aligned}&\delta ^2=0, \\&\delta [X,Y]=[\delta X,Y]+[X,\delta Y]. \end{aligned}$$

$\square$

Remark 11

Note that in the proof of above theorem, we chose the algebra A as $\mathbb {R}^n$ which is also considered as a manifold. We would like to mention that the above proof is applicable to a diffeological A,  that is a choice where A has its algebraic operations which are smooth in the diffeological sense, along the lines of [9]. In this context, $*_{\hbar }$ needs to be smooth.

Remark 12

In fact for any algebra A,  if the algebra of pseudo-differential operators $\Psi (A)$ admits a Lie agebroid structure then by Definition (21), $U(\Psi (A))=U(\Gamma (\Psi (A))\oplus C^\infty (A))$ is the universal envelo** algebra of our Lie algebroid.Therefore, by Theorem 2, we have the bialgebroid $(U(\Psi (A)),C^\infty (A),\alpha ,\beta ,\Delta ,\varepsilon ).$ Now, consider $(U_{\hbar }(\Psi (A)), C^\infty (A)_{\hbar },\alpha _{\hbar },\beta _{\hbar },\Delta _{\hbar },\varepsilon _{\hbar })$ as the quantum universal envelo** algebra of the above bialgebroid. By Theorem 3 and considering Remark 11, this quantum groupoid naturally induces a Lie bialgebroid $(\Psi (A),(\Psi (A))^*)$ as the classical limit. Then one can obtain non-commutative KP equations based on $U_{\hbar }(\Psi (A))$ as we explain in the next section. The interesting application is the choice of A as Connes-Kreimer Hopf algebra, i.e. the algebra A is generated by rooted planar binary trees, as it is described in [8]. If $\Psi (A)$ admits a Lie algebroid structure then by above explanation one can obtain non-commutative KP equations in terms of Connes-Kreimer Hopf algebra.

7 Non-Commutative Kadomtsev–Petviashvili Hierarchy

According to [11], the non-commutativity is arbitrarily introduced for the variables $(x_1 ,x_2,\dots )$ as the following equation.

$$\begin{aligned}{}[x_k, x_l]=i\theta ^{kl}, \end{aligned}$$

where the real constants $\theta ^{kl}$ are called the non-commutative parameters.

Non-commutative field theories are obtained from given commutative field theories by using star-products instead of the ordinary products in the commutative field theories. For instance, on Euclidean spaces, a star-product is given by

$$\begin{aligned} f(x)\star g(x):= f(x)g(x)+\frac{i}{2}\theta ^{kl}\partial _kf(x)\partial _lg(x)+O(\theta ^2), \end{aligned}$$

which is known as Moyal product, where $\partial _k=\frac{\partial }{\partial x_k}.$ The star-product is associative, i.e. $f\star (g \star h)=(f\star g)\star h,$ and it turns to the ordinary product in the commutative limit $\theta ^{kl}\rightarrow 0.$ Using the star product we obtain non-commutativity, i.e.

$$\begin{aligned}{}[x_k, x_l]_{\star }=x_k\star x_l-x_l\star x_k=i\theta ^{kl}. \end{aligned}$$

The usual definition of non-commutative KP hierarchy, i.e.

$$\begin{aligned} \partial _mL=[(L^m)_+,L]_{\star }, \end{aligned}$$

based on this non-commutative condition is given in [11].

Now, assume that our KP equations are defined on the algebra of formal pseudo-differential operators $\Psi (\mathbb {R}^n).$ By Theorem 4, there is a Lie bialgebroid structure on $\Psi (\mathbb {R}^n)$ as the classical limit of the quantum groupoid $(U_{\hbar }(\Psi (\mathbb {R}^n)),C^\infty (\mathbb R^n)_{\hbar },\alpha _{\hbar },\beta _{\hbar },\Delta _{\hbar },\varepsilon _{\hbar }).$ Therefore, we can naturally obtain non-commutative KP equations based on this quantum groupoid. More precisely, imitating the procedure given in [11], we obtain the non-commutative KP equations in terms of $*_{\hbar }$ defined on the mentioned quantum groupoid (see definition of $*_{\hbar }$ in the proof of Theorem 4) instead of Moyal product $\star.$ Moreover, the Lie bracket on $U_{\hbar }(\Psi (\mathbb {R}^n))$ is defined by

$$\begin{aligned}{}[x_k,x_l]_{\hbar }=(x_k *_{\hbar } x_l-x_l *_{\hbar } x_k)+(x_k *_{\hbar } x_l-x_l *_{\hbar } x_k)\hbar +(x_k *_{\hbar } x_l-x_l *_{\hbar } x_k)\hbar ^2+\dots , \end{aligned}$$

where in the classical limit $\hbar \rightarrow 0,$ it turns to the usual Lie bracket on $U(\Psi (\mathbb {R}^n)).$

Consider an Nth order (monic) pseudo-differential operator T as

$$\begin{aligned} T=\partial _x^N+a_{N-1}\partial _x^{N-1}+\dots +a_0+a_{-1}\partial _x^{-1}+a_{-2}\partial _x^{-2}+\dots , \end{aligned}$$

and denote $T_{\ge }:= \partial _x^N+a_{N-1}\partial _x^{N-1}+\dots +a_r\partial _x^r.$ A differential operator $\partial _x^n$ acts formally on a multiplicity operator f as the following generalized Leibniz rule.

$$\begin{aligned} \partial _x ^n f:=\sum _{i\ge 0}\left( \!\!\! \begin{array}{c} n \\ i \end{array} \!\!\!\right) (\partial _x^ if)\partial ^{n-i}, \end{aligned}$$

where the binomial coefficient

$$\begin{aligned} \left( \!\!\! \begin{array}{c} n \\ i \end{array} \!\!\!\right) :=\frac{n(n-1)\dots (n-i+1)}{i(i-1)\dots 1} \end{aligned}$$

is applicable for negative n. For example,

$$\begin{aligned} \partial _x^{-1}f= & {} f\partial _x^{-1}-f'\partial _x^{-2}+f''\partial _x^{-3}-\dots ,\\ \partial _x^{-2}f= & {} f\partial _x^{-2}-2f'\partial _x^{-3}+3f''\partial _x^{-4}-\dots ,\\ \partial _x^{-3}f= & {} f\partial _x^{-3}-3f'\partial _x^{-4}+6f''\partial _x^{-5}-\dots , \end{aligned}$$

where $f':=\frac{\partial f}{\partial x}, f'':=\frac{\partial ^2f}{\partial x^2}.$ Now, consider the following first-order pseudo-differential operator.

$$\begin{aligned} L=\partial _x+u_1+u_2\partial _x^{-1}+u_3\partial _x^{-2}+u_4\partial _x^{-3}+\dots , \end{aligned}$$

where the coefficients $u_k, (k=1,2,\dots )$ are functions of infinite variables $(x_1,x_2,\dots )$ with $x_1=x,$ i.e. $u_k=u_k(x,x_2,\dots ).$ We consider the non-commutativity given by the star-product $*_{\hbar },$ as described in the proof of Theorem 4. Then the Lax hierarchy is defined by

$$\begin{aligned} \partial _m L=[B_m,L]_{*_{\hbar }}, \ \ m=1,2,\dots , \end{aligned}$$

where applying $\partial _m$ on the pseudo-differential operator L is considered coefficient-wise, i.e., $\partial _m L:=[\partial _m,L]$ or $\partial _m\partial _x^k=0.$ Moreover, the operator $B_m$ is defined by $B_m:=(L*_{\hbar }\dots *_{\hbar } L)_{\ge r}$ (for m times). When $u_1=0 (r=0)$ the Lax hierarchy is the non-commutative KP hierarchy which includes the non-commutative equations, see [14,21]. The coefficients of each power of pseudo-differential operators in above Lax hierarchy give a series of infinite non-commutative evolution equations. For instance, for $m=1$ the coefficient of $\partial _x^{1-k}$ gives the equations

$$\begin{aligned} \partial _1u_k=u_k', \ \ \ \ k=2, 3, \dots \Rightarrow x_1=x. \end{aligned}$$

For $m=2$ the coefficient of $\partial _x^{-1}$ leads to

$$\begin{aligned} \partial _2u_2=u_2''+2u_3', \end{aligned}$$

and the coefficient of $\partial _x^{-2}$ gives

$$\begin{aligned} \partial _2u_3 & = u_3''+2u_4'+2u_2 *_{\hbar } u_2'+2[u_2,u_3]_{*_{\hbar }}\\ & = u_3''+2u_4'+\Big (2u_2u_2'+m_1(2u_2,u_2')\hbar +O(\hbar ^2)\Big )\\ & \quad + 2\Big ((u_2u_3+m_1(u_2,u_3)\hbar -u_3u_2-m_1(u_3,u_2)\hbar )\\ & \quad + (u_2u_3+m_1(u_2,u_3)\hbar -u_3u_2-m_1(u_3,u_2)\hbar )\hbar +O(\hbar ^2)\Big ), \end{aligned}$$

where $m_1(u_2,u_3)$ and $m_1(u_3,u_2)$ are elements in $U(\Psi (\mathbb {R}^n)).$

Next, the coefficient of $\partial _x^{-3}$ yields the equation

$$\begin{aligned} \partial _2u_4=u_4''+2u_5'+4u_3 *_{\hbar } u_2'-2u_2 *_{\hbar } u_2''+2[u_2,u_4]_{*_{\hbar }}, \end{aligned}$$

and the same as previous case we can expand it based on $*_{\hbar }.$

For $m=3$ the coefficients of $\partial _x^{-1}, \partial _x^{-2}, \partial _x^{-3},$ and $\partial _x^{-4}$ respectively give the equations

$$\begin{aligned}&\partial _3 u_2=u_2'''+3u_3''+3u_4'+3u_2' *_{\hbar } u_2+3u_2 *_{\hbar } u_2',\\&\partial _3u_3=u_3'''+3u_4''+3u'_5+6u_2 *_{\hbar } u_3'+3u_2' *_{\hbar } u_3+3u_3 *_{\hbar } u_2'+3[u_2,u_4]_{*_{\hbar }},\\&\partial _3 u_4=u_4'''+3u_5''+3u_6'+3u_2' *_{\hbar } u_4+3u_2 *_{\hbar } u_4'+6u_4 *_{\hbar } u_2'-3u_2 *_{\hbar } u_3''\\& \quad -3u_3 *_{\hbar } u_2''+6u_3 *_{\hbar } u_3'+3[u_2,u_5]_{*_{\hbar }}+3[u_3,u_4]_{*_{\hbar }},\\&\partial _3 u_5=\dots. \end{aligned}$$

If we set $2u_2\equiv u, x_2\equiv y, x_3\equiv t$ then we obtain the non-commutative KP equation

$$\begin{aligned} 3u_{yy}=\partial (4u_t-3(\partial u *_{\hbar } u+u *_{\hbar } \partial u)-\partial ^3u)+3\partial [u,\partial ^{-1} u_y]_{*_{\hbar }}. \end{aligned}$$

Note that this equation is analogous to the non-commutative KP equation in [11]. The difference between them is that here the star product is defined by $*_{\hbar }$ instead of Moyal product.

Remark 13

To solve the above non-commutative KP equation, we consider the KP equation on $U_{\hbar }(\Psi (\mathbb {R}^n))$ with the coefficient algebra equipped with $*_{\hbar }.$ In [9] we showed the well-posedness of the KP hierarchy with arbitrary (diffeological) coefficient algebra. Hence, considering diffeological algebraic structures, and Remark 11 we can conclude that the non-commutative KP hierarchy (equipped with $*_{\hbar }$) is well-posed, see also [17].

8 Conclusion

The algebra of formal Pseudo-differential operators $\Psi (\mathbb {R}^n)$ can be naturally equipped with a Lie bialgebroid structure $(\Psi (\mathbb {R}^n), (\Psi (\mathbb {R}^n))^*)$ as the classical limit of the quantum groupoid $(U_{\hbar }(\Psi (\mathbb {R}^n)),C^\infty (\mathbb R^n)_{\hbar },\alpha _{\hbar },\beta _{\hbar },\Delta _{\hbar },\varepsilon _{\hbar })$ associated to it. Then, the KP equations defined on Pseudo-differential operators $\Psi (\mathbb {R}^n)$ are “classical limit” of non-commutative KP equations defined by the above quantum grupoid. Hence, we obtain non-commutative KP equations in a more general sense and not by using non-commutativity of the coordinates i.e., $[x_k, x_l]=i\theta ^{kl}.$ Up to now, in special cases, non-commutative KP equations have been solved or it has been shown the existence and uniqueness of solutions of them using Moyal product, see [18] and references there in. The algebraic procedure described during this paper provides the possibility of solving non-commutative KP equations very generally.