Lie Bialgebroid of Pseudo-differential Operators

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.


Introduction
Consider the formal pseudo-differential operator L as where u 1 , u 2 , … , 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 where (L m ) + denotes the projection of the pseudo-differential operator L m into the space of differential operators, and x 1 , x 2 , … denote independent variables.
The non-commutative KP hierarchy usually has been defined (see e.g. [11]) using non-commutativity of the coordinates in Pseudo-differential operators, for the independent variables x k . Here kl are real constants and called the non-commutative parameters. Hence, the non-commutative KP hierarchy is described by where the star-product ⋆ 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.

Preliminaries
Consider an associative algebra as a triple (A, , ) where A is a vector space over a field k, the map ∶ A ⊗ A → A denotes the multiplication, and ∶ k → A denotes the unit of the algebra. The maps and are linear and satisfy the following associative properties.
If A is commutative then we also have • A,A = where A,A is the flip switching, that is A,A (a ⊗ a � ) = a � ⊗ a. A morphism between algebras (A, , ) and (A � , , � ) is 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, Δ, ) where C is a vector space over a field k and the maps Δ ∶ C → C ⊗ C and ∶ C → k are linear satisfying the co-associative properties If X = G is a finite group with commutative product, then K[G] is a bialgebra with co-multiplication and co-unit (x) = 1 . Also, if X = G is a finite group with convolution prod- 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 The canonical isomorphisms T n (V) ⊗ T m (V) ≅ T n+m (V) induce an associative product on the vector space T(V) = ⨁ n≥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 enveloping algebra of L, with a morphism of Lie algebras i L ∶ L → 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 ∈ L and [., .] is the Lie bracket. We define U(L) ∶= T(L)∕I(L) . If L is a Lie algebra then the universal enveloping algebra of L, denoted by U(L) is a Hopf algebra equipped with co-multiplication 1 3 Journal of Nonlinear Mathematical Physics (2022) 29:869-895 and with co-unit and antipode for X ∈ L. As an example, for the Lie algebra (2) of traceless 2 × 2 matrices, the enveloping algebra U( (2)) is a Hopf algebra, see [12]. Moreover, we can construct a Hopf algebra U q ( (2)) which is a one-parameter deformation of the enveloping algebra of the Lie algebra (2).

Bialgebroids
Definition 4 A bialgebroid is a pair of algebras A and R together with an algebra homomorphism ∶ R → A , an algebra anti-homomorphism ∶ R → A such that the images of and commute in A i.e. (r 1 ) (r 2 ) = (r 2 ) (r 1 ) , for each r 1 , r 2 ∈ R . By these two morphisms we can assign an (R, R)-bimodule structure on A in natural way by r.a = (r)a and a.r = (r)a , r ∈ R, a ∈ A . With this bimodule structure on A we can also consider A ⊗ R A as an (R, R)-bimodule which is given as a left module by r.(a 1 ⊗ a 2 ) = (r)a 1 ⊗ a 2 and as a right module by (a 1 ⊗ a 2 ).r = a 1 ⊗ (r)a 2 , r ∈ R, a 1 , a 2 ∈ A.
We define the co-product Δ ∶ A → A ⊗ R A and the co-unit ∶ A → R as (R, R)bimodule maps satisfying the co-associativity axiom for Δ and co-unit as below The co-product Δ and the algebra structure on A should be compatible in the sense that the kernel of the following map Here we are using the fact that A ⊗ A acts on A ⊗ R A 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 is a left ideal of A. We denote a bialgebroid by (A, R, , , Δ, ) where A is called total algebra, R is called the base algebra, and the maps , 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 s − →G 0 and G 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 ∈ G as an arrow (morphism) from Example 4 If we consider a group G as a groupoid G over a set X = {x} then the source map and target map are as constant maps s, t ∶ G → {x}, s(g) = t(g) = x for each g ∈ G . The arrows or elements of G are as G = s −1 (x) → t −1 (x) = G. We can consider them as G g − →G such that h ↦ gh for each h, g ∈ G . Also the unit for our groupoid is ∶ X → G with (x) = e (e is the identity element of the group G).
Now, we consider the space of smooth functions on G, i.e. C ∞ (G) as the total algebra and R the real numbers as the base algebra. We define = s * and = t * such that In other words s * assigns to each r ∈ R the constant function r ∈ C ∞ (G). So s * (r) = r similarly t * (r) = r for r ∈ R. Moreover, with Therefore * can be considered as the co-unit. Finally, defines the co-product, with Δ(f ) = f ⊗ f . Clearly C ∞ (G) is an (R, R)-bimodule. It is easy to check that , Δ are (R, R)-bimodule maps and they are compatible with the algebra structure on C ∞ (G) . Therefore, we constructed a bialgebroid (C ∞ (G), R, s * , t * , Δ, ) by the groupoid G which is the group G.
Recall that a smooth map f ∶ M → N between smooth manifolds M and N is a local diffeomerphism (or étale map) if (df ) x is an isomorphism for any x ∈ M. An étale groupid is a groupoid with the source map s as a local diffeomorphism. For more details and examples see [19].

Example 6
Again consider the étale groupoid G s → − →G 0 . The Connes algebra C ∞ c (G) of (smooth) complex (or real) functions with compact support on G (see [2,6,7]), together with the base algebra (C ∞ c (G)) 0 defines a bialgebroid, where (C ∞ c (G)) 0 is the subalgebra of C ∞ c (G) of functions with support in G 0 ⊂ G. This subalgebra is commutative and may be identified with the commutative algebra C ∞ c (G 0 ).
The product is the convolution where the sum is over all possible decompositions of g �� ∈ G . We take the co-unit by This sum is over all the elements g of G which satisfy We define Δ as the composition of this inclusion with the inverse of the isomorphism (For the proof that Ω is an isomorphism see [20]).
If a ∈ C ∞ c (G) has the support in an open subset U of G which is so small that s| U is injective, then Δ(a) = a ⊗ = ⊗ a where is any smooth function with compact support in U which constantly equals 1 on the support of a. The functions a ∈ C ∞ c (G) which satisfy the above condition generate the linear space C ∞ c (G).

Example 7 1) If A is any algebra then there is a bialgebroid structure on
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
consists of an algebra morphism T ∶ A → A � and an algebra morphism t ∶ R → 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, , , Δ, ) is a bijective map ∶ A → A which has the following properties 1.
is an algebra anti-isomorphism for A.

Proposition 1 The maps ∶ A → A and id ∶ R → R define a bialgebroid morphism.
Proof In the above definition, condition (1) means that for all x, y ∈ A , (xy) = (y) (x) and condition (2) means that ( (1 R )) = (1 R ) or (1) = 1. On the other hand ( ⊗ )Δ = Δ op , and • = . Moreover by condition (2) commutes with structure maps. Clearly id is a morphism of algebra and it commutes with the structure maps. ◻

Proposition 2 Bialgebras and bialgebroids are equivalent over a field k as a base algebra.
Proof For any bialgebra (A, , , Δ, ) we can consider R = k , where k is the ground field for the algebra A, and , ∶ k → A by (r) = r and (r) = r , (r ∈ k) . Hence, (A, k, , , Δ, ) is a bialgebroid because for r, s ∈ k we have (rs) = rs = (r) (s) so is an algebra homomorphism, also (rs) = rs = sr = (s) (r) so is an antihomomorphism, and finally (r) (s) = rs = sr = (s) (r) so the images of and commute. Conversely, if R is the field k the definition of a bialgebroid over k is reduced to a bialgebra over k, see [15]. ◻ Example 8 1. If L is a Lie algebra then the universal enveloping 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 ( (2)) as mentioned before (there is a duality between SL(2) and U( (2)), see [12]). By the above proposition they are also examples of bialgebroids.
Let (A, R, , , Δ, ) 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 For a bialgebra R = k and End k R ≅ 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, , , Δ, ) is a bialgebroid where = is the embedding R → D and Δ ∶ D → D ⊗ R D is defined by and f , g ∈ R. Also, the usual action of differential operators on C ∞ (P) defines an anchor ∶ D → End k R . The co-unit ∶ D → 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. and  1) and (2) in above proposition. ] equipped with the * -product defined above. Consider ℏ ∶ R ℏ → D ℏ and ℏ ∶ R ℏ → D ℏ given by Moreover, consider the co-product Δ � ∶ D � → D � ⊗ R � D � defined by Δ ℏ = F −1 ΔF, and co-unit as the projection D ℏ → R ℏ . By Theorem 1 we obtain twisted Hopf algebroid (D ℏ , R ℏ , ℏ , ℏ , m, Δ ℏ , ). This twisted Hopf algebroid is called the quantum groupoid associated to the star product * ℏ .

Lie Bialgebras
Definition 9 Let G be a Lie group and ∶ G → GL(V) be the representation of G on the vector space V. Also, let d ∶ → End(V) be the infinitesimal representation of the group representation. then Note that if is 1-cocycle on G then = d e is 1-cocycle on . Also, if G is simply connected then the 1-cocycle on can be integrated to 1-cocycle on G.
Definition 10 Let be a Lie algebra. A Lie bialgebra structure on is a linear map ∶ → ⊗ , called co-commutator, such that 1. * ∶ * ⊗ * → * is a Lie bracket on * 2. is a 1-cocycle of with values in ⊗ .
A homomorphism of Lie bialgebras ∶ → is a homomorphism of Lie algebras such that ( ⊗ )• = • . Definition 11 Let G be a Lie group. A Poisson Lie group is (G, Π) where Π is a Poisson structure such that m ∶ G × G → G is Poisson map. In this case Π is called multiplicative.

Remark 2 A Poisson structure Π is multiplicative if and only if
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 Π = 0. .
1 3 The following proposition gives an example of Lie bialgebras.

Proposition 4
Let (G, Π) be a Poisson Lie group with the Lie algebra . Also, let F ∶ → ∧ be 1-cocycle, and F * Lie bracket then ( , * ) is a Lie bialgebra.
Proof Let (G, Π) be a Poisson Lie group, and F = Π (1) the linear part of Π at the point e. The map Π (1) ∶ → ∧ is linear Poisson structure on where = T e G . Then F * ∶ * ∧ * → * is a Lie algebra structure on * . In fact, for That is if (G, Π) is a Poisson Lie group then F = d ẽ∶ → ∧ such that ◻ Lemma 1 Let Λ be a bivector in the Lie algebra, i.e. Λ ∈ ⋀ 2 and Π g = (l g ) * Λ − (r g ) * Λ . Then Π is multiplicative.

Proof We have
Hence, by Remark 2 in above, Π is multiplicative. ◻
If Λ is such that [Λ, Λ] is Ad-invariant then we call it r-matrix. In the special case, [Λ, Λ] = 0 is called the classical Yang-Baxter equation and Λ is called triangular r-matrix. If ∈ Γ( ⋀ k A * ) , we say that is homogeneous, and its degree is | | = k , and we call it as an A-k-form.

Definition 15 There is a differential operator which takes an A-k-form to an A-(k + 1)-form d A as below
where v 1 , ..., v k+1 ∈ Γ(A) are A-vector fields.

Remark 4 ([4]) The Lie algebroid axioms for A implies that:
The triple (Γ ⋀ * A * , ∧, 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 Γ( ⋀ * A * ) satisfying properties 1-3 in above.

Proof The anchor map is obtained from d A on functions by
The lie bracket [., .] A is determined by .

Remark 5 ([4])
1) The extension of [., .] A to arbitrary A-multivector fields by setting it on homogeneous A-multivector fields v, w is 2) The bracket [., .] A on A-multivector fields has the following properties.
By the Jacobi identity, an arbitrary (not necessarily Poisson) element Θ ∈ Γ( satisfies which is similar to the equation for a flat connection. In the first section we saw how one defines the universal enveloping algebra for a Lie algebra. Now we want to see the definition for Lie algebroids. If (A, , [., .] A ) is a Lie algebroid over the manifold P, then the C ∞ (P) -module C ∞ (P) ⊕ Γ(A) is a Lie algebra over ℝ with the Lie bracket Considering the universal enveloping algebra of this Lie algebra as U = U(C ∞ (P) ⊕ Γ(A)) , we denote f ′ and X ′ as the canonical images of f ∈ C ∞ (P) and X ∈ Γ(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 enveloping algebra of the Lie algebroid A is defined by U(A) = U∕I.

Theorem 2 The universal enveloping algebra U(A) of a Lie algebroid A admits a cocommutative bialgebroid structure.
Proof Let R = C ∞ (P) , and = ∶ R → U(A) be the natural embedding. For the coproduct define This formula extends to a co-product Δ ∶ U(A) → U(A) ⊗ R U(A) by the compatibility condition.
The co-unit map is the projection ∶ U(A) → R . Also, the map ∶ U(A) → End k R defined by is the anchor for our bialgebroid where ∶ U(A) → TP is the algebra homomorphism extending the anchor of the Lie algebroid A. Hence, (U(A), R, , , m, Δ, ) is a co-commutative bialgebroid with anchor . ◻ 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 coming from the structure on A * is a derivation of the Schouten bracket on In other words, ( is a differential Gerstenhaber algebra.

Example 17
For a Poisson manifold (P, Π), the dual pair (TP, T * P) where TP is the standard tangent bundle and T * P is the cotangent Lie algebroid, together with is a Lie bialgebroid. As before we saw, d Π = [Π, .] 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 ⨁ k Γ( ⋀ 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 The same as Lie bialgebras, a useful method of constructing Lie bialgebroids is using r-matrices. Recall that an r-matrix is a section Λ ∈ Γ( ⋀ 2 A) satisfying An r-matrix Λ defines a Lie bialgebroid, where the differential The anchor is

Deformation of Bialgebroids
We consider a topological bialgebra (A, , , Δ, ) where A is a module over the ring 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 ℏ , R ℏ , ℏ , ℏ , m, Δ ℏ , ) which is called the quantum groupoid associated to the star product * ℏ . In this case, we simply say that the quotient A ℏ ∕ℏA ℏ is isomorphic to A as a bialgebroid. Lemma 2 (Xu,[23]) By the conditions (1), (2) in the above definition,

Definition 23
Definition 24 A quantum universal enveloping algebroid (or QUE algebroid), also called a quantum groupoid is a deformation of the standard bialgebroid (UA, R, , , m, Δ, ) of a Lie algebroid A.
] . Then R ℏ defines a star product on the base manifold P for the Lie algebroid A, so that (with f , g ∈ R ) is a Poisson structure on the base P. Define For any f , g ∈ R, x, y ∈ UA write

Remark 10
In [13], Kontsevich has shown that given a Poisson structure {., .} on a Poisson manifold P one can find a * -product such that 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.

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 pseudodifferential 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 zerocharacteristic. Assume that A is equipped with a derivation, i.e. there is a k-linear map D ∶ A → A satisfying the Leibnitz rule D(f .g) = (Df ).g + f .(Dg) for all f , g ∈ A. We say that an element is constant if Df = 0. Consider as a formal variable not in A. The algebra of symbols over A is the vector space which is equipped with the associative multiplication • defined by Note that the algebra A is included in Ψ (A), and for f ∈ A and ∈ Z the multiplication • is given by

Definition 26
The algebra of formal pseudo-differential operators over A is the vector space which is equipped with the unique multiplication which makes the map ∑ ∈ℤ a ↦ ∑ ∈ℤ a D an algebra homomorphism. The algebra Ψ(A) is associative but not commutative. It is a Lie algebra over k with the Lie bracket [P, Q] ∶= PQ − QP. Now, consider A = ℝ n as the commutative ℝ-algebra with unit 1, over the field ℝ, equipped with the Hadamard product (entrywise product) which for n = 1 is the usual product on ℝ.
Theorem 4 There is a natural Lie bialgebroid structure on the algebra of formal pseudo-differential operators Ψ(ℝ n ) as the classical limit of the quantum groupoid associated to it.
Also, if we consider the coordinates (q i , p i ) on Tℝ n we define ).
Consider an Nth order (monic) pseudo-differential operator T as and denote T ≥ ∶= N x + a N−1 N−1 x + ⋯ + a r r x . A differential operator n x acts formally on a multiplicity operator f as the following generalized Leibniz rule.
where the binomial coefficient is applicable for negative n. For example, x 2 . Now, consider the following first-order pseudo-differential operator.
where the coefficients u k , (k = 1, 2, … ) are functions of infinite variables (x 1 , x 2 , … ) with x 1 = x, i.e. u k = u k (x, x 2 , … ). We consider the non-commutativity given by the star-product * ℏ , as described in the proof of Theorem 4. Then the Lax hierarchy is defined by where applying m on the pseudo-differential operator L is considered coefficientwise, i.e., m L ∶= [ m , L] or m k x = 0. Moreover, the operator B m is defined by x + … , m L = [B m , L] * ℏ , m = 1, 2, … , coefficient algebra. Hence, considering diffeological algebraic structures, and Remark 11 we can conclude that the non-commutative KP hierarchy (equipped with * ℏ ) is well-posed, see also [17].

Conclusion
The algebra of formal Pseudo-differential operators Ψ(ℝ n ) can be naturally equipped with a Lie bialgebroid structure (Ψ(ℝ n ), (Ψ(ℝ n )) * ) as the classical limit of the quantum groupoid (U ℏ (Ψ(ℝ n )), C ∞ (ℝ n ) ℏ , ℏ , ℏ , Δ ℏ , ℏ ) associated to it. Then, the KP equations defined on Pseudo-differential operators Ψ(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 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.