Quantum groupoids

PING XU ^{∗}

Max-Planck Institut fur Mathematik Vivatsgasse 7 D-53111 Bonn, GERMANY AND Department of Mathematics The Pennsylvania State University University Park, PA 16802, USA email: ping@math.psu.edu

Abstract

We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, it is shown that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we prove that any regular triangular Lie bialgebroid is quantizable. As an application, we study the dynamical quantum groupoid D⊗_{ℏ}U_{ℏ}𝔤, which gives an interpretation of quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.

1 Introduction

Poisson tensors in many aspects resemble classical triangular r-matrices in quantum group theory. A notion unifying both Poisson structures and Lie bialgebras was introduced in [33], called Lie bialgebroids. Integration theorem for Lie bialgebroids encomposes both Drinfeld theorem of integration of Lie bialgebras on the one hand, and Karasev-Weinstein theorem of existence of local symplectic groupoids for Poisson manifolds on the other hand. Quantization of Lie bialgebras leads to quantum groups, while quantization of Poisson manifolds is the so called star-products. It is therefore natural to expect that there exists some intrinsic connection between these two quantum objects. The purpose of this paper is to fill in this gap by introducing the notion of quantum universal enveloping algebriods (QUE algebriods), or quantum groupoids, as a general framework unifying these two concepts. Part of the results in this paper has been announced in [44] [45].

The general notion of Hopf algebroids was introduced by Lu [31], where the axioms were obtained essentially by translating those of Poisson groupoids to their quantum counterparts. The special case that base algebras are commutative was studied earlier by Maltsiniotis [35] in 1992 based on the work of Deligne on Tannakian category [8]. Hopf algebroids also appeared in Etingof and Varchenko’s work on dynamical quantum groups [14] [15]. In this paper, we will mainly follow Lu’s definition, but some axioms will be modified.

∗Research partially supported by NSF grants DMS97-04391.

As we know, many important examples of Hopf algebras arise as deformations of the universal enveloping algebras of Lie algebras. Given a Lie algebroid A, its universal enveloping algebra UA (see the definition in Section 2) carries a natural cocommutative Hopf algebroid structure. For example, when A is the tangent bundle Lie algebroid T P, one obtains a cocommutative Hopf algebroid structure on D(P), the algebra of differential operators on P. It is natural to expect that deformations of UA, called quantum universal enveloping algebroids or quantum groupoids in this paper, would give us some non-trivial Hopf algebroids. This is the main subject we will undertake the study in the present paper. Examples include the quantum groupoid D_{ℏ}(P) corresponding to a star-product on a Poisson manifold P.

Another important class of quantum groupoids is connected with the so called quantum universal dynamical Yang-Baxter equation, also known as Geravais-Neveu-Felder equation:

R^{12}(λ + ℏh^{(3)})R^{13}(λ)R^{23}(λ + ℏh^{(1)}) = R^{23}(λ)R^{13}(λ + ℏh^{(2)})R^{12}(λ).  (1)

Here R(λ) is a meromorphic function from η^{∗} to U_{ℏ}𝔤⊗U_{ℏ}𝔤, U_{ℏ}𝔤 is a quasi-triangular quantum group, and η ⊂ 𝔤 is an Abelian Lie subalgebra. This equation arises naturally from various contexts in mathematical physics, including quantum Liouville theory, quantum Knizhnik-Zamolodchikov-Bernard equation, and quantum Caloger-Moser model [2] [3] [18]. One approach to this equation due to Babelon et. al. [4] is to use Drinfeld’s theory of quais-Hopf algebras [11]. Consider a meromorphic function F : η^{∗} ⟶ U_{ℏ}𝔤⊗U_{ℏ}𝔤 such that F (λ) is invertible for all λ. Set R(λ) = F^{21}(λ)^{−1}RF^{12}(λ), where R ∈ U_{ℏ}𝔤⊗U_{ℏ}𝔤 is the standard universal R-matrix for the quantum group U_{ℏ}𝔤. One can check that R(λ) satisfies Equation (1) if F (λ) is of zero weight, and satisfies the following shifted cocycle condition:

(Δ_{0}⊗id)F (λ)F^{12}(λ + ℏh^{(3)}) = (id⊗Δ_{0})F (λ)F^{23}(λ),  (2)

where Δ_{0} is the coproduct of U_{ℏ}𝔤. If moreover we assume that

(ϵ_{0}⊗id)F (λ) = 1; (id⊗ϵ_{0})F (λ) = 1,  (3)

where ϵ_{0} is the counit map, one can form an elliptic quantum group, which is a family of quasi-Hopf algebras (U_{ℏ}𝔤, Δ_{λ}) parameterized by λ ∈ η^{∗}: Δ_{λ} = F (λ)^{−1}ΔF (λ). For 𝔤 = 𝔰𝔩_{2}(ℂ), a solution to Equations (2) and (3) was obtained by Babelon [3] in 1991. For general simple Lie algebras, solutions are recently found independently by Arnaudon et. al. [1] and Jimbo et. al. [24] based on the approach of Fronsdal [19].

As we will see in Section 7, Equation (2) arises naturally from the “twistor” equation of a quantum groupoid. This leads to another interpretation of an elliptic quantum group, namely as a quantum groupoid. Roughly speaking, our construction goes as follows. Instead of U_{ℏ}𝔤, we start with the algebra H = D⊗U_{ℏ}𝔤, where D denotes the algebra of meromorphic differential operators on η^{∗}. H is no longer a Hopf algebra. Instead it is a QUE algebroid considered as the Hopf algebroid tensor product of D and U_{ℏ}𝔤. Then the shifted cocycle condition is shown to be equivalent to the equation defining a twistor of this Hopf algebroid. Using this twistor, we obtain a new QUE algebroid D⊗_{ℏ}U_{ℏ}𝔤 (or a quantum groupoid). We note that D⊗_{q}U_{ℏ}𝔤 is co-associative as a Hopf algebroid, although (U_{ℏ}𝔤, Δ_{λ}) is not co-associative. The construction of D⊗_{q}U_{ℏ}𝔤 is in some sense to restore the co-associativity by enlarging the algebra U_{ℏ}𝔤 by tensoring the dynamical part D. The relation between this quantum groupoid and quasi-Hopf algebras (U_{ℏ}𝔤, Δ_{λ}) is, in a certain sense, similar to that between a fiber bundle and its fibers. We expect that this quantum groupoid will be useful in understanding elliptic quantum groups, especially their representation theory. The physical meaning of it, however, still needs to be explored.

This paper is organized as follows. In Section 2, we recall some basic definitions and results concerning Lie bialgebroids. Section 3 is devoted to the definition and basic properties of Hopf algebroids. In particular, for Hopf algebroids with anchor, it is proved that the category of left modules is a monoidal category. As a fundamental construction, in Section 4 we study the twist construction of Hopf algebroids, which generalizes the usual twist construction of Hopf algebras. Despite its complexity comparing to Hopf algebras, the fundamentals are in analogue to those of Hopf algebras. In particular, the monoidal categories of left modules of the twisted and untwisted Hopf algebroids are always equivalent. Section 5 is devoted to the introduction of quantum universal enveloping algebroids. The main part is to show that Lie bialgebroids indeed appear as the classical limit of QUE algebroids, as what is expected. However, unlike the quantum group case, the proof is not trivial and is in fact rather involved. On the other hand, the inverse question: the quantization problem, which would encompose both quantization of Lie bialgebras and deformation quantization of Poisson manifolds as special cases, remains widely open. As a very special case, in Section 6, we show that any regular triangular Lie bialgebroids are always quantizable. The discussion on quantum groupoids associated to universal dynamical R-matrices (i.e. solutions to the quantum dynamical Yang-Baxter equation) occupies Section 7. The last section, Section 8, consists of an appendix, as well as a list of open questions.

We would like to mention the recent work by Etingof and Varchenko [14] [15], where a different approach to the quantum dynamical Yang-Baxter equation in the framework of Hopf algebroid was given.

Acknowledgments. The author would like to thank Giuseppe Dito, Vladimir Drinfeld, Masaki Kashiwara, Jiang-hua Lu, Dale Peterson and Alan Weinstein for useful discussions and comments. In addition to the funding sources mentioned in the first footnote, he would also like to thank RIMS, IHES and Max-Planck Institut for their hospitality and financial support while part of this project was being done.

2 Preliminary on Lie bialgebroids

It is well known that the classical objects corresponding to quantum groups are Lie bialgebras. Therefore, it is not surprising to expect that the classical counterparts of quantum groupoids are Lie bialgebroids. However, unlike Lie bialgebras, Lie bialgebroids were introduced and studied before the invention of quantum groupoids. In fact, they were used mainly in the study of Poisson geometry in connection with symplectic and Poisson groupoids (see [39] [42]).

The purpose of this section is to recall some basic facts concerning Lie bialgebroids. We start with recalling some definitions.

Definition 2.1 A Lie algebroid is a vector bundle A over P together with a Lie algebra structure on the space Γ(A) of smooth sections of A, and a bundle map ρ : A → T P (called the anchor), extended to a map between sections of these bundles, such that

(i) ρ([X, Y]) = [ρ(X), ρ(Y)]; and

(ii) [X, fY] = f[X, Y] + (ρ(X)f)Y for any smooth sections X and Y of A and any smooth function f on P.

Among many examples of Lie algebroids are usual Lie algebras, the tangent bundle of a manifold, and an integrable distribution over a manifold (see [32]). Another interesting example is connected with Poisson manifolds. Let P be a Poisson manifold with Poisson tensor π. Then T^{∗}P carries a natural Lie algebroid structure, called the cotangent bundle Lie algebroid of the Poisson manifold P [7]. The anchor map π^{#} : T^{∗}P → T P is defined by

π^{#} : T_{p}^{∗}P ⟶ T_{p}P : π^{#}(ξ)(η) = π(ξ, η), ∀ξ, η ∈ T_{p}^{∗}P  (4)

and the Lie bracket of 1-forms α and β is given by

[α, β] = L_{π^{#}(α)}β − L_{π^{#}(β)}α − dπ(α, β).  (5)

Given a Lie algebroid A, it is known that ⊕_{k}Γ(∧^{k}A^{∗}) admits a differential d that makes it into a differential graded algebra [26]. Here, d : Γ(∧^{k}A^{∗}) ⟶ Γ(∧^{k+1}A^{∗}) is defined by ([32] [33] [41]):

dω(X_{1}, …, X_{k+1}) = ∑_{i=1}^{k+1}(−1)^{i+1}ρ(X_{i})(ω(X_{1}, ˆ…, X_{k+1}))
+ ∑_{i <j}(−1)^{i+j}ω([X_{i}, X_{j}], X_{1}, ˆ… ˆ…, X_{k+1}),  (6)

for ω ∈ Γ(∧^{k}A^{∗}), X_{i} ∈ ΓA, 1 ≤ i ≤ k + 1. Then d^{2} = 0 and one obtains a cochain complex whose cohomology is called the Lie algebroid cohomology. On the other hand, the Lie bracket on Γ(A) extends naturally to a graded Lie bracket on ⊕_{k}Γ(∧^{k}A) called Schouten bracket, which, together with the usual wedge product ∧, makes it into a Gerstenhaber algebra [46].

As in the case of Lie algebras, associated to any Lie algebroid, there is an associative algebra called the universal enveloping algebra of the Lie algebroid A [22], a concept whose definition we now recall.

Let A → P be a Lie algebroid with anchor ρ. Then the C^{∞}(P)-module direct sum C^{∞}(P)⊕Γ(A) is a Lie algebra over ℝ with the Lie bracket:

[f + X, g + Y] = (ρ(X)g − ρ(Y)f) + [X, Y].

Let U = U(C^{∞}(P) ⊕ Γ(A)) be its standard universal enveloping algebra. For any f ∈ C^{∞}(P) and X ∈ Γ(A), denote by f′ and X′ their canonical image in U. Denote by I the two-sided ideal of U generated by all elements of the form (fg)′ − f′g′ and (fX)′ − f′X′. Define U(A) = U/I, which is called the universal enveloping algebra of the Lie algebroid A. When A is a Lie algebra, this definition reduces to the definition of usual universal enveloping algebras. On the other hand, for the tangent bundle Lie algebroid T P, its universal enveloping algebra is D(P), the algebra of differential operators over P. In between, if A = T P × 𝔤 as the Lie algebroid direct product, then U(A) is isomorphic to D(P)⊗U𝔤. Note that the maps f ↦ f′ and X ↦ X′ considered above descend to linear embedings i_{1} : C^{∞}(P) ↦ U(A), and i_{2} : Γ(A) ↦ U(A); the first map i_{1} is an algebra morphism. These maps have the following properties:

i_{1}(f)i_{2}(X) = i_{2}(fX), [i_{2}(X), i_{1}(f)] = i_{1}(ρ(X)f)), [i_{2}(X), i_{2}(Y)] = i_{2}([X, Y]).  (7)

In fact, U(A) is universal among all triples (B, φ_{1}, φ_{2}) having such properties (see [22] for a proof of this simple fact). Sometimes, it is also useful to think of UA as the algebra of left invariant differential operators on a local Lie groupoid G, which integrates the Lie algebroid A.

The notion of Lie bialgebroids is a natural generalization of that of Lie bialgebras. Roughly speaking, a Lie bialgebroid is a pair of Lie algebroids (A, A^{∗}) satisfying a certain compatibility condition. Such a condition, providing a definition of Lie bialgebroid, was given in [33]. We quote here an equivalent formulation from [23].

Definition 2.2 A Lie bialgebroid is a dual pair (A, A^{∗}) of vector bundles equipped with Lie algebroid structures such that the differential d_{∗} on Γ(∧^{∗}A) coming from the structure on A^{∗} is a derivation of the Schouten bracket on ⊕Γ(∧^{∗}A). Equivalently, d_{∗} is a derivation for sections of A, i.e.,

d_{∗}[X, Y] = [d_{∗}X, Y] + [X, d_{∗}Y], ∀X, Y ∈ Γ(A).  (8)

In other words, (⊕_{k}Γ(∧^{k}A), ∧, [·, ·], d_{∗}) is a strong differential Gerstenhaber algebra [46].

In fact, a Lie bialgebroid is equivalent to a strong differential Gerstenhaber algebra structure on ⊕_{k}Γ(∧^{k}A) (see Proposition 2.3 in [46]).

For a Lie bialgebroid (A, A^{∗}), the base P inherits a natural Poisson structure:

{f, g} =< df, d_{∗}g >, ∀f, g ∈ C^{∞}(P),  (9)

which satisfies the identity: [df, dg] = d{f, g}.

As in the case of Lie bialgebras, a useful method of constructing Lie bialgebroids is via some kind of “r-matrices”. More precisely, by an r-matroid, we mean a section Λ of Γ(∧^{2}A) satisfying

L_{X}[Λ, Λ] = [X, [Λ, Λ]] = 0, ∀X ∈ Γ(A).  (10)

An r-matroid Λ defines a Lie bialgebroid, where the differential d_{∗} : Γ(∧^{∗}A) ⟶ Γ(∧^{∗+1}A) is simply given by d_{∗} = [Λ, ·]. In this case, the bracket on Γ(A^{∗}) is given by

[ξ, η] = L_{Λ^{#}ξ}η − L_{Λ^{#}η}ξ − d[Λ(ξ, η)],  (11)

and the anchor is the composition ρ∘Λ^{#} : A^{∗} ⟶ T P, where Λ^{#} denotes the bundle map A^{∗} ⟶ A defined by Λ^{#}(ξ)(η) = Λ(ξ, η), ∀ξ, η ∈ Γ(A^{∗}). Such a Lie bialgebroid is called a coboundary Lie bialgebroid, in analogue with the Lie algebra case [29] [30]. It is called a triangular Lie bialgebroid if [Λ, Λ] = 0. In particular it is called a regular triangular Lie bialgebroid if Λ is of constant rank.

When P reduces to a point, i.e., A is a Lie algebra, Equation (10) is equivalent to that [Λ, Λ] is ad-invariant, i.e, Λ is an r-matrix in the ordinary sense. On the other hand, when A is the tangent bundle T P with the standard Lie algebroid structure, Equation (10) is equivalent to that [Λ, Λ] = 0, i.e., Λ is a Poisson tensor.

Another interesting class of coboundary Lie bialgebroids is connected with the so called dynamical r-matrices.

Let 𝔤 be a Lie algebra over ℝ (or ℂ) and η ⊂ 𝔤 an Abelian Lie subalgebra. A dynamical r-matrix [13] is a smooth function (or meromorphic function in complex case) r : η^{∗} ⟶ ∧^{2}𝔤 such that

(i). r(λ) is η-invariant, i.e., ad_{h}r(λ) = 0, ∀h ∈ η;

(ii). Altdr − \frac{1}{2}[r, r] is constant over η^{∗} with value in (∧^{2}𝔤)^{𝔤}.

Here dr is considered as a η ⊗ ∧^{2}𝔤-valued function over η^{∗} and Alt denotes the standard skewsymmetrization operator. In particular, if Altdr − \frac{1}{2}[r, r] = 0, it is called a triangular dynamical r-matrix. It is called a regular triangular dynamical r-matrix if r(λ) is of constant rank. The following is a simple example of a dynamical r-matrix.

Example 2.1 Let 𝔤 be a simple Lie algebra with root decomposition 𝔤 = η ⊕ ∑_{α∈Δ_{+}}(𝔤_{α} ⊕ 𝔤_{−α}), where η is a Cartan subalgebra, and Δ_{+} is the set of positive roots. Then

r(λ) = \frac{1}{2} ∑_{α∈Δ_{+}} coth(\frac{1}{2} ≪ α, λ ≫)E_{α} ∧ E_{−α},

is a dynamical r-matrix, where ≪ , ≫ is the Killing form of 𝔤, the E_{α} and E′_{−α}s are standard root vectors, and coth(x) = \frac{e^{x}+e^{−x}}{e^{x}−e^{−x}} is the hyperbolic cotangent function.

A dynamical r-matrix naturally gives rise to a Lie bialgebroid.

Proposition 2.3 [5] [30] Let r : η^{∗} ⟶ ∧^{2}𝔤 be a dynamical r-matrix. Then A = T η^{∗} × 𝔤 equipped with the standard Lie algebroid structure, together with Λ = ∑_{i=1}^{k}(\frac{∂}{∂λ_{i}} ∧h_{i})+r(λ) ∈ Γ(∧^{2}A) defines a coboundary Lie bialgebroid. Here {h_{1}, h_{2}, ⋯, h_{k}} is a basis of η, and (λ_{1}, ⋯, λ_{k}) its induced coordinate system on η^{∗}.

We end this section by recalling the definition of Hamiltonian operators, which will be needed later on. Given a Lie bialgebroid (A, A^{∗}) with associated strong differential Gerstanhaber algebra (⊕_{k}Γ(∧^{k}A), ∧, [·, ·], d_{∗}), one may construct a new Lie bialgebroid via a twist. For that, simply let d̃_{∗} = d_{∗} + [H, ·] for some H ∈ Γ(∧^{2}A). It is easy to check [27] that this still defines a strong differential Gerstanhaber algebra (therefore a Lie bialgebroid), iff

d_{∗}H + \frac{1}{2}[H, H] = 0.  (12)

Such an H is called a Hamiltonian operator of the Lie bialgebroid (A, A^{∗}).

Finally we note that even though we are mainly dealing with real vector bundles and real Lie algebroids in this paper, one may also consider (as already suggested by the early example of dynamical r-matrices) complex Lie algebroids and complex Lie bialgebroids. In this context, one may have to use sheaves of holomorphic sections etc. instead of global ones. Most results in this section are expected to hold with certain suitable modifications.

3 Hopf algebroids

Definition 3.1 A Hopf algebroid (H, R, α, β, m, Δ, ϵ) consists of the following data:

1) a total algebra H with product m, a base algebra R, a source map: an algebra homomorphism α : R ⟶ H, and a target map: an algebra anti-homomorphism β : R ⟶ H such that the images of α and β commute in H, i.e., ∀a, b ∈ R, α(a)β(b) = β(b)α(a). There is then a natural (R, R)-bimodule structure on H given by a·h = α(a)h and h·a = β(a)h. Thus, we can form the (R, R)-bimodule product H⊗_{R}H. It is easy to see that H⊗_{R}H again admits an (R, R)-bimodule structure. This will allow us to form the triple product H⊗_{R}H⊗_{R}H and etc.

2) a co-product: an (R, R)-bimodule map Δ : H ⟶ H⊗_{R}H with Δ(1) = 1⊗1 satisfying the co-associativity:

(Δ⊗_{R}id_{H})Δ = (id_{H}⊗_{R}Δ)Δ : H ⟶ H⊗_{R}H⊗_{R}H;  (13)

3) the product and the co-product are compatible in the following sense:

Δ(h)(β(a)⊗1 − 1⊗α(a)) = 0, in H⊗_{R}H, ∀a ∈ R and h ∈ H, and  (14)

Δ(h_{1}h_{2}) = Δ(h_{1})Δ(h_{2}), ∀h_{1}, h_{2} ∈ H, (see the remark below);  (15)

4) a co-unit map: an (R, R)-bimodule map ϵ : H ⟶ R satisfying ϵ(1_{H}) = 1_{R} (it follows then that ϵβ = ϵα = id_{R}) and

(ϵ⊗_{R}id_{H})Δ = (id_{H}⊗_{R}ϵ)Δ = id_{H} : H ⟶ H.  (16)

Here we have used the identification: R⊗_{R}H ≅ H⊗_{R}R ≅ H (note that both maps on the left hand sides of Equation (16) are well-defined).

Remark It is clear that any left H-module is automatically an (R, R)-bimodule. Now given any left H-modules M_{1} and M_{2}, define,

h · (m_{1}⊗_{R}m_{2}) = Δ(h)(m_{1}⊗m_{2}), ∀h ∈ H, m_{1} ∈ M_{1}, m_{2} ∈ M_{2}.  (17)

The RHS is a well-defined element in M_{1}⊗_{R}M_{2} due to Equation (14). In particular, when taking M_{1} = M_{2} = H, we see that the RHS of Equation (15) makes sense. In fact, Equation (15) implies that M_{1}⊗_{R}M_{2} is again a left H-module under the action defined by Equation (17). Left H-modules are also called representations of the Hopf algebroid H (as an associative algebra). The category of representations of H is denoted by RepH.

There is an equivalent version for the compatibility condition 3), which was used by Lu [31]:

Proposition 3.2 The compatibility condition 3) (Equations (14) and (15)) is equivalent to that the kernel of the map

Ψ : H⊗H⊗H ⟶ H⊗_{R}H : ∑ h_{1}⊗h_{2}⊗h_{3} ⟼ ∑(Δh_{1})(h_{2}⊗h_{3})  (18)

is a left ideal of H⊗H^{op}⊗H^{op}, where H^{op} denotes H with the opposite product.

Proof. Assume that KerΨ is a left ideal. It is clear that for any a ∈ H, 1⊗β(a)⊗1 − 1⊗1⊗α(a) ∈ KerΨ. Hence h⊗β(a)⊗1−h⊗1⊗α(a) = (h⊗1⊗1)(1⊗β(a)⊗1−1⊗1⊗α(a)) belongs to KerΨ. That is, Δ(h)(β(a)⊗1 − 1⊗α(a)) = 0. To prove Equation (15), we assume that Δh_{2} = ∑_{ij} g_{i}⊗_{R}g_{j} for some g_{i}, g_{j} ∈ H. Then h_{2}⊗1⊗1 − ∑_{ij} 1⊗g_{i}⊗g_{j} ∈ KerΨ. This implies that h_{1}h_{2}⊗1⊗1 − ∑_{ij} h_{1}⊗g_{i}⊗g_{j} ∈ KerΨ since KerΨ is a left ideal. Hence Δ(h_{1}h_{2}) − ∑_{ij}(Δh_{1})(g_{i}⊗g_{j}) = 0 in H⊗_{R}H. I.e., Δ(h_{1}h_{2}) = Δ(h_{1})Δ(h_{2}).

Conversely, assume that Equations (14) and (15) hold. Suppose that ∑ h_{1}⊗h_{2}⊗h_{3} ∈ KerΨ. Then for any x, y, z ∈ H, note that in H⊗H^{op}⊗H^{op}, we have (x⊗y⊗z) ∑(h_{1}⊗h_{2}⊗h_{3}) = ∑ xh_{1}⊗h_{2}y⊗h_{3}z. Then

Ψ((x⊗y⊗z) ∑(h_{1}⊗h_{2}⊗h_{3}))
= ∑ Δ(xh_{1})(h_{2}y⊗h_{3}z)
= Δx ∑(Δh_{1})(h_{2}⊗h_{3})(y⊗z)
= 0.

That is, KerΨ is a left ideal.

□

Remark (1) In [31], objects satisfying the above axioms are called bi-algebroids, while the term of Hopf algebroids is reserved to those admitting an antipode. In our definition above, we choose not to require the existence of antipode because many interesting examples, as shown below, often do not admit an antipode.

(2) In the classical case, the compatibility between the Poisson structure and the groupoid structure implies that the base manifold is a coisotropic submanifold of the Poisson groupoid [39]. Being considered as a quantum counterpart, for a Hopf algebroid, it would be natural to expect that the quantum analogue should hold as well, which means that the kernel of ϵ is a left ideal of H. However, we are not able to prove this at the moment (note that this extra condition was required in the definition in [31]).

In most situation, Hopf algebroids are equipped with an additional structure, called an anchor map. Let (H, R, α, β, m, Δ, ϵ) be a Hopf algebroid (over the ground field k of characteristic zero). By End_{k}R, we denote the algebra of linear endmorphisms of R over k. It is clear that End_{k}R is an (R, R)-bimodule, where R acts on it from the left by left multiplication and acts from the right by right multiplication. Assume that R is a left H-module and moreover the representation μ : H ⟶ End_{k}R is an (R, R)-bimodule map. For any x ∈ H and a ∈ R, we denote by x(a) the element μ(x)(a) in R. Define

φ_{α}, φ_{β} : (H⊗_{R}H) ⊗ R ⟶ H,
φ_{α}(x⊗_{R}y ⊗ a) = x(a) · y, and φ_{β}(x⊗_{R}y ⊗ a) = x · y(a).  (19)

Here x, y ∈ H, a ∈ R, and the dot · denotes the (R, R)-bimodule structure on H. Note that φ_{α} and φ_{β} are well defined since μ is an (R, R)-bimodule map.

Proposition 3.3 Under the above assumption, and moreover assume that

φ_{α}(Δx ⊗ a) = xα(a), and φ_{β}(Δx ⊗ a) = xβ(a), ∀x ∈ H, a ∈ R.  (20)

Then the map ϵ̃: H ⟶ R, ϵ̃x = x(1_{R}), satisfies the co-unit property. I.e., it is an (R, R)-bimodule map, ϵ̃(1_{H}) = 1_{R}, and satisfies Equation (16).

Proof. That ϵ̃ is an (R, R)-bimodule map follows from the assumption that the representation μ is an (R, R)-bimodule map. It is clear that ϵ̃(1_{H}) = 1_{R}. To prove Equation (16), assume that Δx = ∑_{i} x_{i}^{(1)}⊗_{R}x_{i}^{(2)}. Then

(ϵ̃⊗_{R}id_{H})Δx = ∑ ϵ̃(x_{i}^{(1)})⊗_{R}x_{i}^{(2)}
= ∑ x_{i}^{(1)}(1_{R})⊗_{R}x_{i}^{(2)}
= φ_{α}(Δx⊗1_{R})
= xα(1_{R})
= x.

Similarly, we have (id_{H}⊗_{R}ϵ̃)Δx = x.

□

It is thus natural to expect that ϵ̃ coincides with the co-unit map.

Definition 3.4 Given a Hopf algebroid (H, R, α, β, m, Δ, ϵ), an anchor map is a representation μ : H ⟶ End_{k}R, which is also an (R, R)-bimodule map satisfying

(i). φ_{α}(Δx ⊗ a) = xα(a) and φ_{β}(Δx ⊗ a) = xβ(a), ∀x ∈ H, a ∈ R;

(ii). x(1_{R}) = ϵx, ∀x ∈ H.

Remark For a Hopf algebra, since R = k and End_{k}R ≅ k, one can simply take μ as the counit. In this case, μ is in fact equivalent to the counit map. However, for a Hopf algebroid, the existence of an anchor map is a stronger assumtion than the existence of a counit. In fact, we can require axioms 1)-3) in Definition (3.1) together with the existence of an anchor map to define a Hopf algebroid with anchor. Then the existence of the counit would be a consequence according to Proposition (3.3).

Given any x = ∑ x_{1}⊗_{R}x_{2} ⋯ ⊗_{R}x_{n} ∈ ⊗_{R}^{n}H and k elements (k ≤ n) a_{i_{1}}, a_{i_{2}}, ⋯, a_{i_{k}} ∈ R, we denote by x(·, ⋯, a_{i_{1}}, ⋯, ·, a_{i_{k}}, ⋯, ·) the element ∑ x_{1}⊗_{R} ⋯ ⊗_{R}x_{i_{1}}(a_{i_{1}})⊗_{R} ⋯ ⊗_{R}x_{i_{k}}(a_{i_{k}})⊗_{R} ⋯ ⊗_{R}x_{n} in ⊗_{R}^{n−k}H. The anchor map assumption gurantees that this is a well-defined element.

Proposition 3.5 For any a, b ∈ R, x ∈ H,

α(a)(b) = ab, β(a)(b) = ba;  (21)
Δ(x)(a, b) = x(ab);  (22)
ϵ(xy) = x(ϵ(y)).  (23)

Proof. To prove Equation (21), we note that α(a)(b) = (a · 1_{H})(b) = a(1_{H}(b)) = ab, where we used the fact that μ is an (R, R)-bimodule map. Similarly, we have β(a)(b) = ba.

For Equation (22), we have

Δ(x)(a, b) = φ_{α}(Δx ⊗ a)(b)
= (xα(a))(b)
= x(α(a)(b))
= x(ab).

Here the last step used Equation (21).

Finally, using (ii) in Definition (3.4), we have

ϵ(xy) = (xy)(1_{R}) = x(y(1_{R})) = x(ϵ(y)).
□

Remark Equation (21) implies that the induced (R, R)-bimodule structure on R, when R being considered as a left H-module, coincides with the usual one by (left and right) multiplications. In fact, this condition is equivalent to requiring that μ is an (R, R)-bimodule map in Definition (3.4). Equation (22) simply means that R⊗_{R}R ≅ R as left H-modules. And the last equation, Equation (23), amounts to saying that ϵ : H ⟶ R is a morphism of left H-modules, where H acts on H by left multiplication.

The following result follows immediately from definition.

Theorem 3.6 Let (H, R, α, β, m, Δ, ϵ) be a Hopf algebroid with anchor μ. Then the category RepH of left H-modules equipped with the tensor product ⊗_{R} as defined by Equation (17), the unit object (R, μ), and the trivial associativity isomorphisms: (M_{1}⊗_{R}M_{2})⊗_{R}M_{3} ⟶ M_{1}⊗_{R}(M_{2}⊗_{R}M_{3}) is a monoidal category.

Example 3.1 Let D denote the algebra of all differential operators on a smooth manifold M, and R the algebra of smooth functions on M. Then D is a Hopf algebroid over R. Here, α = β is the embedding R ⟶ D, while the coproduct Δ : D ⟶ D⊗_{R}D is defined as

Δ(D)(f, g) = D(fg), ∀D ∈ D, and f, g ∈ R.  (24)

Note that D⊗_{R}D is simply the space of bidifferential operators. Clearly, Δ is co-commutative, i.e., Δ^{op} = Δ. The usual action of differential operators on C^{∞}(P) defines an anchor μ : H ⟶ End_{k}R. In this case, the co-unit ϵ : D ⟶ R is the natural projection from a differential operator to its 0th-order part. It is easy to see that left D-modules are D-modules in the usual sense, and the tensor product is the usual tensor product of D-modules over R. We note, however, that this Hopf algebroid does not admit an antipode in any natural sense. Given a differential operator D, its antipode, if it exists, would be the dual operator D^{∗}. However, the latter is a differential operator on 1-densities, which does not possess any canonical identification with a differential operator on R.

The construction above can be generalized to show that the universal enveloping algebra UA of a Lie algebroid A admits a co-commutative Hopf algebroid structure.

Again we take R = C^{∞}(P), and let α = β : R ⟶ UA be the natural embedding. For the co-product, we set

Δ(f) = f⊗_{R}1, ∀f ∈ R;
Δ(X) = X⊗_{R}1 + 1⊗_{R}X, ∀X ∈ Γ(A).

This formula extends to a co-product Δ : UA ⟶ UA⊗_{R}UA by the compatibility condition: Equations (14) and (15). Alternatively, we may identify UA as the subalgebra of D(G) consisting of left invariant differential operators on a (local) Lie groupoid G integrating A, and then restrict the co-product Δ_{G} on D(G) to this subalgebra. This is well defined since Δ_{G} maps left invariant differential operators to left invariant bidifferential operators. Finally, the map (μx)(f) = (ρx)(f), ∀x ∈ UA, f ∈ R defines an anchor, and the co-unit map is then the projection ϵ : UA ⟶ R, where ρ : UA ⟶ D(P) denotes the algebra homomorphism extending the anchor of the Lie algebroid anchor (denoted by the same symbol ρ).

Theorem 3.7 (UA, R, α, β, m, Δ, ϵ) is a co-commutative Hopf algebroid with anchor μ.

4 Twist construction

As in the Hopf algebra case, twist construction is an important method of producing new Hopf algebroids. This section is devoted to the study on this useful construction. We start with the following

Proposition 4.1 Let (H, R, α, β, m, Δ, ϵ) be a Hopf algebroid with anchor μ, and let φ_{α} and φ_{β} be the maps defined by Equation (19). Then for any x, y, z ∈ H and a ∈ R,

φ_{α}((Δx)(y⊗_{R}z) ⊗ a) = xα(y(a))z;  (25)
φ_{β}((Δx)(y⊗_{R}z) ⊗ a) = xβ(z(a))y.  (26)

Proof. Assume that Δx = ∑_{i} x_{i}^{(1)}⊗_{R}x_{i}^{(2)}. Then

φ_{α}((Δx)(y⊗_{R}z) ⊗ a) = φ_{α}(∑_{i}(x_{i}^{(1)}y⊗_{R}x_{i}^{(2)}z) ⊗ a)
= ∑_{i} α((x_{i}^{(1)}y)(a))x_{i}^{(2)}z
= ∑_{i} α(x_{i}^{(1)}(y(a)))x_{i}^{(2)}z.

On the other hand, using (i) in Definition 3.4,

xα(y(a))z = φ_{α}(Δx ⊗ y(a))z
= φ_{α}(∑_{i} x_{i}^{(1)}⊗_{R}x_{i}^{(2)} ⊗ y(a))z
= ∑_{i} α(x_{i}^{(1)}(y(a)))x_{i}^{(2)}z.

Hence, φ_{α}((Δx)(y⊗_{R}z) ⊗ a) = xα(y(a))z. Equation (26) can be proved similarly.

□

Now let F be an element in H⊗_{R}H. Define α_{F}, β_{F} : R ⟶ H, respectively, by

α_{F}(a) = φ_{α}(F⊗a), β_{F}(a) = φ_{β}(F⊗a), ∀a ∈ R.  (27)

And for any a, b ∈ R, set

a ∗_{F} b = α_{F}(a)(b).  (28)

More explicitly, if F = ∑_{i} x_{i}⊗_{R}y_{i} for x_{i}, y_{i} ∈ H, then ∀a, b ∈ R,

α_{F}(a) = ∑_{i} x_{i}(a) · y_{i} = ∑_{i} α(x_{i}(a))y_{i},  (29)
β_{F}(a) = ∑_{i} x_{i} · y_{i}(a) = ∑_{i} β(y_{i}(a))x_{i}, and  (30)
a ∗_{F} b = ∑_{i} x_{i}(a)y_{i}(b).  (31)

Proposition 4.2 Assume that F ∈ H⊗_{R}H satisfies:

(Δ⊗_{R}id)FF^{12} = (id⊗_{R}Δ)FF^{23} in H⊗_{R}H⊗_{R}H; and  (32)
(ϵ⊗_{R}id)F = 1_{H}; (id⊗_{R}ϵ)F = 1_{H}.  (33)

Here F^{12} = F ⊗1 ∈ (H⊗_{R}H)⊗H, F^{23} = 1⊗F ∈ H⊗(H⊗_{R}H), and in Equation (33) we have used the identification: R⊗_{R}H ≅ H⊗_{R}R ≅ H (note that both maps on the left hand sides of Equation (33) are well-defined). Then

(i). (R, ∗_{F}) is an associative algebra, and 1_{R} ∗_{F} a = a ∗_{F} 1_{R} = a, ∀a ∈ R.

(ii). α_{F} : R_{F} ⟶ H is an algebra homomorphism, and β_{F} : R_{F} ⟶ H is anti-homomorphism. Here R_{F} stands for the algebra (R, ∗_{F}).

(iii). α_{F}(a)β_{F}(b) = β_{F}(b)α_{F}(a), ∀a, b ∈ R

Proof. As a first step, we prove that for any a, b ∈ R,

α_{F}(a ∗_{F} b) = (α_{F}a)(α_{F}b),  (34)
β_{F}(a ∗_{F} b) = (β_{F}b)(β_{F}a).  (35)

Assume that F = ∑_{i} x_{i}⊗_{R}y_{i} for x_{i}, y_{i} ∈ H. Then

(Δ⊗_{R}id)FF^{12} = ∑_{ij} Δx_{i}(x_{j}⊗_{R}y_{j})⊗_{R}y_{i},  (36)
(id⊗_{R}Δ)FF^{23} = ∑_{ij} x_{i}⊗_{R}Δy_{i}(x_{j}⊗_{R}y_{j}).  (37)

Thus

[(Δ⊗_{R}id)FF^{12}](a, b, ·) = ∑_{ij} Δx_{i}(x_{j} ⊗ y_{j})(a, b)⊗_{R}y_{i}
= ∑_{ij} x_{i}(x_{j}(a)y_{j}(b))⊗_{R}y_{i}
= ∑_{ij} α[x_{i}(x_{j}(a)y_{j}(b))]y_{i}
= α_{F}(∑_{j} x_{j}(a)y_{j}(b))
= α_{F}(a ∗_{F} b),

where the second equality used Equation (22).

On the other hand,

[(id⊗_{R}Δ)FF^{23}](a, b, ·) = ∑_{ij} x_{i}(a)⊗_{R}φ_{α}(Δy_{i}(x_{j}⊗_{R}y_{j}) ⊗ b)
= ∑_{ij} x_{i}(a)⊗_{R}y_{i}α(x_{j}(b))y_{j}
= ∑_{ij} α(x_{i}(a))y_{i}α(x_{j}(b))y_{j}
= (α_{F}a)(α_{F}b).

Thus Equation (34) follows from Equation (32). The equation β_{F}(a ∗_{F} b) = (β_{F}b)(β_{F}a) can be proved similarly.

Now for any a, b, c ∈ R, [(α_{F}a)(α_{F}b)](c) = (α_{F}a)((α_{F}b)(c)) = a ∗_{F} (b ∗_{F} c). On the other hand, α_{F}(a ∗_{F} b)(c) = (a ∗_{F} b) ∗_{F} c. The associativity of R_{F} thus follows from Equation (34).

Finally, we have α_{F}(1_{R}) = ∑_{i} x_{i}(1_{R}) · y_{i} = ∑_{i} ϵ(x_{i}) · y_{i} = (ϵ ⊗_{R} id)F = 1_{H}. Similarly, β_{F}(1_{R}) = 1_{H}. It thus follows that 1_{R} ∗_{F} a = α_{F}(1_{R})(a) = 1_{H}(a) = a. Similarly, a ∗_{F} 1_{R} = a.

For the last statement, a similar computation leads to [(Δ⊗_{R}id)FF^{12}](a, ·, b) = β_{F}b ∗_{F} α_{F}a, and

[(id⊗_{R}Δ)FF^{23}](a, ·, b) = α_{F}a ∗_{F} β_{F}b.

Thus (iii) follows immediately. This concludes the proof.

□

Proposition 4.3 Under the same hypothesis as in Proposition 4.2, we have

F(β_{F}(a) ⊗ 1 − 1⊗α_{F}(a)) = 0 in H⊗_{R}H, ∀a ∈ R.  (38)

Proof.

F(β_{F}(a) ⊗ 1 − 1⊗α_{F}(a))
= (∑_{i} x_{i}⊗_{R}y_{i})(∑_{j} β(y_{j}(a))x_{j} ⊗ 1 − ∑_{j} 1 ⊗ α(x_{j}(a))y_{j})
= ∑_{ij}[x_{i}β(y_{j}(a))x_{j}⊗_{R}y_{i} − x_{i}⊗_{R}y_{i}α(x_{j}(a))y_{j}].

Now using Equations (36) (37) and (25) (26), we obtain

[(Δ⊗_{R}id)FF^{12}](·, a, ·)

= ∑_{ij} φ_{β}(Δx_{i}(x_{j}⊗_{R}y_{j}) ⊗ a)⊗_{R}y_{i}
= ∑_{ij} x_{i}β(y_{j}(a))x_{j}⊗_{R}y_{i},

and

[(id⊗_{R}Δ)FF^{23}](·, a, ·)
= ∑_{ij} x_{i}⊗_{R}φ_{α}(Δy_{i}(x_{j}⊗_{R}y_{j}) ⊗ a)
= ∑_{ij} x_{i}⊗_{R}y_{i}α(x_{j}(a))y_{j}.

Thus the conclusion follows immediately from Equation (32).

□

As an immediate consequence, we have Corollary 4.4 Let M_{1} and M_{2} be any left H-modules. Then

F^{#} : M_{1} ⊗_{R_{F}} M_{2} ⟶ M_{1}⊗_{R}M_{2}
(m_{1} ⊗_{R_{F}} m_{2}) ⟶ F · (m_{1}⊗m_{2}), m_{1} ∈ M_{1}, and m_{2} ∈ M_{2},  (39)

is a well defined linear map.

Note that M_{1}⊗_{R}M_{2} is automatically an (R, R)-bimodule since both M_{1} and M_{2} are (R, R)- bimodules. Similarly, M_{1} ⊗_{R_{F}} M_{2} is an (R_{F}, R_{F})-bimodule. In the meantime, M_{1}⊗_{R}M_{2} is also a left H-module. The next lemma indicates how these module structures are related.

Lemma 4.5 For any a ∈ R and m ∈ M_{1} ⊗_{R_{F}} M_{2},

F^{#}(a ·_{F} m) = α_{F}(a) · F^{#}(m);  (40)
F^{#}(m ·_{F} a) = β_{F}(a) · F^{#}(m),  (41)

where the dot on the RHS means the left H-action on M_{1}⊗_{R}M_{2}, and the dot ·_{F} on the LHS refers to the both left and right R_{F}-actions on M_{1} ⊗_{R_{F}} M_{2}.

Proof. For simplicity, let us assume that m = m_{1} ⊗_{R_{F}} m_{2} for m_{1} ∈ M_{1} and m_{2} ∈ M_{2}. Then

F^{#}(a ·_{F} (m_{1} ⊗_{R_{F}} m_{2}))
= F^{#}((α_{F}(a)m_{1}) ⊗_{R_{F}} m_{2})
= F · (∑_{i} α(x_{i}(a))y_{i}m_{1} ⊗_{R_{F}} m_{2})
= ∑_{ij} x_{j}α(x_{i}(a))y_{i}m_{1}⊗_{R}y_{j}m_{2}
= (Δ⊗_{R}id)FF^{12}(a, ·, ·) · (m_{1}⊗m_{2}).

On the other hand,

α_{F}(a) · F^{#}(m_{1} ⊗_{R_{F}} m_{2})
= α_{F}(a) · (∑_{j} x_{j}m_{1}⊗_{R}y_{j}m_{2})
= ∑_{j} Δ(α_{F}(a))(x_{j}m_{1}⊗_{R}y_{j}m_{2})
= ∑_{ij} Δ(α(x_{i}(a))y_{i})(x_{j}m_{1}⊗_{R}y_{j}m_{2})
= ∑_{ij}((x_{i}(a) · Δy_{i})(x_{j}m_{1}⊗_{R}y_{j}m_{2})
= ∑_{ij}[(x_{i}(a)⊗_{R}Δy_{i}(x_{j}⊗_{R}y_{j})](m_{1} ⊗ m_{2})
= (id⊗_{R}Δ)FF^{23}(a, ·, ·) · (m_{1} ⊗ m_{2}).

The conclusion thus follows again from Equation (32).

□

We say that F is invertible if F^{#} defined by Equation (39) is a vector space isomorphism for any left H-modules M_{1} and M_{2}. In this case, in particular we can take M_{1} = M_{2} = H so that we have an isomorphism

F^{#} : H ⊗_{R_{F}} H ⟶ H⊗_{R}H.  (42)

An immediate consequence of Lemma 4.5 is the following Corollary 4.6 If F is invertible, then for any a ∈ R_{F} and n ∈ M_{1}⊗_{R}M_{2},

F^{#−1}(α_{F}(a) · n) = a ·_{F} F^{#−1}(n);

F^{#−1}(β_{F}(a) · n) = F^{#−1}(n) ·_{F} a.

Definition 4.7 An element F ∈ H⊗_{R}H is called a twistor if it is invertible and satisfies Equations (32) and (33).

Now assume that F is a twistor. Define a new coproduct Δ_{F} : H ⟶ H ⊗_{R_{F}} H by

Δ_{F} = F^{−1}ΔF,  (43)

where Equation (43) means that Δ_{F}(x) = F^{#−1}(Δ(x)F), ∀x ∈ H. In what follows, we will prove that Δ_{F} is indeed a Hopf algebroid co-product. Lemma 4.8 For any x ∈ H and a ∈ R_{F},

Δ(a ·_{F} x) = α_{F}(a) · Δx;  (44)
Δ(x ·_{F} a) = β_{F}(a) · Δx,  (45)

where ·_{F} refers to the R_{F} (both the left and the right) actions on H. Proof. We have

Δ(a ·_{F} x)
= Δ(α_{F}(a)x)
= Δ(α_{F}(a))Δx
= α_{F}(a) · Δx.

Equation (45) can be proved similarly.

□

Proposition 4.9 Δ_{F} : H ⟶ H ⊗_{R_{F}} H is an (R_{F}, R_{F})-bimodule map. Proof. For any a ∈ R_{F} and x ∈ H, using Lemma 4.8 and Corollary 4.6, we have

Δ_{F}(a ·_{F} x)

= F^{#−1}[Δ(a ·_{F} x)F]
= F^{#−1}[α_{F}(a) · ΔxF]
= a ·_{F} F^{#−1}(ΔxF)
= a ·_{F} Δ_{F}(x).

Similarly, we can show that Δ_{F}(x ·_{F} a) = Δ_{F}x ·_{F} a.

□

Proposition 4.10 The comultiplication Δ_{F} : H ⟶ H⊗_{R_{F}} H is compatible with the multiplication in H.

Proof. Consider the maps

Ψ : H⊗H⊗H ⟶ H⊗_{R}H : ∑ h_{1}⊗h_{2}⊗h_{3} ⟼ ∑(Δh_{1})(h_{2}⊗h_{3}),  (46)

and

Ψ_{F} : H⊗H⊗H ⟶ H ⊗_{R_{F}} H : ∑ h_{1}⊗h_{2}⊗h_{3} ⟼ ∑(Δ_{F}h_{1})(h_{2}⊗h_{3}).  (47)

We first prove that

F^{#}∘Ψ_{F} = Ψ∘F_{23}^{#}.  (48)
(F^{#}∘Ψ_{F})(∑ h_{1}⊗h_{2}⊗h_{3}) = ∑ F^{#}((Δ_{F}h_{1})(h_{2}⊗h_{3}))
= ∑ F^{#}[F^{#}^{−1}((Δh_{1})F)(h_{2}⊗h_{3})]
= ∑ Δh_{1}F(h_{2}⊗h_{3})
= (Ψ∘F_{23}^{#})(∑ h_{1}⊗h_{2}⊗h_{3}).

Equation (48) implies that KerΨ_{F} = KerΨ. To see this, note that x ∈ KerΨ_{F}, i.e., Ψ_{F}(x) = 0 is equivalent to (F^{#}∘Ψ_{F})(x) = 0, which is equivalent to (Ψ∘F_{23}^{#})(x) = 0, or F_{23}^{#}(x) ∈ KerΨ. Since KerΨ is a left ideal in H⊗H^{op}⊗H^{op}, the latter is equivalent to that x ∈ KerΨ. The final conclusion thus follows from Proposition 3.2.

□

Proposition 4.10 implies that M_{1}⊗_{R_{F}} M_{2} is still a left H-module for any left H-modules M_{1}, M_{2}. Moreover, it is easy to see that F^{#} : M_{1}⊗_{R_{F}} M_{2} ⟶ M_{1}⊗_{R}M_{2} is an isomorphism of left H-modules. Now we are ready to prove that Δ_{F} is coassociative.

Proposition 4.11 Δ_{F} : H ⟶ H ⊗_{R_{F}} H is coassociative.

Proof. Suppose that M_{1}, M_{2}, M_{3} are any left H-modules. It suffices to prove that the natural identification φ_{F} : (M_{1} ⊗_{R_{F}} M_{2}) ⊗_{R_{F}} M_{3} ⟶ M_{1} ⊗_{R_{F}} (M_{2} ⊗_{R_{F}} M_{3}) is an isomorphism of left H-modules.

Consider the following diagram:

(M_{1} ⊗_{R_{F}} M_{2}) ⊗_{R_{F}} M_{3} ⟶^{(F^{12})^{#}} (M_{1}⊗_{R}M_{2}) ⊗_{R_{F}} M_{3} ⟶^{[(Δ⊗_{R}id)F]^{#}} (M_{1}⊗_{R}M_{2})⊗_{R}M_{3}
φ_{F} ↓  ↓ φ
M_{1} ⊗_{R_{F}} (M_{2} ⊗_{R_{F}} M_{3}) ⟶^{(F^{23})^{#}} M_{1} ⊗_{R_{F}} (M_{2}⊗_{R}M_{3}) ⟶^{[(id⊗_{R}Δ)F]^{#}} M_{1}⊗_{R}(M_{2}⊗_{R}M_{3})

Equation (32) implies that the above diagram commutes. Since all the other maps involved in the diagram above are isomorphisms of left H-modules, φ_{F} is an H-module isomorphism as well. This concludes the proof.

□

By now, we have actually proved all the Hopf algebroid axioms for (H, R_{F}, α_{F}, β_{F}, m, Δ_{F}, ϵ) except for the condition on counit ϵ. Instead of proving this last condition directly, here we show that μ is still an anchor after the twist, and therefore axiom 4) in Definition 3.1 would be a consequence according to the remark following Definition 3.4. Note that R_{F} can still be considered as a left H-module under the representation μ : H ⟶ EndR_{F} (here only the underlying vector space structure on R_{F} is involved). We prove that μ still satisfies the anchor axioms.

Lemma 4.12 For any x, y ∈ H and a ∈ R,

φ_{α}^{F}((x ⊗_{R_{F}} y) ⊗ a) = φ_{α}(F^{#}(x ⊗_{R_{F}} y) ⊗ a);  (49)
φ_{β}^{F}((x ⊗_{R_{F}} y) ⊗ a) = φ_{β}(F^{#}(x ⊗_{R_{F}} y) ⊗ a).  (50)

Proof.

φ_{α}^{F}((x ⊗_{R_{F}} y) ⊗ a) = x(a) ·_{F} y
= α_{F}(x(a))y
= ∑_{i} α(x_{i}(x(a)))y_{i}y
= ∑_{i} α(x_{i}x(a))y_{i}y.

On the other hand,

φ_{α}(F^{#}(x ⊗_{R_{F}} y) ⊗ a)
= φ_{α}(∑_{i} x_{i}x⊗_{R}y_{i}y⊗a)
= ∑_{i} α(x_{i}x(a))y_{i}y.

Hence, φ_{α}^{F}((x⊗_{R_{F}} y)⊗a) = φ_{α}(F^{#}(x⊗_{R_{F}} y)⊗a). Similarly, one can prove that φ_{β}^{F}((x⊗_{R_{F}} y)⊗a) = φ_{β}(F^{#}(x ⊗_{R_{F}} y) ⊗ a).

□

Proposition 4.13 The map μ : H ⟶ EndR_{F} satisfies the anchor axioms in Definition 3.4 for (H, R_{F}, α_{F}, β_{F}, m, Δ_{F}, ϵ).

Proof. First we need to show that μ is an (R_{F}, R_{F})-bimodule map. This can be checked easily since (a ·_{F} x)(b) = (α_{F}(a)x)(b) = α_{F}(a)(x(b)) = a ∗_{F} x(b). Similarly, (x ·_{F} a)(b) = x(b) ∗_{F} a.

The axioms (i) and (ii) in Definition 3.4 hold automatically since τ is an anchor for (H, R, α, β, m, Δ, ϵ). Now according to Lemma 4.12

φ_{α}^{F}(Δ_{F}x⊗a)
= φ_{α}(ΔxF⊗a)
= ∑_{i} φ_{α}(Δx(x_{i}⊗_{R}y_{i})⊗a)
= ∑_{i} xα(x_{i}(a))y_{i}
= xα_{F}(a).

Here the second from the last equality used Equation (25). Similarly, φ_{β}^{F}(Δ_{F}x⊗a) = xβ_{F}(a). This concludes the proof.

□

In summary, we have proved

Theorem 4.14 Assume that (H, R, α, β, m, Δ, ϵ) is a Hopf algebroid with anchor μ, and F ∈ H⊗_{R}H a twistor. Then (H, R_{F}, α_{F}, β_{F}, m, Δ_{F}, ϵ) is a Hopf algebroid, which still admits μ as an anchor. Moreover, its corresponding monoidal category of left H-modules is equivalent to that of (H, R, α, β, m, Δ, ϵ).

We say that (H, R_{F}, α_{F}, β_{F}, m, Δ_{F}, ϵ) is obtained from (H, R, α, β, m, Δ, ϵ) by twisting via F.

The following theorem generalizes a standard result in Hopf algebras [9].

Theorem 4.15 If F_{1} ∈ H⊗_{R}H is a twistor to the Hopf algebroid H, and F_{2} ∈ H⊗_{R_{F_{1}}} H a twistor for the twisted Hopf algebroid H_{F_{1}}, then the Hopf algebroid obtained by twisting H via F_{1} then via F_{2} is equivalent to that obtained by twisting via F_{1}F_{2}. Here F_{1}F_{2} ∈ H⊗_{R}H is understood as F_{1}^{#}(F_{2}), where F_{1}^{#} : H⊗_{R_{F_{1}}} H ⟶ H⊗_{R}H is the map as defined in Equation (42).

Proof. Clearly, F = F_{1}F_{2} = F_{1}^{#}(F_{2}) is a well defined element in H⊗_{R}H. We only need to verify that F is still a twistor. The rest of the result is a routine verification. For this purpose, it suffices to show that F satisfies both Equations (32) and (33). To check these equations, in fact we may think of F_{1} and F_{2} as elements in H⊗H by taking some representatives. Now

(Δ⊗_{R}id)FF^{12}
= (Δ⊗_{R}id)F_{1}(Δ⊗_{R}id)F_{2}F_{1}^{12}F_{2}^{12}
= (Δ⊗_{R}id)F_{1}F_{1}^{12}[(F_{1}^{12})^{−1}(Δ⊗_{R}id)F_{2}F_{1}^{12}]F_{2}^{12}
= [(Δ⊗_{R}id)F_{1}F_{1}^{12}][(Δ_{F_{1}} ⊗_{R_{F}} id)F_{2}F_{2}^{12}]
= [(id⊗_{R}Δ)F_{1}F_{1}^{23}][(id ⊗_{R_{F}} Δ_{F_{1}})F_{2}F_{2}^{23}]
= (id⊗_{R}Δ)F_{1}(id⊗_{R}Δ)F_{2}F_{1}^{23}F_{2}^{23}
= (id⊗_{R}Δ)FF^{23}.

To prove Equation (33), assume that F_{1} = ∑_{i} x_{i}^{(1)}⊗_{R}y_{i}^{(1)}, and F_{2} = ∑_{i} x_{i}^{(2)}⊗_{R_{F_{1}}} y_{i}^{(2)}. Then

F_{1}F_{2} = ∑_{ij} x_{i}^{(1)}x_{j}^{(2)}⊗_{R}y_{i}^{(1)}y_{j}^{(2)}. And

(ϵ⊗_{R}id)F
= ∑_{ij} ϵ(x_{i}^{(1)}x_{i}^{(2)})⊗_{R}y_{i}^{(1)}y_{j}^{(2)}
= ∑_{ij} ϵ(x_{i}^{(1)}x_{i}^{(2)}) · y_{i}^{(1)}y_{j}^{(2)} (using Equation (23))
= ∑_{ij} x_{i}^{(1)}(ϵx_{i}^{(2)}) · y_{i}^{(1)}y_{j}^{(2)}
= ∑_{ij} φ_{α}(x_{i}^{(1)}⊗_{R}y_{i}^{(1)}⊗ϵx_{j}^{(2)})y_{j}^{(2)}
= ∑_{j} φ_{α}(F_{1}⊗ϵx_{j}^{(2)})y_{j}^{(2)}
= ∑_{j} α_{F_{1}}(ϵx_{j}^{(2)})y_{j}^{(2)}
= (ϵ⊗_{R_{F_{1}}} id)F_{2}
= 1_{H}.

Similarly, we prove that (id⊗_{R}ϵ)F = 1_{H}.

□

We end this section by the following:

Example 4.1 Let P be a smooth manifold, D the algebra of differential operators on P, and R = C^{∞}(P). Let D[[ℏ]] denote the space of formal power series in ℏ with coefficients in D. The Hopf algebroid structure on D naturally extends to a Hopf algebroid structure on D[[ℏ]] over the base R[[ℏ]], which also admits a natural anchor map.

Let F = 1⊗_{R}1 + ℏB_{1} + ⋯ ∈ D⊗_{R}D[[ℏ]](≅ D[[ℏ]]⊗_{R[[ℏ]]}D[[ℏ]]) be a formal power series of bidifferential operators. It is easy to see that F is a twistor iff the multiplication on R[[ℏ]] defined by:

f ∗_{ℏ} g = F(f, g), ∀f, g ∈ R[[ℏ]]  (51)

is associative with identity being the constant function 1, i.e., ∗_{ℏ} is a star product on P. In this case, the bracket {f, g} = B_{1}(f, g) − B_{1}(g, f), ∀f, g ∈ C^{∞}(P) defines a Poisson bracket on P, and f ∗_{ℏ} g = F(f, g) is simply a deformation quantization of such a Poisson structure [6].

The twisted Hopf algebroid can be easily described. Here D_{ℏ} = D[[ℏ]] is equipped with the usual multiplication, R_{ℏ} = R[[ℏ]] is the ∗-product defined by Equation (51), α : R_{ℏ} ⟶ D_{ℏ} and β : R_{ℏ} ⟶ D_{ℏ} are given, respectively, by

α_{ℏ}(f)g = f ∗_{ℏ} g, β(f)g = g ∗_{ℏ} f, ∀f, g ∈ R.

The co-product Δ_{ℏ} : D_{ℏ} ⟶ D_{ℏ}⊗_{R_{ℏ}}D_{ℏ} is

Δ_{ℏ} = F^{−1}ΔF,

and the co-unit ϵ remains the same, i.e., the projection D_{ℏ} ⟶ R_{ℏ}. This twisted Hopf algebroid (D_{ℏ}, R_{ℏ}, α, β, m, Δ_{ℏ}, ϵ) is called the quantum groupoid associated to the star product ∗_{ℏ}.

5 Quantum groupoids and their classical limits

The main purpose of this section is to introduce quantum universal enveloping algebroids (QUE algebroids), also called quantum groupoids in the paper, as a deformation of the standard Hopf algebroid UA.

Definition 5.1 A deformation of a Hopf algebroid (H, R, α, β, m, Δ, ϵ) over a field k is a topological Hopf algebroid (H_{ℏ}, R_{ℏ}, α_{ℏ}, β_{ℏ}, m_{ℏ}, Δ_{ℏ}, ϵ_{ℏ}) over the ring k[[ℏ]] of formal power series in ℏ such that

(i). H_{ℏ} is isomorphic to H[[ℏ]] as k[[ℏ]] module with identity 1_{H}, and R_{ℏ} is isomorphic to R[[ℏ]] as k[[ℏ]] module with identity 1_{R};

(ii). α_{ℏ} = α(mod ℏ), β_{ℏ} = β(mod ℏ), m_{ℏ} = m(mod ℏ), ϵ_{ℏ} = ϵ(mod ℏ);

(iii). Δ_{ℏ} = Δ(mod ℏ).

In this case, we simply say that the quotient H_{ℏ}/ℏH_{ℏ} is isomorphic to H as a Hopf algebroid.

Here the meaning of (i) and (ii) is clear. However, for Condition (iii), we need the following simple fact:

Lemma 5.2 Under the hypothesis (i) and (ii) as in Definition 5.1, H_{ℏ}⊗_{R_{ℏ}}H_{ℏ}/ℏ(H_{ℏ}⊗_{R_{ℏ}}H_{ℏ}) is isomorphic to H⊗_{R}H as a vector space.

Proof. Define τ : H_{ℏ}⊗H_{ℏ} ⟶ H⊗_{R}H by

(∑_{i} x_{i}ℏ^{i})⊗(∑_{i} y_{i}ℏ^{i}) ⟶ x_{0}⊗_{R}y_{0}.

For any a ∈ R and x, y ∈ H, since

(β_{ℏ}a⊗1 − 1⊗α_{ℏ}a)(x⊗y)
= (βa)x⊗y − x⊗(αa)y + O(ℏ),

then τ[(β_{ℏ}a⊗1 − 1⊗α_{ℏ}a)(x⊗y)] = 0. In other words, τ descends to a well defined map from H_{ℏ}⊗_{R_{ℏ}}H_{ℏ} to H⊗_{R}H. It is easy to see that τ is surjective and Kerτ = ℏ(H_{ℏ}⊗_{R_{ℏ}}H_{ℏ}). The conclusion thus follows immediately.

□

By abuse of notation, we will still use τ to denote the induced map H_{ℏ}⊗_{R_{ℏ}}H_{ℏ} ⟶ H⊗_{R}H. We shall also use the notation ℏ ↦ 0 to denote this map whenever the meaning is clear from the context. Then, Condition (iii) means that lim_{ℏ↦0}Δ_{ℏ}(x) = Δ(x) for any x ∈ H.

Definition 5.3 A quantum universal enveloping algebroid (or a QUE algebroid), also called a quantum groupoid, is a deformation of the standard Hopf algebroid (UA, R, α, β, m, Δ, ϵ) of a Lie algebroid A.

Let U_{ℏ}A = UA[[ℏ]] and R_{ℏ} = R[[ℏ]]. Assume that (U_{ℏ}A, R_{ℏ}, α_{ℏ}, β_{ℏ}, m_{ℏ}, Δ_{ℏ}, ϵ_{ℏ}) is a quantum groupoid. Then R_{ℏ} defines a star product on P so that the equation

\{f, g\} = lim_{ℏ↦0} \frac{1}{ℏ}(f ∗_{ℏ} g − g ∗_{ℏ} f), ∀f, g ∈ R

defines a Poisson structure on the base space P.

Now define

δf = lim_{ℏ↦0} \frac{1}{ℏ}(α_{ℏ}f − β_{ℏ}f) ∈ UA, ∀f ∈ R,
Δ^{1}X = lim_{ℏ↦0} \frac{1}{ℏ}(Δ_{ℏ}X − (1⊗_{R_{ℏ}}X + X⊗_{R_{ℏ}}1)) ∈ UA⊗_{R}UA, ∀X ∈ Γ(A), and
δX = Δ^{1}X − (Δ^{1}X)_{21} ∈ UA⊗_{R}UA.

Here for T = ∑ x⊗_{R}y ∈ UA⊗_{R}UA, T_{21} is defined to be ∑ y⊗_{R}x. For the convenience of notations, we introduce

AltT = T − T_{21}, ∀T ∈ UA⊗_{R}UA

so that δX = AltΔ^{1}X.

Below we will use ∗_{ℏ} to denote both the multiplications in U_{ℏ}A and in R_{ℏ} provided there is no confusion. For any f, g ∈ R, x, y ∈ UA, write

α_{ℏ}f = f + ℏα_{1}f + ℏ^{2}α_{2}f + O(ℏ^{3});
β_{ℏ}f = f + ℏβ_{1}f + ℏ^{2}β_{2}f + O(ℏ^{3});
f ∗_{ℏ} g = fg + ℏB_{1}(f, g) + O(ℏ^{2});
x ∗_{ℏ} y = xy + ℏm_{1}(x, y) + O(ℏ^{2}),

where α_{1}f, β_{1}f, α_{2}f, β_{2}f and m_{1}(x, y) are elements in UA. Hence,

{f, g} = B_{1}(f, g) − B_{1}(g, f), and
δf = α_{1}f − β_{1}f.

Lemma 5.4 For any f, g ∈ R,

(i). α_{1}(fg) = g(α_{1}f) + f(α_{1}g) + [α_{1}f, g] + m_{1}(f, g) − B_{1}(f, g);

(ii). β_{1}(fg) = g(β_{1}f) + f(β_{1}g) + [β_{1}f, g] + m_{1}(f, g) − B_{1}(g, f);

(iii). [α_{1}f, g] − [β_{1}g, f] = m_{1}(g, f) − m_{1}(f, g).

Proof. From the identity α_{ℏ}(f ∗_{ℏ} g) = α_{ℏ}f ∗_{ℏ} α_{ℏ}g, it follows, by considering its ℏ-terms, that

α_{1}(fg) + B_{1}(f, g) = m_{1}(f, g) + (α_{1}f)g + f(α_{1}g).

Thus (i) follows immediately. And (ii) can be proved similarly.

On the other hand, we know, from the definition of Hopf algebroids, that (α_{ℏ}f) ∗_{ℏ} (β_{ℏ}g) = (β_{ℏ}g) ∗_{ℏ} (α_{ℏ}f). By considering its ℏ-terms, we obtain

(α_{1}f)g + f(β_{1}g) + m_{1}(f, g) = g(α_{1}f) + (β_{1}g)f + m_{1}(g, f).

This proves (iii).

□

Corollary 5.5 For any f, g ∈ R, (i). δ(fg) = fδg + gδf; (ii). [δf, g] = {f, g}.

Proof. By symmetrizing the third identity in Lemma 5.4, we obtain that

[α_{1}f − β_{1}f, g] − [β_{1}g − α_{1}g, f] = 0,

i.e.,

[δf, g] = −[δg, f].

Now subtracting Equation (ii) from Equation (i) in Lemma 5.4, one obtains that

δ(fg) = gδf + fδg + [δf, g] − {f, g}.

I.e.,

δ(fg) − (gδf + fδg) = [δf, g] − {f, g}.

Note that the LHS of this equation is symmetric with respect to f and g, whereas the RHS is skew-symmetric, so both sides must vanish. The conclusion thus follows immediately.

□

Lemma 5.6 For any f ∈ R, (i). Δ_{ℏ}f = f⊗_{R_{ℏ}}1 + ℏ(α_{1}f⊗_{R_{ℏ}}1 − Δ_{ℏ}α_{1}f) + ℏ^{2}(α_{2}f⊗_{R_{ℏ}}1 − Δ_{ℏ}α_{2}f) + O(ℏ^{3}); (ii). Δ_{ℏ}f = 1⊗_{R_{ℏ}}f + ℏ(1⊗_{R_{ℏ}}β_{1}f − Δ_{ℏ}β_{1}f) + ℏ^{2}(1⊗_{R_{ℏ}}β_{2}f − Δ_{ℏ}β_{2}f) + O(ℏ^{3}); (iii). f⊗_{R_{ℏ}}1 − 1⊗_{R_{ℏ}}f = ℏ(1⊗_{R_{ℏ}}α_{1}f − β_{1}f⊗_{R_{ℏ}}1) + ℏ^{2}(1⊗_{R_{ℏ}}α_{2}f − β_{2}f⊗_{R_{ℏ}}1) + O(ℏ^{3}).

Proof. Since Δ_{ℏ} : U_{ℏ}A ⟶ U_{ℏ}A⊗_{R_{ℏ}}U_{ℏ}A is an (R_{ℏ}, R_{ℏ})-bimodule map, it follows that

Δ_{ℏ}(f ·_{ℏ} 1) = f ·_{ℏ} Δ_{ℏ}1.

Here, as well as in the sequel, ·_{ℏ} denotes the R_{ℏ} (both left and right) actions on U_{ℏ}A, or on its appropriate tensor powers.

Now

f ·_{ℏ} 1 = α_{ℏ}f = f + ℏα_{1}f + ℏ^{2}α_{2}f + O(ℏ^{3}),

while

f ·_{ℏ} Δ_{ℏ}1 = f ·_{ℏ} (1⊗_{R_{ℏ}}1) = α_{ℏ}f⊗_{R_{ℏ}}1 = (f + ℏα_{1}f + ℏ^{2}α_{2}f)⊗_{R_{ℏ}}1 + O(ℏ^{3}).

Thus it follows that

Δ_{ℏ}f = f⊗_{R_{ℏ}}1 + ℏ(α_{1}f⊗_{R_{ℏ}}1 − Δ_{ℏ}α_{1}f) + ℏ^{2}(α_{2}f⊗_{R_{ℏ}}1 − Δ_{ℏ}α_{2}f) + O(ℏ^{3}).

Similarly, one can prove (ii).

Finally, since 1⊗_{R_{ℏ}}α_{ℏ}f = β_{ℏ}f⊗_{R_{ℏ}}1, we have

1⊗_{R_{ℏ}}(f + ℏα_{1}f + ℏ^{2}α_{2}f + O(ℏ^{3})) = (f + ℏβ_{1}f + ℏ^{2}β_{2}f + O(ℏ^{3}))⊗_{R_{ℏ}}1.

This implies (iii).

□

Corollary 5.7 For any f, g ∈ R,

(i). Δ(δf) = δf⊗_{R}1 + 1⊗_{R}δf;

(ii). Δ^{1}(δf) = ν_{2}f⊗_{R}1 + 1⊗_{R}ν_{2}f − Δν_{2}f, where ν_{2}f = α_{2}f − β_{2}f.

Proof. Combining the three identities in Lemma 5.6 ((i)-(ii)+(iii)), we obtain that

δf⊗_{R_{ℏ}}1 + 1⊗_{R_{ℏ}}δf − Δ_{ℏ}δf
+ℏ[α_{2}f⊗_{R_{ℏ}}1 + 1⊗_{R_{ℏ}}α_{2}f − β_{2}f⊗_{R_{ℏ}}1 − 1⊗_{R_{ℏ}}β_{2}f − Δ_{ℏ}(α_{2}f − β_{2}f)] + O(ℏ^{2}) = 0.

I.e.,

δf⊗_{R_{ℏ}}1 + 1⊗_{R_{ℏ}}δf − Δ_{ℏ}δf + ℏ[ν_{2}f⊗_{R_{ℏ}}1 + 1⊗_{R_{ℏ}}ν_{2}f − Δ_{ℏ}ν_{2}f] + O(ℏ^{2}).  (52)

By letting ℏ ↦ 0, this implies that δf⊗_{R}1 + 1⊗_{R}δf − Δ(δf) = 0. This concludes the proof of (i).

Now writing Δ_{ℏ}δf = δf⊗_{R_{ℏ}}1+1⊗_{R_{ℏ}}δf + ℏΔ1_{ℏ}(δf), and substituting it back into Equation (52), we obtain

Δ1_{ℏ}(δf) = ν_{2}f⊗_{R_{ℏ}}1 + 1⊗_{R_{ℏ}}ν_{2}f − Δ_{ℏ}ν_{2}f + O(ℏ).

(ii) thus follows immediately by letting ℏ ↦ 0.

□

An immediate consequence is Corollary 5.8 For any f ∈ R, δf ∈ Γ(A) and δ^{2}f = 0. Proof. From Corollary 5.7 (i), it follows that δf is primitive, i.e., δf ∈ Γ(A). According to Corollary 5.7 (ii), Δ^{1}(δf) is symmetric. Therefore, δ^{2}f = δ(δf), being the skew-symmetric part of Δ^{1}(δf), equals zero.

□

Lemma 5.9 For any X ∈ Γ(A),

Δ^{1}X⊗_{R}1 + (Δ⊗_{R}id)Δ^{1}X = 1⊗_{R}Δ^{1}X + (id⊗_{R}Δ)Δ^{1}X.  (53)

Proof. For any X ∈ Γ(A), denote

Δ1_{ℏ}X = \frac{1}{ℏ}[Δ_{ℏ}X − (1⊗_{R_{ℏ}}X + X⊗_{R_{ℏ}}1)].

Thus Δ^{1}X = lim_{ℏ↦0}Δ1_{ℏ}X and

Δ_{ℏ}X = 1⊗_{R_{ℏ}}X + X⊗_{R_{ℏ}}1 + ℏΔ1_{ℏ}X.

Then

(Δ_{ℏ}⊗_{R_{ℏ}}id)Δ_{ℏ}X = 1⊗_{R_{ℏ}}1⊗_{R_{ℏ}}X + Δ_{ℏ}X⊗_{R_{ℏ}}1 + ℏ(Δ_{ℏ}⊗_{R_{ℏ}}id)Δ1_{ℏ}X
= 1⊗_{R_{ℏ}}1⊗_{R_{ℏ}}X + 1⊗_{R_{ℏ}}X⊗_{R_{ℏ}}1 + X⊗_{R_{ℏ}}1⊗_{R_{ℏ}}1 + ℏ[Δ1_{ℏ}X⊗_{R_{ℏ}}1 + (Δ_{ℏ}⊗_{R_{ℏ}}id)Δ1_{ℏ}X)];

and

(id⊗_{R_{ℏ}}Δ_{ℏ})Δ_{ℏ}X = 1⊗_{R_{ℏ}}Δ_{ℏ}X + X⊗_{R_{ℏ}}1⊗_{R_{ℏ}}1 + ℏ(id⊗_{R_{ℏ}}Δ_{ℏ})Δ1_{ℏ}X
= 1⊗_{R_{ℏ}}1⊗_{R_{ℏ}}X + 1⊗_{R_{ℏ}}X⊗_{R_{ℏ}}1 + X⊗_{R_{ℏ}}1⊗_{R_{ℏ}}1 + ℏ[1⊗_{R_{ℏ}}Δ1_{ℏ}X + (id⊗_{R_{ℏ}}Δ_{ℏ})Δ1_{ℏ}X].

It thus follows that

Δ1_{ℏ}X⊗_{R_{ℏ}}1 + (Δ_{ℏ}⊗_{R_{ℏ}}id)Δ1_{ℏ}X = 1⊗_{R_{ℏ}}Δ1_{ℏ}X + (id⊗_{R_{ℏ}}Δ_{ℏ})Δ1_{ℏ}X.  (54)

The conclusion thus follows immediately by letting ℏ ↦ 0.

□

According to Proposition 8.1 in Appendix, immediately we have the following Corollary 5.10 For any X ∈ Γ(A), δX ∈ Γ(∧^{2}A).

Lemma 5.11 For any f ∈ R and X ∈ Γ(A),

δ(fX) = fδX + δf ∧ X.

Proof. For any X ∈ Γ(A), again we let

Δ1_{ℏ}X = \frac{1}{ℏ}[Δ_{ℏ}X − (1⊗_{R_{ℏ}}X + X⊗_{R_{ℏ}}1)].

Now

f ·_{ℏ} X = α_{ℏ}f ∗_{ℏ} X = fX + ℏ[(α_{1}f)X + m_{1}(f, X)] + O(ℏ^{2}).

Hence

Δ_{ℏ}(f ·_{ℏ} X)
= Δ_{ℏ}(fX) + ℏ[Δ_{ℏ}((α_{1}f)X) + Δ_{ℏ}m_{1}(f, X)] + O(ℏ^{2})
= 1⊗_{R_{ℏ}}fX + fX⊗_{R_{ℏ}}1 + ℏ[Δ1_{ℏ}(fX) + Δ_{ℏ}((α_{1}f)X) + Δ_{ℏ}m_{1}(f, X)] + O(ℏ^{2}).  (55)

On the other hand,

f ·_{ℏ} Δ_{ℏ}X

= f ·_{ℏ} (X⊗_{R_{ℏ}}1 + 1⊗_{R_{ℏ}}X + ℏΔ1_{ℏ}X)
= [fX + ℏ((α_{1}f)X + m_{1}(f, X))]⊗_{R_{ℏ}}1 + (f + ℏα_{1}f)⊗_{R_{ℏ}}X + ℏf ·_{ℏ} Δ1_{ℏ}X + O(ℏ^{2}). (56)

From Equations (55) and (56), it follows that

1⊗_{R_{ℏ}}fX − f⊗_{R_{ℏ}}X = ℏ[(α_{1}f)X⊗_{R_{ℏ}}1 + m_{1}(f, X)⊗_{R_{ℏ}}1 + α_{1}f⊗_{R_{ℏ}}X + f ·_{ℏ} Δ1_{ℏ}X
−Δ1_{ℏ}(fX) − Δ_{ℏ}((α_{1}f)X) − Δ_{ℏ}m_{1}(f, X)] + O(ℏ^{2}).  (57)

From the identity 1⊗_{R_{ℏ}}f ·_{ℏ} X = (1 ·_{ℏ} f)⊗_{R_{ℏ}}X, it follows that

1⊗_{R_{ℏ}}(fX + ℏ((α_{1}f)X + m_{1}(f, X))) = (f + ℏβ_{1}f)⊗_{R_{ℏ}}X + O(ℏ^{2}).

That is,

1⊗_{R_{ℏ}}fX − f⊗_{R_{ℏ}}X = ℏ[β_{1}f⊗_{R_{ℏ}}X − 1⊗_{R_{ℏ}}(α_{1}f)X − 1⊗_{R_{ℏ}}m_{1}(f, X)] + O(ℏ^{2}).  (58)

By comparing Equations (57) and (58), one obtains that

Δ1_{ℏ}(fX) = f ·_{ℏ} Δ1_{ℏ}X + (α_{1}f − β_{1}f)⊗_{R_{ℏ}}X + (α_{1}f)X⊗_{R_{ℏ}}1 + 1⊗_{R_{ℏ}}(α_{1}f)X − Δ_{ℏ}((α_{1}f)X)

+m_{1}(f, X)⊗_{R_{ℏ}}1 + 1⊗_{R_{ℏ}}m_{1}(f, X) − Δ_{ℏ}m_{1}(f, X) + O(ℏ).

Taking the limit by letting ℏ ↦ 0, we obtain

Δ^{1}(fX) = fΔ^{1}X + δf⊗_{R}X + (α_{1}f)X⊗_{R}1 + 1⊗_{R}(α_{1}f)X − Δ((α_{1}f)X)

+m_{1}(f, X)⊗_{R}1 + 1⊗_{R}m_{1}(f, X) − Δm_{1}(f, X).

The conclusion thus follows by taking the skew-symmetrization.

□

In summary, we have proved the following

Proposition 5.12 For any f, g ∈ R and X ∈ Γ(A), (i). δf ∈ Γ(A) and δX ∈ Γ(∧^{2}A); (ii). δ(fg) = fδg + gδf; (iii). δ(fX) = fδX + δf ∧ X; (iv). [δf, g] = {f, g}; (v). δ^{2}f = 0.

Properties (i)-(iii) above allow us to extend δ to a well-defined degree 1 derivation δ : Γ(∧^{∗}A) ⟶ Γ(∧^{∗+1}A). Below we will show that (⊕Γ(∧^{∗}A), ∧, [·, ·], δ) is a strong differential Gersterhaber algebra. For this purpose, it suffices to show that δ is a derivation with respect to [·, ·], and δ^{2} = 0. We will prove these facts in two separate propositions below.

Proposition 5.13 For any X, Y ∈ Γ(A),

δ[X, Y] = [δX, Y] + [X, δY].  (59)

Proof. ∀X, Y ∈ Γ(A),

Δ_{ℏ}(X ∗_{ℏ} Y) = Δ_{ℏ}X ∗_{ℏ} Δ_{ℏ}Y = (1⊗_{R_{ℏ}}X + X⊗_{R_{ℏ}}1 + ℏΔ1_{ℏ}X) ∗_{ℏ} (1⊗_{R_{ℏ}}Y + Y ⊗_{R_{ℏ}}1 + ℏΔ1_{ℏ}Y).

It thus follows that

Δ_{ℏ}[X, Y]_{ℏ} = 1⊗_{R_{ℏ}}[X, Y]_{ℏ} + [X, Y]_{ℏ}⊗_{R_{ℏ}}1 + ℏ[(Δ1_{ℏ}X)(1⊗Y + Y ⊗1) + (1⊗X + X⊗1)Δ1_{ℏ}Y
−(Δ1_{ℏ}Y)(1⊗X + X⊗1) − (1⊗Y + Y ⊗1)Δ1_{ℏ}X] + O(ℏ^{2}).  (60)

Here [X, Y]_{ℏ} = X ∗_{ℏ} Y − Y ∗_{ℏ} X. Then [X, Y]_{ℏ} = [X, Y] + ℏl_{1}(X, Y) + O(ℏ^{2}), where l_{1}(X, Y) = m_{1}(X, Y) − m_{1}(Y, X). Hence

Δ_{ℏ}[X, Y]_{ℏ} = Δ_{ℏ}[X, Y] + ℏΔ_{ℏ}l_{1}(X, Y) + O(ℏ^{2})
= 1⊗_{R_{ℏ}}[X, Y] + [X, Y]⊗_{R_{ℏ}}1 + ℏ(Δ1_{ℏ}[X, Y] + Δ_{ℏ}l_{1}(X, Y)) + O(ℏ^{2}).  (61)

Comparing Equation (61) with Equation (60), we obtain

Δ1_{ℏ}[X, Y] = −Δ_{ℏ}l_{1}(X, Y) + 1⊗_{R_{ℏ}}l_{1}(X, Y) + l_{1}(X, Y)⊗_{R_{ℏ}}1

+(Δ1_{ℏ}X)(1⊗Y + Y ⊗1) + (1⊗X + X⊗1)Δ1_{ℏ}Y

−(Δ1_{ℏ}Y)(1⊗X + X⊗1) − (1⊗Y + Y ⊗1)Δ1_{ℏ}X + O(ℏ).

This implies, by letting h ↦ 0, that

Δ^{1}[X, Y] = −Δl_{1}(X, Y) + 1⊗_{R}l_{1}(X, Y) + l_{1}(X, Y)⊗_{R}1
+(Δ^{1}X)(1⊗Y + Y ⊗1) + (1⊗X + X⊗1)Δ^{1}Y
−(Δ^{1}Y)(1⊗X + X⊗1) − (1⊗Y + Y ⊗1)Δ^{1}X.

Equation (59) thus follows immediately by taking the skew-symmetrization.

□

Proposition 5.14 For any X ∈ Γ(A),

δ^{2}X = 0.

Proof. Let

J_{ℏ} = (Δ_{ℏ}⊗_{R_{ℏ}}id)Δ1_{ℏ}X − (id⊗_{R_{ℏ}}Δ_{ℏ})Δ1_{ℏ}X − 1⊗_{R_{ℏ}}Δ1_{ℏ}X + Δ1_{ℏ}X⊗_{R_{ℏ}}1.  (62)

From Equation (54) we know that J_{ℏ} = 0. Let {e_{i} ∈ UA} (e_{0} = 1) be a local basis of UA over the left module R. Assume that δX = ∑ Y_{i} ∧ Z_{i} with Y_{i}, Z_{i} ∈ Γ(A), and

Δ^{1}X = ∑ f^{ij}e_{i}⊗_{R}e_{j} + ∑(Y_{i}⊗_{R}Z_{i} − Z_{i}⊗_{R}Y_{i}),

where f^{ij} ∈ R are symmetric: f^{ij} = f^{ji}. We also assume that Δe_{i} = ∑ g_{i}^{kl}e_{k}⊗_{R}e_{l} with g_{i}^{kl} = g_{i}^{lk} ∈ R since Δ is co-commutative. Let us write

Δ1_{ℏ}X = ∑ f^{ij} ·_{ℏ} e_{i}⊗_{R_{ℏ}}e_{j} + ∑(Y_{i}⊗_{R_{ℏ}}Z_{i} − Z_{i}⊗_{R_{ℏ}}Y_{i}) + ℏΔ2_{ℏ}X; and  (63)
Δ_{ℏ}e_{i} = ∑ g_{i}^{kl} ·_{ℏ} e_{k}⊗_{R_{ℏ}}e_{l} + ℏΔ1_{ℏ}e_{i},  (64)

for some Δ2_{ℏ}X and Δ1_{ℏ}e_{i} ∈ U_{ℏ}A⊗_{R_{ℏ}}U_{ℏ}A.

Then

(id⊗_{R_{ℏ}}Δ_{ℏ})Δ1_{ℏ}X = ∑(α_{ℏ}f^{ij} ∗_{ℏ} β_{ℏ}g_{j}^{kl} ∗_{ℏ} e_{i})⊗_{R_{ℏ}}e_{k}⊗_{R_{ℏ}}e_{l} + ∑(Y_{i}⊗_{R_{ℏ}}Z_{i}⊗_{R_{ℏ}}1 + Y_{i}⊗_{R_{ℏ}}1⊗_{R_{ℏ}}Z_{i}

−Z_{i}⊗_{R_{ℏ}}Y_{i}⊗_{R_{ℏ}}1 − Z_{i}⊗_{R_{ℏ}}1⊗_{R_{ℏ}}Y_{i}) + ℏ[∑ f^{ij} ·_{ℏ} e_{i}⊗_{R_{ℏ}}Δ1_{ℏ}e_{j}

+ ∑ Y_{i}⊗_{R_{ℏ}}Δ1_{ℏ}Z_{i} − ∑ Z_{i}⊗_{R_{ℏ}}Δ1_{ℏ}Y_{i} + (id⊗_{R_{ℏ}}Δ_{ℏ})Δ2_{ℏ}X],

and

1⊗_{R_{ℏ}}Δ1_{ℏ}X = ∑ β_{ℏ}f^{ij}⊗_{R_{ℏ}}e_{i}⊗_{R_{ℏ}}e_{j} + ∑(1⊗_{R_{ℏ}}Y_{i}⊗_{R_{ℏ}}Z_{i} − 1⊗_{R_{ℏ}}Z_{i}⊗_{R_{ℏ}}Y_{i}) + ℏ(1⊗_{R_{ℏ}}Δ2_{ℏ}X).

Similarly, we may write

Δ1_{ℏ}X = ∑ e_{i}⊗_{R_{ℏ}}e_{j} ·_{ℏ} f^{ij} + ∑(Y_{i}⊗_{R_{ℏ}}Z_{i} − Z_{i}⊗_{R_{ℏ}}Y_{i}) + ℏΔ̃2_{ℏ}X; and  (65)
Δ_{ℏ}e_{i} = ∑ e_{k}⊗_{R_{ℏ}}e_{l} ·_{ℏ} g_{i}^{kl} + ℏΔ̃1_{ℏ}e_{i},  (66)

for some Δ̃2_{ℏ}X and Δ̃1_{ℏ}e_{i} ∈ U_{ℏ}A⊗_{R_{ℏ}}U_{ℏ}A.

Hence,

(Δ_{ℏ}⊗_{R_{ℏ}}id)Δ1_{ℏ}X = ∑ e_{k}⊗_{R_{ℏ}}e_{l}⊗_{R_{ℏ}}(α_{ℏ}g_{i}^{kl} ∗_{ℏ} β_{ℏ}f^{ij} ∗_{ℏ} e_{j}) + ∑(Y_{i}⊗_{R_{ℏ}}1⊗_{R_{ℏ}}Z_{i} + 1⊗_{R_{ℏ}}Y_{i}⊗_{R_{ℏ}}Z_{i}

−Z_{i}⊗_{R_{ℏ}}1⊗_{R_{ℏ}}Y_{i} − 1⊗_{R_{ℏ}}Z_{i}⊗_{R_{ℏ}}Y_{i}) + ℏ[∑ Δ̃1_{ℏ}e_{i}⊗_{R_{ℏ}}e_{j} ·_{ℏ} f^{ij}

+ ∑ Δ1_{ℏ}Y_{i}⊗_{R_{ℏ}}Z_{i} − ∑ Δ1_{ℏ}Z_{i}⊗_{R_{ℏ}}Y_{i} + (Δ_{ℏ}⊗_{R_{ℏ}}id)Δ̃2_{ℏ}X],

and

Δ1_{ℏ}X⊗_{R_{ℏ}}1 = ∑ e_{i}⊗_{R_{ℏ}}e_{j}⊗_{R_{ℏ}}α_{ℏ}f^{ij} + ∑_{i}(Y_{i}⊗_{R_{ℏ}}Z_{i}⊗_{R_{ℏ}}1 − Z_{i}⊗_{R_{ℏ}}Y_{i}⊗_{R_{ℏ}}1) + ℏΔ̃2_{ℏ}X⊗_{R_{ℏ}}1.

Thus we have J_{ℏ} = I_{ℏ} + ℏK_{ℏ}, where

I_{ℏ} = ∑ e_{i}⊗_{R_{ℏ}}e_{j}⊗_{R_{ℏ}}(α_{ℏ}g_{l}^{ij} ∗_{ℏ} β_{ℏ}f^{lk} ∗_{ℏ} e_{k}) − ∑(α_{ℏ}f^{il} ∗_{ℏ} β_{ℏ}g_{l}^{jk} ∗_{ℏ} e_{i})⊗_{R_{ℏ}}e_{j}⊗_{R_{ℏ}}e_{k}
− ∑ β_{ℏ}f^{ij}⊗_{R_{ℏ}}e_{i}⊗_{R_{ℏ}}e_{j} + ∑ e_{i}⊗_{R_{ℏ}}e_{j}⊗_{R_{ℏ}}α_{ℏ}f^{ij},

and

K_{ℏ} = ∑ Δ̃1_{ℏ}e_{i}⊗_{R_{ℏ}}e_{j} ·_{ℏ} f^{ij} + ∑ Δ1_{ℏ}Y_{i}⊗_{R_{ℏ}}Z_{i} − ∑ Δ1_{ℏ}Z_{i}⊗_{R_{ℏ}}Y_{i} + (Δ_{ℏ}⊗_{R_{ℏ}}id)Δ̃2_{ℏ}X + Δ̃2_{ℏ}X⊗_{R_{ℏ}}1

−[∑ f^{ij} ·_{ℏ} e_{i}⊗_{R_{ℏ}}Δ1_{ℏ}e_{j} + ∑ Y_{i}⊗_{R_{ℏ}}Δ1_{ℏ}Z_{i} − ∑ Z_{i}⊗_{R_{ℏ}}Δ1_{ℏ}Y_{i} + (id⊗_{R_{ℏ}}Δ_{ℏ})Δ2_{ℏ}X + 1⊗_{R_{ℏ}}Δ2_{ℏ}X].

By Alt, we denote the standard skew-symmetrization operator on UA⊗_{R}UA⊗_{R}UA defined by

Alt(x_{1}⊗_{R}x_{2}⊗_{R}x_{3}) = ∑_{σ∈S_{3}} \frac{1}{3!}(signσ)x_{σ(1)}⊗_{R}x_{σ(2)}⊗_{R}x_{σ(3)},

where x_{1}, x_{2}, x_{3} ∈ UA. It is tedious but straightforward to verify that Alt(lim_{ℏ↦0} \frac{1}{ℏ}I_{ℏ}) = 0. Therefore, Alt(lim_{ℏ↦0}K_{ℏ}) = 0, i.e.,

Alt[∑ f^{ij}Δ̃^{1}e_{i}⊗_{R}e_{j} + ∑ Δ^{1}Y_{i}⊗_{R}Z_{i} − ∑ Δ^{1}Z_{i}⊗_{R}Y_{i} + (Δ⊗_{R}id)Δ̃^{2}X + Δ̃^{2}X⊗_{R}1

−(∑ f^{ij}e_{i}⊗_{R}Δ^{1}e_{j} + ∑ Y_{i}⊗_{R}Δ^{1}Z_{i} − ∑ Z_{i}⊗_{R}Δ^{1}Y_{i} + (id⊗_{R}Δ)Δ^{2}X + 1⊗_{R}Δ^{2}X)] = 0.

The final conclusion thus follows immediately by applying the skew-symmetrization operator Alt to the equation above and using the following simple fact:

Lemma 5.15 (i). Alt(Δ̃^{2}X⊗_{R}1 − 1⊗_{R}Δ^{2}X) = 0.

(ii). Alt ∑(f^{ij}Δ̃^{1}e_{i}⊗_{R}e_{j} − f^{ij}e_{i}⊗_{R}Δ^{1}e_{j}) = 0.

(iii). Alt((id⊗_{R}Δ)Δ^{2}X) = Alt((Δ⊗_{R}id)Δ̃^{2}X) = 0.

Proof. It follows from Equations (63) and (65) that

ℏΔ2_{ℏ}X − ℏΔ̃2_{ℏ}X = ∑(f^{ij} ·_{ℏ} e_{i}⊗_{R_{ℏ}}e_{j} − e_{i}⊗_{R_{ℏ}}e_{j} ·_{ℏ} f^{ij})
= ∑(f^{ij} ·_{ℏ} e_{i}⊗_{R_{ℏ}}e_{j} − e_{i}⊗_{R_{ℏ}}f^{ij} ·_{ℏ} e_{j} + e_{i}⊗_{R_{ℏ}}f^{ij} ·_{ℏ} e_{j} − e_{i}⊗_{R_{ℏ}}e_{j} ·_{ℏ} f^{ij})
= ℏ ∑ δf^{ij} ∗_{ℏ} e_{i}⊗_{R_{ℏ}}e_{j} + ℏ ∑ e_{i}⊗_{R_{ℏ}}δf^{ij} ∗_{ℏ} e_{j} + O(ℏ^{2}).

Hence,

Δ^{2}X − Δ̃^{2}X
= ∑(δf^{ij})e_{i}⊗_{R}e_{j} + ∑ e_{i}⊗_{R}(δf^{ij})e_{j}
= ∑ Δ(δf^{ij})e_{i}⊗_{R}e_{j},

which is symmetric. It thus follows that

Alt(Δ̃^{2}X⊗_{R}1 − 1⊗_{R}Δ^{2}X)
= Alt(Δ^{2}X⊗_{R}1 − 1⊗_{R}Δ^{2}X − ∑ Δ(δf^{ij})e_{i}⊗_{R}e_{j}⊗_{R}1)
= 0.

Similarly, one can show that

Δ^{1}e_{i} − Δ̃^{1}e_{i} = ∑_{kl} Δ(δg_{i}^{kl})(e_{k}⊗_{R}e_{l}).

Thus, (ii) follows immediately. Finally, (iii) is obvious since Δ is co-commutative.

□

Combining Propositions 5.12, 5.13 and 5.14, we conclude that (⊕Γ(∧^{∗}A), ∧, [·, ·], δ) is indeed a strong differential Gersterhaber algebra. Hence (A, A^{∗}) is a Lie bialgebroid, which will be called the classical limit of the quantum groupoid U_{ℏ}A. In summary, we have proved

Theorem 5.16 A quantum groupoid (U_{ℏ}A, R_{ℏ}, α_{ℏ}, β_{ℏ}, m_{ℏ}, Δ_{ℏ}, ϵ_{ℏ}) naturally induces a Lie bialgebroid (A, A^{∗}) as a classical limit. The induced Poisson structure of this Lie bialgebroid on the base space P coincides with the one obtained as the classical limit of the base ∗-algebra R_{ℏ}.

As an example, in what follows, we will examine the classical limit of those quantum groupoids obtained from the standard Hopf algebroid UA[[ℏ]] by twists. Consider (UA[[ℏ]], R[[ℏ]], α, β, m, Δ, ϵ) equipped with the standard Hopf algebroid structure induced from that on UA. Assume that

F_{ℏ} = 1⊗_{R}1 + ℏΛ̄ + O(ℏ^{2}) ∈ UA⊗_{R}UA[[ℏ]],  (67)

with Λ̄ ∈ UA⊗_{R}UA, is a twistor, and let (U_{ℏ}A, R_{ℏ}, α_{ℏ}, β_{ℏ}, m_{ℏ}, Δ_{ℏ}, ϵ_{ℏ}) be its resulting twisted QUE algebroid.

Lemma 5.17 Assume that F_{ℏ} ∈ UA⊗_{R}UA[[ℏ]] given by Equation (67) is a twistor. Then

Λ̄⊗_{R}1 + (Δ⊗_{R}id)Λ̄ = 1⊗_{R}Λ̄ + (id⊗_{R}Δ)Λ̄.  (68)

Proof. This follows immediately from the ℏ^{1}-expansion of Equation (32).

□

Using Proposition 8.1 in Appendix, we have

Corollary 5.18 Under the same hypothesis as in Lemma 5.17, then Λ = AltΛ̄( =^{def} Λ̄ − Λ̄_{21}) is a section of ∧^{2}A.

It is not surprising to expect the following:

Theorem 5.19 Let (U_{ℏ}A, R_{ℏ}, α_{ℏ}, β_{ℏ}, m_{ℏ}, Δ_{ℏ}, ϵ_{ℏ}) be the quantum groupoid obtained from (UA[[ℏ]], R[[ℏ]], α, β, m, Δ, ϵ) by twisting via F_{ℏ} as given by Equation (67). Then its classical limit is a coboundary Lie bialgebroid (A, A^{∗}, Λ), where Λ = AltΛ̄. In particular, its induced Poisson structure on the base space is ρΛ, which admits R_{ℏ} as a deformation quantization.

Proof. It suffices to prove that δf = [f, Λ] and δX = [X, Λ], ∀f ∈ R and X ∈ Γ(A).

Write

Λ̄ = ∑_{i} d_{i}⊗_{R}e_{i} ∈ UA⊗_{R}UA.

It is easy to see that for any f ∈ R,

α_{ℏ}f = f + ℏ ∑_{i}((ρd_{i})f)e_{i} + O(ℏ^{2}), and
β_{ℏ}f = f + ℏ ∑_{i}((ρe_{i})f)d_{i} + O(ℏ^{2}).

It thus follows that

δf = ∑_{i}[((ρd_{i})f)e_{i} − ((ρe_{i})f)d_{i}] = [f, Λ].

Now we will prove the second identity δX = [X, Λ]. From the definition of F #_{ℏ}, it follows that

F #_{ℏ}[1⊗_{R_{ℏ}}X + X⊗_{R_{ℏ}}1 + ℏ ∑_{i}(Xd_{i}⊗_{R_{ℏ}}e_{i} + d_{i}⊗_{R_{ℏ}}Xe_{i} − d_{i}X⊗_{R_{ℏ}}e_{i} − d_{i}⊗_{R_{ℏ}}e_{i}X)]

= F_{ℏ}[1⊗X + X⊗1 + ℏ ∑_{i}(Xd_{i}⊗e_{i} + d_{i}⊗Xe_{i} − d_{i}X⊗e_{i} − d_{i}⊗e_{i}X)]
= 1⊗_{R}X + X⊗_{R}1 + ℏ ∑_{i}(Xd_{i}⊗_{R}e_{i} + d_{i}⊗_{R}e_{i}X) + O(ℏ^{2})
= (ΔX)F_{ℏ} + O(ℏ^{2}).

It thus follows that

Δ_{ℏ}X
= F #_{ℏ}^{−1}((ΔX)F_{ℏ})
= 1⊗_{R_{ℏ}}X + X⊗_{R_{ℏ}}1 + ℏ ∑_{i}(Xd_{i}⊗_{R_{ℏ}}e_{i} + d_{i}⊗_{R_{ℏ}}Xe_{i} − d_{i}X⊗_{R_{ℏ}}e_{i} − d_{i}⊗_{R_{ℏ}}e_{i}X) + O(ℏ^{2}).

Therefore

Δ^{1}X = (1⊗X + X⊗1)Λ̄ − Λ̄(1⊗X + X⊗1).

This immediately implies that

δX = Δ^{1}X − (Δ^{1}X)_{21}
= [(1⊗X + X⊗1)Λ̄ − Λ̄(1⊗X + X⊗1)] − [(1⊗X + X⊗1)Λ̄_{21} − Λ̄_{21}(1⊗X + X⊗1)]
= (1⊗X + X⊗1)Λ − Λ(1⊗X + X⊗1)
= [X, Λ].

This concludes the proof.

□

More generally, we have

Theorem 5.20 Assume that (U_{ℏ}A, R_{ℏ}, α_{ℏ}, β_{ℏ}, m_{ℏ}, Δ_{ℏ}, ϵ_{ℏ}) is a quantum groupoid with classical limit (A, A^{∗}). Let F_{ℏ} ∈ U_{ℏ}A⊗_{R_{ℏ}}U_{ℏ}A be a twistor such that F_{ℏ} = 1⊗_{R_{ℏ}}1(modℏ). Then Λ = Alt(lim_{ℏ↦0}ℏ^{−1}(F_{ℏ} − 1⊗_{R_{ℏ}}1)) is a section of Γ(∧^{2}A), and is a Hamiltonian operator of the Lie bialgebroid (A, A^{∗}). If (U_{ℏ}A, R̃_{ℏ}, α̃_{ℏ}, β̃_{ℏ}, m_{ℏ}, Δ̃_{ℏ}, ϵ_{ℏ}) is obtained from (U_{ℏ}A, R_{ℏ}, α_{ℏ}, β_{ℏ}, m_{ℏ}, Δ_{ℏ}, ϵ_{ℏ}) by twisting via F_{ℏ}, its corresponding Lie bialgebroid is obtained from (A, A^{∗}) by twisting via Λ. In particular, if (A, A^{∗}) is a coboundary Lie bialgebroid with r-matroid Λ_{0}, the latter is still a coboundary Lie bialgebroid and its r-matroid is Λ_{0} + Λ.

Proof. The proof is similar to that of Theorem 5.19, even though it is a little bit more complicated. We omit it here.

□

Remark. It is easy to see that the classical limit of the quantum groupoid (D_{ℏ}, R_{ℏ}, α, β, m, Δ_{ℏ}, ϵ) in Example 4.1 is the standard Lie bialgebroid (T P, T^{∗}P) associated to the Poisson manifold P. It, however, would be interesting to explore its “dual” quantum groupoid, namely the one with the Lie bialgebroid (T^{∗}P, T P).

6 Quantization of Lie bialgebroids

Definition 6.1 A quantization of a Lie bialgebroid (A, A^{∗}) is a quantum groupoid (U_{ℏ}A, R_{ℏ}, α_{ℏ}, β_{ℏ}, m_{ℏ}, Δ_{ℏ}, ϵ_{ℏ}) whose classical limit is (A, A^{∗}).

It is a deep theorem of Etingof and Kazhdan [12] that every Lie bialgebra is quantizable. On the other hand, the existence of ∗-products for arbitrary Poisson manifolds was recently proved by Kontsevich [25]. In terms of Hopf algebroids, this amounts to saying that the Lie bialgebroid (T P, T^{∗}P) associated to a Poisson manifold P is always quantizable. It is therefore natural to expect:

Conjecture Every Lie bialgebroid is quantizable.

Below we will prove a very special case of this conjecture by using Fedosov quantization [16] [43].

Theorem 6.2 Any regular triangular Lie bialgebroid is quantizable.

We need some preparation first. Recall that given a Lie algebroid A ⟶ P with anchor ρ, an A-connection on a vector bundle E ⟶ P is an ℝ-linear map:

Γ(A) ⊗ Γ(E) ⟶ Γ(E)
X ⊗ s ⟶ ∇_{X}s,

satisfying the axioms resembling those of the usual linear connections, i.e., ∀f ∈ C^{∞}(P), X ∈ Γ(A), s ∈ Γ(E),

∇_{fX}s = f∇_{X}s;
∇_{X}(fs) = (ρ(X)f)s + f∇_{X}s.

In particular, if E = A, an A-connection is called torsion-free if

∇_{X}Y − ∇_{Y} X = [X, Y].

A torsion-free connection always exists for any Lie algebroid.

Let ω ∈ Γ(∧^{2}A^{∗}) be a closed two-form, i.e., dω = 0. An A-connection on A is said to be compatible with ω if ∇_{X}ω = 0. If ω is non-degenerate, a compatible torsion-free connection always exists.

Lemma 6.3 If ω ∈ Γ(∧^{2}A^{∗}) is a closed non-degenerate two-form, there exists a compatible torsionfree A-connection on A.

Proof. This result is standard (see [36] [40]). The proof is simply just a repetition of that of the existence of a symplectic connection for a symplectic manifold. For completeness, we sketch a proof here.

First, take any torsion-free A-connection ∇. Then any other A-connection can be written as

˜∇_{X}Y = ∇_{X}Y + S(X, Y), ∀X, Y ∈ Γ(A),  (69)

where S is a (2, 1)-tensor. Clearly, ˜∇ is torsion-free iff S is symmetric, i.e., S(X, Y) = S(Y, X) for any X, Y ∈ Γ(A).

˜∇ is compatible with ω ∈ Γ(∧^{2}A^{∗}) iff ˜∇_{X}ω = 0. The latter is equivalent to

ω(S(X, Y), Z) − ω(S(X, Z), Y) = (∇_{X}ω)(Y, Z).  (70)

Let S be the (2, 1)-tensor defined by the equation:

ω(S(X, Y), Z) = \frac{1}{3}[(∇_{X}ω)(Y, Z) + (∇_{Y} ω)(X, Z)].  (71)

Clearly, S(X, Y), defined in this way, is symmetric with respect to X and Y. Now

ω(S(X, Y), Z) − ω(S(X, Z), Y)
= \frac{1}{3}[(∇_{X}ω)(Y, Z) + (∇_{Y} ω)(X, Z)] − \frac{1}{3}[(∇_{X}ω)(Z, Y) + (∇_{Z}ω)(X, Y)]
= \frac{1}{3}[(∇_{X}ω)(Y, Z) + (∇_{Y} ω)(X, Z) + (∇_{X}ω)(Y, Z) + (∇_{Z}ω)(Y, X)]
= (∇_{X}ω)(Y, Z).

Here the last step follows from the identity:

(∇_{X}ω)(Y, Z) + (∇_{Y} ω)(Z, X) + (∇_{Z}ω)(X, Y) = 0,

which is equivalent to that dω = 0. This implies that ˜∇ is a torsion-free symplectic connection.

□

Proof of Theorem 6.2 Let (A, A^{∗}, Λ) be a regular triangular Lie bialgebroid. Then Λ^{#} : A^{∗} ⟶ A is a Lie algebroid morphism [33]. Therefore its image Λ^{#}A^{∗} is a Lie subalgebroid of A, and Λ can be considered as a section of ∧^{2}(Λ^{#}A^{∗}). Hence, by restricting to Λ^{#}A^{∗} if necessary, one may always assume that Λ is nondegenerate. Let ω = Λ^{−1} ∈ Γ(∧^{2}A^{∗}). Then ω is closed: dω = 0. Let ∇ be a compatible torsion-free A-connection on A, which always exists according to Lemma 6.3. Let (G ⟶ ⟶ P, α, β) be a local groupoid corresponding to the Lie algebroid A. Let Λ^{l} denote the left translation of Λ, so Λ^{l} defines a left invariant Poisson structure on G. This is a regular Poisson structure, whose symplectic leaves are simply α-fibers. The A-connection ∇ induces a leafwise linear connection ˜∇ for the α-fibration. To see this, simply define for any X, Y ∈ Γ(A),

˜∇_{X^{l}}Y^{l} = (∇_{X}Y)^{l},  (72)

where X^{l}, Y^{l} and (∇_{X}Y)^{l} denote their corresponding left invariant vector fields on G. Since left invariant vector fields span the tangent space of α-fibers, this indeed defines a linear connection on each α-fiber α^{−1}(u), ∀u ∈ P. Such a connection will be denoted by ˜∇_{u}. Clearly, ˜∇_{u} is torsion-free since ∇ is torsion-free. Moreover, ˜∇ preserves the Poisson structure Λ^{l}, and is left-invariant in the sense that

L_{x}^{∗} ˜∇_{u} = ˜∇_{v}, ∀x ∈ G such that β(x) = u, α(x) = v.  (73)

Applying Fedosov quantization to this situation, one obtains a ∗-product on G

f ∗_{ℏ} g = fg + \frac{1}{2}ℏΛ^{l}(f, g) + ⋯ + ℏ^{k}B_{k}(f, g) + ⋯  (74)

quantizing the Poisson structure Λ^{l}. In fact, this ∗-product is given by a family of leafwise ∗- products indexed by u ∈ P quantizing the leafwise symplectic structures on α-fibers. Since the Poisson structure Λ^{l} is left invariant, which means that the leafwise symplectic structures are invariant by left translations L_{x}, and the symplectic connections ˜∇_{u} are left-invariant, the resulting Fedosov ∗-products are invariant under left translations. In other words, the bidifferential operators B_{k}(·, ·) are all left invariant, and therefore belong to UA⊗_{R}UA. In this way, we obtain an element F_{ℏ} = 1 + ℏΛ + O(ℏ^{2}) ∈ UA⊗_{R}UA[[ℏ]] so that the ∗-product on G is

f ∗_{ℏ} g = F_{ℏ}(f, g), ∀f, g ∈ C^{∞}(G).

The associativity of ∗_{ℏ} implies that F_{ℏ} satisfies Equation (32):

(Δ⊗_{R}id)F_{ℏ}F 12_{ℏ} = (id⊗_{R}Δ)F_{ℏ}F 23_{ℏ}.  (75)

The identity 1 ∗_{ℏ} f = f ∗_{ℏ} 1 = f implies that

(ϵ⊗_{R}id)F_{ℏ} = 1_{H}; (id⊗_{R}ϵ)F_{ℏ} = 1_{H}.  (76)

Thus F_{ℏ} ∈ UA⊗_{R}UA[[ℏ]] is a twistor, and the resulting twisted Hopf algebroid (U_{ℏ}A, R_{ℏ}, α_{ℏ}, β_{ℏ}, m_{ℏ}, Δ_{ℏ}, ϵ_{ℏ}) is a quantization of the triangular Lie bialgebroid (A, A^{∗}, Λ). This concludes the proof of the theorem.

□

In particular, when the base P reduces to a point, Theorem 6.2 implies that every finite dimensional triangular r-matrix is quantizable. Of course, there is no need to use Fedosov method in this special case. There is a very nice short proof due to Drinfeld [10].

Remark. We note that the induced Poisson structure on the base P of a non-degenerate triangular Lie bialgebroid was studied by Nest-Tsygan [36] and Weinstein [40], for which a ∗-product was constructed. In fact our algebra R_{ℏ} provides a ∗-product for such a Poisson structure, so here we obtain an alternative proof of their result.

7 Universal dynamical R-matrices and dynamical quantum groupoids

Let U_{ℏ}𝔤 be a quasi-triangular quantum universal enveloping algebra over ℂ with R-matrix R ∈ U_{ℏ}𝔤⊗U_{ℏ}𝔤, η ⊂ 𝔤 a finite dimensional Abelian Lie subalgebra such that Uη[[ℏ]] is a commutative subalgebra of U_{ℏ}𝔤. By M(η^{∗}), we denote the space of meromorphic functions on η^{∗}, and by D the algebra of meromorphic differential operators on η^{∗}. Let us consider H = D⊗U_{ℏ}𝔤. Then H is a Hopf algebroid over ℂ with base algebra R = M(η^{∗})[[ℏ]], whose coproduct and counit are denoted respectively by Δ and ϵ. Moreover the map

μ(D ⊗ u)(f) = (ϵ_{0}u)D(f), ∀D ∈ D, u ∈ U_{ℏ}𝔤, f ∈ M(η^{∗})  (77)

is an anchor map. Here ϵ_{0} is the counit of the Hopf algebra U_{ℏ}𝔤. Let us fix a basis in η, say {h_{1}, ⋯, h_{k}}, and let {ξ_{1}, ⋯, ξ_{k}} be its dual basis, which in turn defines a coordinate system (λ_{1}, ⋯, λ_{k}) on η^{∗}.

Set

θ = ∑_{i=1}^{k}(\frac{∂}{∂λ_{i}} ⊗h_{i}) ∈ H⊗H, and Θ = exp ℏθ ∈ H⊗H.  (78)

Note that θ, and hence Θ, is independent of the choice of the basis in η. The following fact can be easily verified.

Lemma 7.1 Θ satisfies Equations (32) and (33).

Proof. Consider A = D^{inv}⊗Uη[[ℏ]], where D^{inv} consists of holomorphic differential operators on η^{∗} invariant under the translations. Then A is a Hopf subalgebroid of H, which is in fact a Hopf algebra. Clearly, θ ∈ A⊗A, so Θ ∈ A⊗A. It thus suffices to prove that

(Δ⊗id)ΘΘ^{12} = (id⊗Δ)ΘΘ^{23} in A⊗A⊗A.

Now

(Δ⊗id)ΘΘ^{12}
= ((Δ⊗id) exp ℏθ) exp ℏθ^{12}
= exp ℏ((Δ⊗id)θ + θ^{12})
= exp ℏ ∑_{i=1}^{k}(\frac{∂}{∂λ_{i}} ⊗1⊗h_{i} + 1⊗\frac{∂}{∂λ_{i}} ⊗h_{i} + \frac{∂}{∂λ_{i}} ⊗h_{i}⊗1).

Here in the second equality we used the fact that (Δ⊗id)θ and θ^{12} commute in A⊗A⊗A.

Similarly, we have

(id⊗Δ)ΘΘ^{23} = exp ℏ ∑_{i=1}^{k}(\frac{∂}{∂λ_{i}} ⊗1⊗h_{i} + 1⊗\frac{∂}{∂λ_{i}} ⊗h_{i} + \frac{∂}{∂λ_{i}} ⊗h_{i}⊗1).

This proves Equation (32). Finally, Equation (33) follows from a straightforward verification.

□

Remark There is a more intrinsic way of proving this fact in terms of deformation quantization. Consider T^{∗}η^{∗} with the standard cotangent bundle symplectic structure ∑_{i=1}^{k} dλ_{i} ∧ dp_{i}. It is wellknown that

f ∗_{ℏ} g = fe^{ℏ(∑_{i=1}^{k} \overleftarrow{\frac{∂}{∂λ_{i}}}⊗\overrightarrow{\frac{∂}{∂p_{i}}})}g  (79)

defines a ∗-product on T^{∗}η^{∗}, called the Wick type ∗-product corresponding to the normal order

quantization. Hence exp ℏ(∑_{i=1}^{k} \overleftarrow{\frac{∂}{∂λ_{i}}}⊗\overrightarrow{\frac{∂}{∂p_{i}}}), as a (formal) bidifferential operator on T^{∗}η^{∗} (i.e. as an

element in D(T^{∗}η^{∗})⊗_{M(T^{∗}η^{∗})}D(T^{∗}η^{∗})[[ℏ]] satisfies Equations (32) and (33) according to Example 4.1. Note that elements in (Dη^{∗}⊗Uη)⊗_{M(η^{∗})}(Dη^{∗}⊗Uη)[[ℏ]] can be considered as (formal) bidifferential operators on T^{∗}η^{∗} invariant under the p-translations, so (Dη^{∗}⊗Uη)⊗_{M(η^{∗})}(Dη^{∗}⊗Uη)[[ℏ]] is nat-

urally a subspace of D(T^{∗}η^{∗})⊗_{M(T^{∗}η^{∗})}D(T^{∗}η^{∗})[[ℏ]]. Clearly exp ℏ(∑_{i=1}^{k} \overleftarrow{\frac{∂}{∂λ_{i}}}⊗\overrightarrow{\frac{∂}{∂p_{i}}}) is a p-invariant

bidifferential operator on T^{∗}η^{∗}, and is equal to Θ under the above identification. Equations (32) and (33) thus follows immediately.

In other words, Θ is a twistor of the Hopf algebroid H. As we shall see below, it is this Θ that links a shifted cocycle F (λ) and a Hopf algebroid twistor.

Now let F ∈ M(η^{∗}, U_{ℏ}𝔤⊗U_{ℏ}𝔤), define F^{12}(λ + ℏh^{(3)}) ∈ M(η^{∗}, U_{ℏ}𝔤⊗U_{ℏ}𝔤⊗U_{ℏ}𝔤) by

F^{12}(λ + ℏh^{(3)}) = F (λ)⊗1 + ℏ ∑_{i} \frac{∂F}{∂λ_{i}} ⊗h_{i} + \frac{1}{2!}ℏ^{2} ∑_{i_{1}i_{2}} \frac{∂^{2}F}{∂λ_{i_{1}}∂λ_{i_{2}}} ⊗h_{i_{1}}h_{i_{2}}

+ ⋯ + \frac{ℏ^{k}}{k!} ∑ \frac{∂^{k}F}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} ⊗h_{i_{1}} ⋯ h_{i_{k}} + ⋯,  (80)

similarly for F^{23}(λ + ℏh^{(1)}) etc.

Lemma 7.2 Let X be a meomorphic vector field on η^{∗}, and F ∈ M(η^{∗}, U_{ℏ}𝔤⊗U_{ℏ}𝔤). Then

(ΔX)F = F (ΔX) + X(F), in H⊗_{R}H.  (81)

Proof. Note that F, being considered as an element in H⊗_{R}H, clearly satisfies the condition that ∀f ∈ R,

F (f⊗1 − 1⊗f) = 0, in H⊗_{R}H.

So both (ΔX)F and F (ΔX) are well defined elements in H⊗_{R}H. For any f, g ∈ R, considering both sides of Equation (81) as U_{ℏ}𝔤⊗U_{ℏ}𝔤-valued bidifferential operators and applying them to f⊗g, one obtains that

[(ΔX)F − F (ΔX)](f⊗g)
= X(F fg) − F X(fg)
= X(F)fg.

Thus Equation (81) follows.

□

An element F ∈ M(η^{∗}, U_{ℏ}𝔤⊗U_{ℏ}𝔤) is said to be of zero weight if

[F (λ), 1⊗h + h⊗1] = 0, ∀λ ∈ η^{∗}, h ∈ η.  (82)

Lemma 7.3 Assume that F ∈ M(η^{∗}, U_{ℏ}𝔤⊗U_{ℏ}𝔤) is of zero weight. Then ∀n ∈ ℕ,

[(Δ⊗_{R}id)θ^{n}]F^{12} = ∑_{k=0}^{n} ∑_{0≤i_{1}···i_{k}≤n} C_{n}^{k}(\frac{∂^{k}F}{∂λ_{i_{1}} ⋯ ∂λ_{i_{k}}} ⊗h_{i_{1}} ⋯ h_{i_{k}})(Δ⊗_{R}id)θ^{n−k};  (83)
[(id⊗_{R}Δ)θ]F^{23} = F^{23}(id⊗_{R}Δ)θ.  (84)

Proof. To prove Equation (83), let’s first consider n = 1. Then

[(Δ⊗_{R}id)θ]F^{12}
= ∑_{i}(Δ\frac{∂}{∂λ_{i}} ⊗_{R}h_{i})F^{12}
= ∑_{i}(Δ\frac{∂}{∂λ_{i}})F ⊗_{R}h_{i}
= ∑_{i}(F Δ\frac{∂}{∂λ_{i}} + \frac{∂F}{∂λ_{i}})⊗_{R}h_{i}
= F^{12}(Δ⊗_{R}id)θ + ∑_{i} \frac{∂F}{∂λ_{i}} ⊗_{R}h_{i}.

The general conclusion follows from induction by using the equation above repeatedly.

For Equation (84), we have

[(id⊗_{R}Δ)θ]F^{23}
= ∑_{i}(\frac{∂}{∂λ_{i}} ⊗_{R}Δh_{i})F^{23}
= ∑_{i} \frac{∂}{∂λ_{i}} ⊗_{R}(Δh_{i})F
= ∑_{i} \frac{∂}{∂λ_{i}} ⊗_{R}(F Δh_{i})
= F^{23}(id⊗_{R}Δ)θ,

where the second from the last equality follows from the fact that F is of zero weight. Equation (84) thus follows.

□

Proposition 7.4 Assume that F ∈ M(η^{∗}, U_{ℏ}𝔤⊗U_{ℏ}𝔤) is of zero weight. Then

[(Δ⊗_{R}id)Θ]F^{12}(λ) = F^{12}(λ + ℏh^{(3)})(Δ⊗_{R}id)Θ;  (85)
[(id⊗_{R}Δ)Θ]F^{23}(λ) = F^{23}(λ)(id⊗_{R}Δ)Θ.  (86)

Proof. Note that (Δ⊗_{R}id)Θ = exp ℏ((Δ⊗_{R}id)θ). Equations (85) and (86) thus follow immediately from Lemma 7.3.

□

Remark One may rewrite Equation (85) as

F^{12}(λ + ℏh^{(3)}) = [(Δ⊗_{R}id)Θ]F^{12}(λ)[(Δ⊗_{R}id)Θ]^{−1}
= e^{ℏ ∑_{i=1}^{k}(Δ \frac{∂}{∂λ_{i}} ⊗h_{i})}F^{12}(λ)e^{−ℏ ∑_{i=1}^{k}(Δ \frac{∂}{∂λ_{i}} ⊗h_{i})}.

This is essentially the definition of F^{12}(λ + ℏh^{(3)}) used in [2], where the operator ∑_{i=1}^{k}(Δ\frac{∂}{∂λ_{i}} ⊗h_{i})

was denoted by D = ∑_{i=1}^{k} \frac{∂}{∂λ_{i}} h_{i}^{(3)}.

Now set

F = F (λ)Θ ∈ H⊗_{R}H.  (87)

Theorem 7.5 Assume that F ∈ M(η^{∗}, U_{ℏ}𝔤⊗U_{ℏ}𝔤) is of zero weight. Then F is a twistor (i.e. satisfies Equations (32) and (33)) iff

[(Δ_{0}⊗id)F (λ)]F^{12}(λ + ℏh^{(3)}) = [(id⊗Δ_{0})F (λ)]F^{23}(λ),  (88)
(ϵ_{0}⊗id)F (λ) = 1; (id⊗ϵ_{0})F (λ) = 1,  (89)

where Δ_{0} is the coproduct of U_{ℏ}𝔤, and ϵ_{0} is the counit map.

Proof. Using Proposition 7.4, we have

(Δ⊗_{R}id)FF^{12}
= [(Δ⊗_{R}id)F (λ)Θ]F^{12}(λ)Θ^{12}
= [(Δ_{0}⊗id)F (λ)][(Δ⊗_{R}id)Θ]F^{12}(λ)Θ^{12}
= [(Δ_{0}⊗id)F (λ)]F^{12}(λ + ℏh^{(3)})(Δ⊗_{R}id)ΘΘ^{12},

and

(id⊗_{R}Δ)FF^{23}
= [(id⊗_{R}Δ)F (λ)Θ]F^{23}(λ)Θ^{23}
= [(id⊗Δ_{0})F (λ)](id⊗_{R}Δ)ΘF^{23}(λ)Θ^{23}
= [(id⊗Δ_{0})F (λ)]F^{23}(λ)(id⊗_{R}Δ)ΘΘ^{23}.

Thus it follows from Lemma 7.1 that Equation (32) and Equation (88) are equivalent.

For Equation (33), we note that F = F (λ)Θ = ∑_{k=0}^{∞} \frac{ℏ^{k}}{k!} F (λ)θ^{k}. It is easy to see that for k ≥ 1, (ϵ⊗_{R}id)(F (λ)θ^{k}) = (id⊗_{R}ϵ)(F (λ)θ^{k}) = 0 since ϵ(\frac{∂^{k}}{∂λ_{i_{1}}···∂λ_{i_{k}}}) = 0 and ϵ(h_{i_{1}} ⋯ h_{i_{k}}) = 0. Thus it is immediate that Equations (33) and (89) are equivalent. This concludes the proof of the theorem.

□

Now assume that F (λ) is a solution to Equations (88) and (89) so that we can form a quantum groupoid by twisting D⊗U_{ℏ}𝔤 via F. The resulting quantum groupoid is denoted by D⊗_{ℏ}U_{ℏ}𝔤, and is called a dynamical quantum groupoid.

As an immediate consequence of Theorem 4.14, we have

Theorem 7.6 As a monoidal category, the category of D⊗_{ℏ}U_{ℏ}𝔤-modules is equivalent to that of D⊗U_{ℏ}𝔤-modules, and therefore is a braided monoidal category.

Remark. Representations of a quantum dynamical R-matrix can be understood using the representations of the dynamical quantum groupoid D⊗_{ℏ}U_{ℏ}𝔤. The discussion on this topic will appear somewhere else.

In what follows, we describe various structures of D⊗_{ℏ}U_{ℏ}𝔤 more explicitly.

Proposition 7.7 (i). f ∗_{F} g = fg, ∀f, g ∈ M(η^{∗}), i.e., R_{F} is isomorphic to M(η^{∗})[[ℏ]] with pointwise multiplication;

(ii). α_{F}f = exp (ℏ ∑_{i=1}^{k} h_{i} \frac{∂}{∂λ_{i}})f = ∑_{1≤i_{1},···,i_{n}≤k} \frac{ℏ^{n}}{n!} \frac{∂^{n}f}{∂λ_{i_{1}}···∂λ_{in}} h_{i_{1}} ⋯ h_{i_{n}}, ∀f ∈ R;

(iii). β_{F}f = f, ∀f ∈ R.

Proof. Assume that F (λ) = ∑_{i} F_{i}(λ)u_{i}⊗v_{i}, with u_{i}, v_{i} ∈ U_{ℏ}𝔤. Let

F_{n} = F (λ)θ^{n} = ∑_{i} ∑_{1≤i_{1},···,i_{n}≤k} F_{i}(λ)u_{i} \frac{∂^{n}}{∂λ_{i_{1}} ⋯ ∂λ_{i_{n}}} ⊗_{R}v_{i}h_{i_{1}} ⋯ h_{i_{n}}.

Then

α_{F_{n}}f = ∑_{i} ∑_{1≤i_{1},···,i_{n}≤k} F_{i}(λ)(ϵ_{0}u_{i})\frac{∂^{n}f}{∂λ_{i_{1}} ⋯ ∂λ_{i_{n}}} v_{i}h_{i_{1}} ⋯ h_{i_{n}}

= ∑_{1≤i_{1},···,i_{n}≤k} \frac{∂^{n}f}{∂λ_{i_{1}} ⋯ ∂λ_{i_{n}}} [∑_{i} F_{i}(λ)(ϵ_{0}u_{i})v_{i}]h_{i_{1}} ⋯ h_{i_{n}}

= ∑_{1≤i_{1},···,i_{n}≤k} \frac{∂^{n}f}{∂λ_{i_{1}} ⋯ ∂λ_{i_{n}}} h_{i_{1}} ⋯ h_{i_{n}},

where the last equality used the fact that ∑_{i} F_{i}(λ)(ϵ_{0}u_{i})v_{i} = (ϵ_{0}⊗id)F (λ) = 1.

Similarly, we have β_{F_{n}}f = f if n = 0, and otherwise β_{F_{n}}f = 0.

Combining these equations, one obtains immediately that

α_{F}f = ∑_{n=0}^{∞} \frac{ℏ^{n}}{n!} α_{F_{n}}f = ∑_{1≤i_{1},···,i_{n}≤k} \frac{ℏ^{n}}{n!} \frac{∂^{n}f}{∂λ_{i_{1}} ⋯ ∂λ_{i_{n}}} h_{i_{1}} ⋯ h_{i_{n}},

β_{F}f = ∑_{n=0}^{∞} \frac{ℏ^{n}}{n!} β_{F_{n}}f = f, and

f ∗_{F} g = (α_{F}f)(g) = fg.

□

To describe the relation between Δ_{F} and the quasi-Hopf coproducts Δ_{λ}, we need a kind of “projection” map from H ⊗_{R_{F}} H to M(η^{∗}, U_{ℏ}𝔤⊗U_{ℏ}𝔤). This can be defined as follows. Let Ad_{Θ} : H⊗H ⟶ H⊗H be the adjoint operator: Ad_{Θ}w = ΘwΘ^{−1}, ∀w ∈ H⊗H. Composing with the natural projection, one obtains a map, denoted by the same symbol Ad_{Θ}, from H⊗H to H⊗_{R}H. Since α_{Θ} = α_{F}, β_{Θ} = β_{F} and Θ(β_{Θ}f⊗1 − 1⊗α_{Θ}f) = 0 in H⊗_{R}H, then Θ(β_{F}f⊗1 − 1⊗α_{F}f) = 0 in H⊗_{R}H. This implies that Ad_{Θ} descends to a map from H ⊗_{R_{F}} H to H⊗_{R}H. On the other hand, there exists an obvious projection P r from H⊗_{R}H to M(η^{∗}, U_{ℏ}𝔤⊗U_{ℏ}𝔤), which is just to take the 0th-order component. Now composing with this projection, one obtains a map from H ⊗_{R_{F}} H to M(η^{∗}, U_{ℏ}𝔤⊗U_{ℏ}𝔤), which is denoted by T. The following proposition gives an explicit description of this map T.

An element x = D⊗_{ℏ}u ∈ H is said to be of order k if D ∈ D[[ℏ]] is of order k.

Proposition 7.8 (i). T (x ⊗_{R_{F}} y) = 0 if either x or y is of order greater than zero.

(ii). T (fu ⊗_{R_{F}} gv) = u⊗g(α_{F}f)v, ∀f, g ∈ M(η^{∗}, U_{ℏ}𝔤⊗U_{ℏ}𝔤) and u, v ∈ U_{ℏ}𝔤.

Proof. (i) is obvious. We prove (ii) below.

T (fu ⊗_{R_{F}} gv)
= P r(e^{ℏ ∑_{i=1}^{k}( \frac{∂}{∂λ_{i}} ⊗h_{i})}(fu⊗gv)e^{−ℏ ∑_{i=1}^{k}( \frac{∂}{∂λ_{i}} ⊗h_{i})})
= P r(e^{ℏ ∑_{i=1}^{k}( \frac{∂}{∂λ_{i}} ⊗h_{i})}(fu⊗gv))
= P r ∑_{1≤i_{1},···,i_{n}≤k} \frac{ℏ^{n}}{n!} (\frac{∂^{n}fu}{∂λ_{i_{1}} ⋯ ∂λ_{i_{n}}})⊗gh_{i_{1}} ⋯ h_{i_{n}}v

= ∑_{1≤i_{1},···,i_{n}≤k} \frac{ℏ^{n}}{n!} \frac{∂^{n}f}{∂λ_{i_{1}} ⋯ ∂λ_{i_{n}}} u⊗gh_{i_{1}} ⋯ h_{i_{n}}v
= u⊗g(α_{F}f)v.
□

Proposition 7.9 The following diagram:

\begin{aligned}& & Δ_{F} \\ & H & ⟶ & H ⊗_{R_{F}} H \\ i & ↑ & & ↓ & T \\ & U_{ℏ}𝔤 & ⟶ & M(η^{∗}, U_{ℏ}𝔤⊗U_{ℏ}𝔤) \\ & & Δ_{λ} \end{aligned}  (90)

commutes, where i : U_{ℏ}𝔤 ⟶ H is the natural embedding. I.e., Δ_{λ} = T ∘Δ_{F} ∘i.

Proof. For any u ∈ U_{ℏ}𝔤, (Δ_{F} ∘i)(u) = Δ_{F}(u) = F^{−1}(Δ_{0}u)F = Θ^{−1}F (λ)^{−1}(Δ_{0}u)F (λ)Θ. Then (T ∘Δ_{F} ∘i)(u) = T [(Δ_{F} ∘i)(u)] = P r[F (λ)^{−1}(Δ_{0}u)F (λ)] = F (λ)^{−1}(Δ_{0}u)F (λ) = Δ_{λ}u. The conclusion thus follows.

□

The following theorem describes the classical limit of this quantum groupoid D⊗_{ℏ}U_{ℏ}𝔤.

Theorem 7.10 Let (U_{ℏ}𝔤, R) be a quasitriangular quantum universal enveloping algebra, and R = 1 + ℏr_{0}(modℏ). Assume that F (λ) ∈ U_{ℏ}𝔤⊗U_{ℏ}𝔤 is a shifted cocycle that F (λ) = 1 + ℏf(λ)(modℏ). Then the classical limit of the corresponding dynamical quantum groupoid D⊗_{ℏ}U_{ℏ}𝔤 is a coboundary Lie bialgebroid (A, A^{∗}, Λ), where A = T η^{∗} × 𝔤 and Λ = ∑_{i=1}^{k} \frac{∂}{∂λ_{i}} ∧ h_{i} + \frac{1}{2}Alt(r_{0}) + Altf(λ).

Proof. It is well known [9] that r_{0} ∈ 𝔤⊗𝔤, and the operator δ : 𝔤 ⟶ ∧^{2}𝔤, δa = [1⊗a + a⊗1, r_{0}], ∀a ∈ 𝔤, defines the cobracket of the corresponding Lie bialgebra of (U_{ℏ}𝔤, R). Thus the Lie bialgbroid corresponding to D⊗U_{ℏ}𝔤 is a coboundary Lie bialgbroid (T η^{∗}×𝔤, T^{∗}η^{∗}×𝔤^{∗}) with r-matroid \frac{1}{2}Alt(r_{0}). On the other hand, it is obvious that Altlim_{ℏ↦0}ℏ^{−1}(F−1) = AltF (λ)+∑_{i=1}^{k} \frac{∂}{∂λ_{i}} ∧h_{i}. The conclusion follows from Theorem 5.20.

□

As a consequence, we have

Corollary 7.11 Under the same hypothesis as in Theorem 7.10, r(λ) = \frac{1}{2}Alt(r_{0}) + Altf(λ) is a dynamical r-matrix.

We refer the interested reader to [1] [19] [24] for the construction of shifted cocycles F (λ) for simple Lie algebras. In this case, the corresponding dynamical r-matrix is the standard one r(λ) = \frac{1}{2} ∑_{α∈Δ_{+}} coth(\frac{1}{2} ≪ α, λ ≫)E_{α} ∧ E_{−α}.

Remark We may replace θ in Equation (78) by θ̃ = ∑_{i=1}^{k} \frac{1}{2}(\frac{∂}{∂λ_{i}} ⊗h_{i} − h_{i}⊗\frac{∂}{∂λ_{i}}) ∈ H⊗H and set Θ̃ = exp ℏθ̃ ∈ H⊗H. In this case, one can show that F̃ = F (λ)Θ̃ satisfies Equation (32 ) is equivalent to the following condition for F (λ):

[(Δ_{0}⊗id)F (λ)]F^{12}(λ + \frac{1}{2}ℏh^{(3)}) = [(id⊗Δ_{0})F (λ)]F^{23}(λ − \frac{1}{2}ℏh^{(1)}).

Then R(λ) = F^{21}(λ)^{−1}RF^{12}(λ) satisfies

R^{12}(λ + \frac{1}{2}ℏh^{(3)})R^{13}(λ − \frac{1}{2}ℏh^{(2)})R^{23}(λ + \frac{1}{2}ℏh^{(1)}) = R^{23}(λ − \frac{1}{2}ℏh^{(1)})R^{13}(λ + \frac{1}{2}ℏh^{(2)})R^{12}(λ − \frac{1}{2}ℏh^{(3)}).

(91)

In fact, both Θ and Θ̃ can be obtained from the quantization of the cotangent bundle symplectic structure T^{∗}η^{∗}, using the normal order and the Weyl order respectively, so they are equivalent. This indicates that solutions to Equation (1) and Equation (91) should be equivalent as well.

8 Appendix and open questions

Given any permutation (i_{1}i_{2}i_{3}) of (123), by σ_{i_{1}i_{2}i_{3}} we denote the permutation operator on UA⊗_{R}UA⊗_{R}UA given by

σ_{i_{i}i_{2}i_{3}}(∑ x_{1}⊗_{R}x_{2}⊗_{R}x_{3}) = ∑ x_{i_{1}}⊗_{R}x_{i_{2}}⊗_{R}x_{i_{3}}.

Proposition 8.1 Assume that T ∈ UA⊗_{R}UA satisfies

T ⊗_{R}1 + (Δ⊗_{R}id)T = 1⊗_{R}T + (id⊗_{R}Δ)T.  (92)

Then AltT =^{def} T − T_{21} is a section in Γ(∧^{2}A).

Proof. First we show that

(Δ⊗_{R}id)AltT = 1⊗_{R}AltT + σ_{132}(AltT ⊗_{R}1).  (93)

To see this, write T = ∑_{i} u_{i}⊗_{R}v_{i}, where u_{i}, v_{i} ∈ UA. Then Equation (92) becomes

∑_{i} u_{i}⊗_{R}v_{i}⊗_{R}1 + ∑_{i} Δu_{i}⊗_{R}v_{i} = ∑_{i} 1⊗_{R}u_{i}⊗_{R}v_{i} + ∑_{i} u_{i}⊗_{R}Δv_{i}.  (94)

Applying the permutation operators σ_{231} and σ_{132}, respectively, to both sides of the above equation, one leads to

∑_{i} v_{i}⊗_{R}1⊗_{R}u_{i} + ∑_{i} σ_{231}(Δu_{i}⊗_{R}v_{i}) = ∑_{i} u_{i}⊗_{R}v_{i}⊗_{R}1 + ∑_{i} Δv_{i}⊗_{R}u_{i};  (95)

∑_{i} u_{i}⊗_{R}1⊗_{R}v_{i} + ∑_{i} σ_{132}(Δu_{i}⊗_{R}v_{i}) = ∑_{i} 1⊗_{R}v_{i}⊗_{R}u_{i} + ∑_{i} u_{i}⊗_{R}Δv_{i}.  (96)

Combining Equations (94), (95) and (96) ((94+(95)-(96)), we obtain:

∑(Δu_{i}⊗_{R}v_{i} − Δv_{i}⊗_{R}u_{i}) = ∑_{i}(1⊗_{R}u_{i}⊗_{R}v_{i} − 1⊗_{R}v_{i}⊗_{R}u_{i} + u_{i}⊗_{R}1⊗_{R}v_{i} − v_{i}⊗_{R}1⊗_{R}u_{i}),

which is equivalent to Equation (93). Here we have used the identity: σ_{231}(Δu_{i}⊗_{R}v_{i}) = σ_{132}(Δu_{i}⊗_{R}v_{i}), which can be easily verified using the fact that Δu_{i} is symmetric.

The conclusion essentially follows from Equation (93). To see this, let us write AltT = ∑_{i} u_{i}⊗_{R}v_{i}, where {v_{i} ∈ UA} are assumed to be R-linearly independent.

From Equation (93), it follows that

∑_{i} Δu_{i}⊗_{R}v_{i} = ∑_{i}(1⊗_{R}u_{i}⊗_{R}v_{i} + u_{i}⊗_{R}1⊗_{R}v_{i}).

I.e., ∑_{i}(Δu_{i} − 1⊗_{R}u_{i} − u_{i}⊗_{R}1)⊗_{R}v_{i} = 0. Hence Δu_{i} = 1⊗_{R}u_{i} + u_{i}⊗_{R}1, which implies that u_{i} ∈ Γ(A). Since AltT is skew symmetric, we conclude that AltT ∈ Γ(∧^{2}A).

□

Remark It might be useful to consider the following cochain complex

0 → R →^{∂} UA →^{∂} UA⊗_{R}UA →^{∂} UA⊗_{R}UA⊗_{R}UA →^{∂}  (97)

where ∂ : ⊗_{R}^{n}UA ⟶ ⊗_{R}^{n+1}UA, ∂ = ∂^{0} − ∂^{1} + ⋯ + (−1)^{n+1}∂^{n+1}, ∂^{i}(x_{1}⊗_{R} ⋯ ⊗_{R}x_{n}) = x_{1}⊗_{R} ⋯ ⊗_{R}x_{i−1}⊗_{R}Δx_{i}⊗_{R}x_{i+1}⊗_{R} ⋯ ⊗_{R}x_{n} for 1 ≤ i ≤ n, and ∂^{0}x = 1⊗_{R}x, ∂^{n+1}x = x⊗_{R}1. It is simple to check that ∂^{2} = 0. In fact, this is the subcomplex of the Hochschild cochain complex of the algebra C^{∞}(G) (G is a local groupoid integrating the Lie algebroid A) by restricting to the left invariant muti-differential operators. It is natural to expect that the cohomology of this complex is isomorphic to Γ(∧^{∗}A), where the isomorphism from H^{∗} (more precisely, cocycles) to Γ(∧^{∗}A) is the usual skew-symmetrization map. This is known to be true for Lie algebras [11] and the tangent bundle Lie algebroid. However we could not find such a general result in the literature. In terms of this cochain complex, it is simple to describe what we have proved in Proposition 8.1. It simply means that Alt : UA⊗_{R}UA ⟶ UA⊗_{R}UA maps 2-cocycles into Γ(∧^{2}A).

We end this paper by a list of open questions.

Question 1: We believe that techniques in [12] would be useful to prove the conjecture in Section 6. While the proof of Etingof and Kazhdan relies heavily on the double of a Lie bialgebra, the double of a Lie bialgeboid is no longer a Lie algeboid. Instead it is a Courant algebroid [27], where certain anomalies are inevitable. As a first step, it is natural to ask: what is the universal enveloping algebra of a Courant algebroid? Roytenberg and Weinstein proved that Courant algebroids give rise to homotopy Lie algebras [37]. It is expected that these homotopy Lie algebras are useful to understand this question as well as the quantization problem.

Question 2: One can form a Kontsevich’s formality-type conjecture for Lie algebroids, where one simply replace in Kontsevich formality theorem [25] polyvector fields by sections of ∧^{∗}A and multi-differential operators by UA⊗_{R} ⋯ ⊗_{R}UA for a Lie algebroid A. Does this conjecture hold? It is not clear if Kontsevich’s proof in [25] can be generalized to the general context of Lie algebroids. If this conjecture holds, it would imply that any triangular Lie bialgebroid is quantizable.

Question 3: Given a solution r : η^{∗} ⟶ 𝔤⊗𝔤 of the classical dynamical Yang-Baxter equation:

Altdr + [r^{12}, r^{13}] + [r^{12}, r^{23}] + [r^{13}, r^{23}] = 0  (98)

(in this case Altr satisfies Condition (ii) of a dynamical r-matrix (see Introduction) if r + r^{21} is ad-invariant), a quantization of r is a quantum dynamical R-matrix R : η^{∗} ⟶ U_{ℏ}𝔤⊗U_{ℏ}𝔤 such that R(λ) = 1+hr(modℏ^{2}), where U_{ℏ}𝔤 is a quantum universal enveloping algebra. Is every dynamical rmatrix quantizable? Our result in Section 6 should imply that every regular triangular dynamical r-matrix is quantizable. However, it is not even clear whether a general triangular dynamical r-matrix is quantizable.

Question 4: According to the general spirit of deformation theory, any deformation corresponds to a certain cohomology. In particular, the deformation of Hopf algebras is controlled by the cohomology of a certain double complex arising from the Hopf algebra structure [20] [21]. It is natural to ask what is the proper cohomology theory controlling the deformation of a Hopf algebroid, and in particular what is the premier obstruction to the quantization problem.

Question 5: What is the connection between dynamical quantum groupoids and quantum Virasoro algebra or quantum W-algebras [17] [38]?

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