Triangular dynamical r-matrices and quantization

PING XU ^{∗}

Department of Mathematics The Pennsylvania State University University Park, PA 16802, USA email: ping@math.psu.edu

Abstract

We provide a general study for triangular dynamical r-matrices using Poisson geometry. We show that a triangular dynamical r-matrix always gives rise to a regular Poisson manifold. Using the Fedosov method, we prove that non-degenerate (i.e., the corresponding Poisson manifolds are symplectic) triangular dynamical r-matrices r : 𝔥^{∗} ⟶ ∧^{2}𝔤 are quantizable, and the quantization is classified by the relative Lie algebra cohomology H^{2}(𝔤, 𝔥)[[ℏ]].

1 Introduction

In the last two decades, the theory of quantum groups has undergone tremendous development. The classical counterpart of quantum groups are Lie bialgebras [12]. Many interesting quantum groups were found and studied by various authors, but the existence proof of quantization for arbitrary Lie bialgebras was obtained only recently by Etingof and Kazhdan [14]. For triangular Lie bialgebras, however, an elementary proof of quantization was obtained by Drinfeld in 1983 [13]. Drinfeld’s idea can be outlined as follows. A triangular r-matrix on a Lie algebra 𝔤 defines a left invariant Poisson structure on its corresponding Lie group G. By restricting to a Lie subalgebra if necessary, one may in fact assume that this is symplectic. One may then quantize the r-matrix by finding a G-invariant ∗-product on G, of which there may be several. In [13], Drinfeld identified the symplectic manifold with a coadjoint orbit of a central extension of 𝔤, then applying Berezin quantization [6].

Recently, there is growing interest in the so called quantum dynamical Yang-Baxter equation (see Equation (12)). This equation arises naturally from various contexts in mathematical physics. It first appeared in the work of Gervais-Neveu in their study of quantum Liouville theory [24]. Recently it reappeared in Felder’s work on the quantum Knizhnik-Zamolodchikov-Bernard equation. It also has been found to be connected with the quantum Caloger-Moser systems [2] [19]. As the quantum Yang-Baxter equation is connected with quantum groups, the quantum dynamical Yang-Baxter equation is known to be connected with elliptic quantum groups [22], as well as with Hopf algebroids or quantum groupioids [16] [17] [40] [42].

∗Research partially supported by NSF grants DMS97-04391.

The classical counterpart of the quantum dynamical Yang-Baxter equation was first considered by Felder [22], and then studied by Etingof and Varchenko [15]. This is the so called classical dynamical Yang-Baxter equation, and a solution to such an equation (plus some other reasonable conditions) is called a classical dynamical r-matrix. More precisely, given a Lie algebra 𝔤 over ℝ (or over ℂ) with an Abelian Lie subalgebra 𝔥, a classical dynamical r-matrix is a smooth (or meromorphic) function r(λ) : 𝔥^{∗} ⟶ 𝔤⊗𝔤 satisfying the following conditions:

(i). (zero weight condition) [h⊗1 + 1⊗h, r(λ)] = 0, ∀h ∈ 𝔥;

(ii). (normal condition) r + r^{21} = Ω, where Ω ∈ (S^{2}𝔤)^{𝔤} is a Casimir element;

(iii). (classical dynamical Yang-Baxter equation)

Alt(dr) + [r^{12}, r^{13}] + [r^{12}, r^{23}] + [r^{13}, r^{23}] = 0,  (1)

where Altdr = ∑(h_{i}^{(1)} \frac{∂r^{23}}{∂λ^{i}} − h_{i}^{(2)} \frac{∂r^{13}}{∂λ^{i}} + h_{i}^{(3)} \frac{∂r^{12}}{∂λ^{i}}).

A fundamental question is whether a classical dynamical r-matrix is always quantizable. There have appeared many results in this direction. For the standard classical dynamical r-matrix in 𝔰𝔩_{2}(ℂ), a quantization was obtained by Babelon [3] in 1991. For general simple Lie algebras, quantizations were recently found independently by Arnaudon et al. [1] and Jimbo et al. [25] based on the approach of Fronsdal [23]. Similar results were also found by Etingof and Varchenko [17] using intertwining operators. Very recently, using a method similar to [1] [23] [25], Etingof et al.

[18] obtained a quantization for all the classical dynamical r-matrices of semi-simple Lie algebras in Schiffmann’s classification list [35]. However, the general quantization problem still remains open, and a general quantization recipe has yet to be found. On the other hand, the classification problem has seen little progress.

In this paper, we study the quantization problem for general triangular classical dynamical r-matrices. Triangular classical dynamical r-matrices are those satisfying the skew-symmetric condition r^{21}(λ)+r^{12}(λ) = 0. In this case, Equation (1) is equivalent to ∑_{i} h_{i} ∧ \frac{∂r}{∂λ^{i}} + \frac{1}{2}[r, r] = 0. These r-matrices are in one-one correspondence with regular Poisson structures π = ∑_{i} \overrightarrow{h_{i}} ∧ \frac{∂}{∂λ^{i}} + \overrightarrow{r(λ)} on the manifold 𝔥^{∗} × G invariant under the left G and right H-actions. Thus one may expect to quantize a classical dynamical r-matrix by finding a certain special type of star-product [5] on the corresponding Poisson manifold. This is exactly the route we take in the present paper. In some sense, this is also a natural generalization of the quantization method used by Drinfeld in [13] as outlined at the beginning of the introduction. In fact, in the present paper, we mainly deal with non-degenerate triangular classical dynamical r-matrices (i.e., the corresponding Poisson manifolds are in fact symplectic). Berezin quantization no longer works in this situation. However, one may use the Fedosov method to obtain the desired star-products as we will see later. It is well-known that star products on a symplectic manifold are classified by the second cohomology group of the manifold with coefficients in the formal ℏ-power series. In light of this result, we are able to classify the quantization of a non-degenerate triangular classical dynamical r-matrix and prove that the quantization is parameterized by the relative Lie algebra cohomology H^{2}(𝔤, 𝔥)[[ℏ]].

For a general triangular classical dynamical r-matrix, it is natural to ask whether it is possible to reduce it to a non-degenerate one by restricting to a Lie subalgebra. This is always true in the non-dynamical case [13]. Unfortunately, in general this fails in the dynamical case, and we will study the conditions under which this is possible. In this case, these r-matrices are called splittable. Splittable triangular classical dynamical r-matrices resemble in many ways non-degenerate ones. And in particular, they can be quantized by the Fedosov method.

The paper is outlined as follows. After Section 1 (this introduction), in Section 2, we study general properties of triangular classical dynamical r-matrices. It is proved that triangular classical dynamical r-matrices correspond to some special Poisson structures on 𝔥^{∗} × G, which are always regular. This may seem surprising at first glance since the rank of r(λ) may depend on the point λ. The main tool in Section 2 is the method of Lie groupoids and Lie algebroids. In particular, we show how gauge transformations, first introduced by Etingof and Varchenko [15], enter naturally from the viewpoint of Lie algebroids. The study of the tangent space of the modular space of dynamical r-matrices naturally leads to dynamical r-matrix cohomology, which is shown to be isomorphic to the relative Lie algebra cohomology when r is non-degenerate. Section 3 is devoted to the proof of the equivalence between quantization of triangular classical dynamical r-matrices and the so called compatible star products on their corresponding Poisson manifolds 𝔥^{∗} × G. In Section 4, we study symplectic connections on such symplectic manifolds (M = 𝔥^{∗} × G). In particular, we show that there exists a G × H-invariant (i.e. left G-invariant and right H-invariant) torsion free symplectic connection on M such that the left invariant vector fields \overrightarrow{h}, ∀h ∈ 𝔥 are all parallel. The main result of Section 5 is that the Fedosov quantization with such a symplectic connection and some suitable choice of Weyl curvatures gives rise to compatible ∗-products on M = 𝔥^{∗} × G. Therefore, as a consequence, we prove the existence of a quantization of non-degenerate triangular classical dynamical r-matrices. The presentation in Section 5, however, is made in a more general setting, which is of its own interest. Section 6 is devoted to the classification of quantization. In particular, we show that the equivalence classes of quantizations of a non-degenerate triangular dynamical r-matrix r : 𝔥^{∗} ⟶ ∧^{2}𝔤 are parameterized by the relative Lie algebra cohomology with coefficients in the formal ℏ-power series H^{2}(𝔤, 𝔥)[[ℏ]]. Some speculation on the classification of quantization of a general triangular classical dynamical r-matrix is given as a conjecture, which is consistent with Kontesvich’s formality theorem [26]. Section 7 is an Appendix, where we recall some basic ingredients of the Fedosov quantization, which are used throughout the paper.

Finally, some remarks are in order. Quantization of dynamical r-matrices is related to quantization of Lie bialgebroids as shown in [42]. However, for simplicity, we will avoid using quantum groupoids in the present paper even though many ideas are rooted there. Also in this paper, we work in the smooth case. Namely, Lie algebras are finite dimensional Lie algebras over ℝ, all manifolds and maps are smooth, but our approach works for the complex category as well. For simplicity, we assume that a dynamical r-matrix is always defined on 𝔥^{∗}. In reality, it may only be defined on an open neighbourhood U ⊂ 𝔥^{∗} of the origin, but all our results hold in this situation as well.

Acknowledgments. The author would like to thank Martin Bordemann and Boris Tysgan for useful discussions. He is also grateful to Pavel Etingof for suggesting to him to write this up. In addition to the funding sources mentioned in the first footnote, he would also like to thank the Max-Planck Institut for the hospitality and financial support while part of this project was being done.

2 Triangular dynamical r-matrices

In this section, we study some general aspects of triangular dynamical r-matrices. As a useful tool, we shall utilize the method of Lie algebroids and Lie groupoids. Let 𝔤 be a Lie algebra and 𝔥 ⊂ 𝔤 an Abelian Lie subalgebra of dimension l. By a triangular dynamical r-matrix, we mean a smooth function r : 𝔥^{∗} ⟶ ∧^{2}𝔤 satisfying:

(i). the zero weight condition: [h, r(λ)] = 0, ∀h ∈ 𝔥, and

(ii). the classical dynamical Yang-Baxter equation (CDYBE):

∑_{i} h_{i} ∧ \frac{∂r}{∂λ^{i}} + \frac{1}{2}[r, r] = 0,  (2)

where the bracket [·, ·] refers to the Schouten type bracket: ∧^{k}𝔤⊗∧^{l} 𝔤 ⟶ ∧^{k+l−1}𝔤 induced from the Lie algebra bracket on 𝔤. Here {h_{1}, ⋯, h_{l}} is a basis in 𝔥, and (λ^{1}, ⋯, λ^{l}) its induced coordinate system on 𝔥^{∗}. It is known [4] [30] that the CDYBE is closely related to Lie bialgebroids. Recall that a Lie bialgebroid is a pair of Lie algebroids (A, A^{∗}) satisfying the following compatibility condition (see [32] [33] and [27]):

d_{∗}[X, Y] = [d_{∗}X, Y] + [X, d_{∗}Y], ∀X, Y ∈ Γ(A),  (3)

where the differential d_{∗} on Γ(∧^{∗}A) comes from the Lie algebroid structure on A^{∗}.

Given a Lie algebroid A over P with anchor a, and a section Λ of Γ(∧^{2}A) satisfying the condition [Λ, Λ] = 0, one may define a Lie algebroid structure on A^{∗} by simply requiring the differential d_{∗} : Γ(∧^{k}A) ⟶ Γ(∧^{k+1}A) to be d_{∗} = [Λ, ·]. More explicitly, denote by Λ^{#} the bundle map A^{∗} ⟶ A defined by Λ^{#}(ξ)(η) = Λ(ξ, η), ∀ξ, η ∈ Γ(A^{∗}). Then the bracket on Γ(A^{∗}) is defined by

[ξ, η] = L_{Λ^{#}ξ}η − L_{Λ^{#}η}ξ − d[Λ(ξ, η)],  (4)

and the anchor a_{∗} is the composition a ∘ Λ^{#} : A^{∗} ⟶ T P. It is easy to show that (A, A^{∗}) is a Lie bialgebroid, which is called a triangular Lie bialgebroid [32].

Now consider A = T 𝔥^{∗} × 𝔤 and equip A with the standard product Lie algebroid structure. Then the anchor a : T 𝔥^{∗} × 𝔤 ⟶ T 𝔥^{∗} is simply the projection. The relation between triangular dynamical r-matrices and triangular Lie bialgebroids are described by the following [4] [30].

Proposition 2.1 Given a smooth function r : 𝔥^{∗} ⟶ ∧^{2}𝔤, r is a triangular dynamical r-matrix iff the Lie algebroid (A, a) together with Λ = ∑_{i} h_{i} ∧ \frac{∂}{∂λ^{i}} + r(λ) ∈ Γ(∧^{2}A) defines a triangular Lie bialgebroid.

Proof. By a straightforward computation, we have [Λ, Λ] = 2(∑_{i} h_{i} ∧ \frac{∂r}{∂λ^{i}} + \frac{1}{2}[r, r]+ ∑_{i}[r, h_{i}]∧ \frac{∂}{∂λ^{i}}). It thus follows that [Λ, Λ] = 0 iff ∑_{i} h_{i} ∧ \frac{∂r}{∂λ^{i}} + \frac{1}{2}[r, r] = 0 and [r, h_{i}] = 0, i = 1, ⋯ l, i.e., r is a triangular dynamical r-matrix.

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Let G be a Lie group integrating the Lie algebra 𝔤 and H ⊂ G an Abelian Lie subgroup with Lie algebra 𝔥. Consider M = 𝔥^{∗} × G. Let G act on M from the left by left multiplication on G, and H act from the right by right multiplication on G. An equivalent version of Proposition 2.1 is the following

Proposition 2.2 For a smooth function r : 𝔥^{∗} ⟶ ∧^{2}𝔤, r is a triangular dynamical r-matrix iff π = ∑_{i} \overrightarrow{h_{i}} ∧ \frac{∂}{∂λ^{i}} + \overrightarrow{r(λ)} defines a G × H-invariant Poisson structure on M = 𝔥^{∗} × G, where \overrightarrow{h_{i}} ∈ X (M) is the left invariant vector field on M generated by h_{i} and similarly \overrightarrow{r(λ)} ∈ Γ(∧^{2}T M) is the left invariant bivector field on M corresponding to r(λ).

Theorem 2.3 If r : 𝔥^{∗} ⟶ ∧^{2}𝔤 is a triangular dynamical r-matrix, then 𝔥 + r(λ)^{#}𝔥^{⊥} is a Lie subalgebra of 𝔤. Moreover for any λ, μ ∈ 𝔥^{∗}, the Lie subalgebras 𝔥 + r(λ)^{#}𝔥^{⊥} and 𝔥 + r(μ)^{#}𝔥^{⊥} are isomorphic.

Proof. For any λ ∈ 𝔥^{∗}, A_{λ} = T_{λ}𝔥^{∗} ⊕ 𝔤 ≅ 𝔥^{∗} ⊕ 𝔤 and A_{λ}^{∗} ≅ 𝔥 ⊕ 𝔤^{∗}. Under these identifications, the bundle map Λ_{λ}^{#} : A_{λ}^{∗} ⟶ A_{λ} is given by

(h, ξ) ↦ (i^{∗}ξ, −h + r(λ)^{#}ξ), ∀h ∈ 𝔥 and ξ ∈ 𝔤^{∗},  (5)

where i : 𝔥 ⟶ 𝔤 is the inclusion. Set B = Λ^{#}(A^{∗}) = ∪_{λ∈𝔥^{∗}}Λ_{λ}^{#}(A_{λ}^{∗}) ⊂ A. Since (A, Λ) defines a triangular Lie bialgebroid, B is integrable. I.e., Γ(B) is closed under the Lie algebroid bracket on Γ(A). Hence kera|_{B_{λ}} is a Lie subalgebra of kera|_{A_{λ}}. Now it is easy to see that kera|_{B_{λ}} = 𝔥+r(λ)^{#}𝔥^{⊥} and kera|_{A_{λ}} = 𝔤. It thus follows that 𝔥 + r(λ)^{#}𝔥^{⊥} is a Lie subalgebra of 𝔤. On the other hand, from Equation (5), it is easy to see that a(B_{λ}) = T_{λ}𝔥^{∗}. Hence a : B ⟶ T 𝔥^{∗} is surjective, which implies that B is in fact a transitive Lie algebroid (also called a gauge Lie algebroid [31]). Thus the dimension of B_{λ} must be independent of λ, B is a subbundle of A, and the isotropic Lie algebras of B at different points of 𝔥^{∗} are all isomorphic. This implies that, for any λ, μ ∈ 𝔥^{∗}, 𝔥 + r(λ)^{#}𝔥^{⊥} is isomorphic to 𝔥 + r(μ)^{#}𝔥^{⊥}.

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For the sake of simplicity, we denote by 𝔤_{λ} the Lie subalgebra 𝔥 + r(λ)^{#}𝔥^{⊥}. Define the rank of a triangular dynamical r-matrix r to be dim𝔤_{λ} − dim𝔥, which is denoted as rank r. We say a triangular dynamical r-matrix r is non-degenerate if rank r = dim𝔤 − dim𝔥.

An immediate consequence of Theorem 2.3 is

Corollary 2.4 Under the same hypothesis as in Theorem 2.3, rank r is independent of the point λ and therefore is a well-defined even number. Moreover B = Λ^{#}A^{∗} ⊂ A is a Lie subalgebroid of rank 2dim 𝔥+rank r, and (M, π) is a regular Poisson manifold of rank 2dim 𝔥+rank r.

In particular, we have the following

Corollary 2.5 Given a triangular dynamical r-matrix r : 𝔥^{∗} ⟶ ∧^{2}𝔤, the following statements are equivalent:

(i). r is non-degenerate;

(ii). the bundle map Λ^{#} : A^{∗} ⟶ A is nondegenerate;

(iii). 𝔤_{λ} = 𝔤, ∀λ ∈ 𝔥^{∗};

(iv). (M, π) is a symplectic manifold.

If we choose a decomposition 𝔤 = 𝔥 ⊕ 𝔪, where 𝔪 is a subspace of 𝔤, and choose a basis {h_{1}, ⋯, h_{l}} for 𝔥 and a basis {e_{1}, ⋯, e_{m}} for 𝔪, we may write

r(λ) = ∑ a^{ij}(λ)h_{i} ∧ h_{j} + ∑ b^{ij}(λ)h_{i} ∧ e_{j} + ∑ c^{ij}(λ)e_{i} ∧ e_{j}.  (6)

Then it is simple to see that 𝔤_{λ} = 𝔥 ⊕ Span{∑_{j} c^{ij}(λ)e_{j}|i = 1, ⋯, m}, and rank r is the rank of the matrix (c^{ij}(λ)). Therefore, we immediately see that the rank of (c^{ij}(λ)) is independent of λ. Clearly r is non-degenerate iff the matrix (c^{ij}(λ)) is non-degenerate.

A natural question arises as to whether it is possible to make an arbitrary triangular dynamical r-matrix non-degenerate by considering it to be valued in a Lie subalgebra of 𝔤 if necessary. This is true in the non-dynamical case [13], for example. However, in the dynamical case, this is not always possible as we will see below. Nevertheless we will single out those r-matrices possessing this property, which will be called splittable. Splittable triangular dynamical r-matrices contain a large class of interesting dynamical r-matrices, which in fact include almost all examples we know, e.g., those as classified in [15] when 𝔤 is a simple Lie algebra. More precisely,

Definition 2.6 A triangular dynamical r-matrix r : 𝔥^{∗} ⟶ ∧^{2}𝔤 is said to be splittable if for any λ ∈ 𝔥^{∗}, i^{∗}(r(λ)^{#−1}𝔥) = 𝔥^{∗}, where i : 𝔥 ⟶ 𝔤 is the inclusion.

Proposition 2.7 Suppose that r is a triangular dynamical r-matrix. Then the following statements are equivalent:

(i). r is splittable;

(ii). for any λ ∈ 𝔥^{∗}, r(λ)^{#}𝔤^{∗} ⊂ 𝔤_{λ};

(iii). if r(λ) is given as in Equation (6) under a decomposition 𝔤 = 𝔥 ⊕ 𝔪, then for any i, ∑_{j} b^{ij}(λ)e_{j} ∈ Span{∑_{j} c^{ij}(λ)e_{j}|i = 1, ⋯, m};

(iv). for any fixed λ ∈ 𝔥^{∗}, there exists a decomposition 𝔤 = 𝔥 ⊕ 𝔪, under which

r(λ) = ∑ a^{ij}(λ)h_{i} ∧ h_{j} + ∑ c^{ij}(λ)e_{i} ∧ e_{j};  (7)

(v). T 𝔥^{∗} × {0} ⊂ B.

Let us first prove the following simple lemma in linear algebra.

Lemma 2.8 Let V = 𝔥 ⊕ 𝔪 be a decomposition of vector spaces, and let {h_{1}, ⋯, h_{l}} be a basis of 𝔥, and {e_{1}, ⋯, e_{m}} a basis of 𝔪. Let r ∈ ∧^{2}V be any element such that r = ∑ a^{ij}h_{i} ∧ h_{j} + ∑ h_{i} ∧ x^{i} + ∑ c^{ij}e_{i} ∧ e_{j}, where x^{i} ∈ 𝔪, and a_{ij}, c_{ij} are skew-symmetric, i.e., a_{ij} = −a_{ji} and c_{ij} = −c_{ji}. If I ⊂ {1, ⋯ l} is a subset of index such that for any i_{0} ∈ I, x^{i_{0}} ∈ Span{∑_{j} c^{ij}e_{j}|i = 1, ⋯, m}. Then one can change the decomposition V = 𝔥 ⊕ 𝔪̃ so that under a suitable basis {ẽ_{1}, ⋯, ẽ_{m}} of 𝔪̃, we have r = ∑ ã^{ij}h_{i} ∧ h_{j} + ∑_{i∉I} h_{i} ∧ x^{i} + ∑ c^{ij}ẽ_{i} ∧ ẽ_{j}.

Proof. ∀i_{0} ∈ I, by assumption, there are constants γ_{i}^{i_{0}}, i = 1, ⋯, m, such that x^{i_{0}} = 2 ∑_{ij} γ_{i}^{i_{0}}c^{ij}e_{j}. Let ẽ_{i} = e_{i} + ∑_{i_{0}∈I} γ_{i}^{i_{0}}h_{i_{0}}, ∀i = 1, ⋯, m. Then

∑ c^{ij}ẽ_{i} ∧ ẽ_{j}
= ∑ c^{ij}(e_{i} + ∑_{i_{0}∈I} γ_{i}^{i_{0}}h_{i_{0}}) ∧ (e_{j} + ∑_{i_{0}∈I} γ_{j}^{i_{0}}h_{i_{0}})
= ∑ c^{ij}e_{i} ∧ e_{j} + 2 ∑ c^{ij}γ_{i}^{i_{0}}h_{i_{0}} ∧ e_{j} (mod ∧^{2} 𝔥)
= ∑ c^{ij}e_{i} ∧ e_{j} + ∑_{i_{0}∈I} h_{i_{0}} ∧ x^{i_{0}} (mod ∧^{2} 𝔥).

Hence r = ∑ c^{ij}ẽ_{i} ∧ ẽ_{j} + ∑_{i∉I} h_{i} ∧ x^{i} (mod ∧^{2} 𝔥). This concludes the proof.

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Proof of Proposition 2.7

(i)⇒(ii) Let us fix a basis {h_{1}, ⋯, h_{l}} of 𝔥, and let {h_{∗}^{1}, ⋯, h_{∗}^{l}} be its dual basis in 𝔥^{∗}. By assumption, for any 1 ≤ i ≤ l, there is a ξ^{i} ∈ 𝔤^{∗} such that i^{∗}ξ^{i} = h_{∗}^{i} and r(λ)^{#}ξ^{i} ∈ 𝔥. Given any ξ ∈ 𝔤^{∗}, take a_{i} =< ξ, h_{i} > and η = ξ − ∑ a_{i}ξ^{i}. Then it is easy to see that η ∈ 𝔥^{⊥}. Hence r(λ)^{#}ξ = ∑ a_{i}r(λ)^{#}ξ^{i} + r(λ)^{#}η ∈ 𝔥 + r(λ)^{#}𝔥^{⊥} = 𝔤_{λ}.

(ii)⇒(iii) Let {h_{1}, ⋯, h_{l}} be a basis of 𝔥, {e_{1}, ⋯, e_{m}} a basis of 𝔪, and {h_{∗}^{1}, ⋯, h_{∗}^{l}, e_{∗}^{1}, ⋯, e_{∗}^{m}} the dual basis of {h_{1}, ⋯, h_{l}, e_{1}, ⋯, e_{m}} in 𝔤^{∗}. It is trivial to see that r(λ)^{#}e_{∗}^{i} = − ∑_{j} b^{ji}(λ)h_{j} + 2 ∑_{j} c^{ij}(λ)e_{j}. Hence we have

𝔤_{λ} = 𝔥 ⊕ Span{∑_{j} c^{ij}(λ)e_{j}|i = 1, ⋯, m}.

Now r(λ)^{#}h_{∗}^{i} = ∑_{j} 2a^{ij}(λ)h_{j} + ∑_{j} b^{ij}(λ)e_{j}. Since r(λ)^{#}h_{∗}^{i} ∈ 𝔤_{λ} by assumption, it follows that ∑_{j} b^{ij}(λ)e_{j} ∈ Span{∑_{j} c^{ij}(λ)e_{j}|i = 1, ⋯, m}.

(iii)⇒(vi) This follows from Lemma 2.8.

(vi)⇒(v) If r(λ) = ∑ a^{ij}(λ)h_{i} ∧ h_{j} + ∑ c^{ij}(λ)e_{i} ∧ e_{j}, then r(λ)^{#}h_{∗}^{i} = 2 ∑_{j} a^{ij}(λ)h_{j}. Thus according to Equation (5), Λ_{λ}^{#}(2 ∑_{j} a^{ij}(λ)h_{j}, h_{∗}^{i}) = (h_{∗}^{i}, 0). Hence, (h_{∗}^{i}, 0) ∈ B_{λ}. This implies that T_{λ}𝔥^{∗} × {0} ⊂ B_{λ}.

(v)⇒(i) Given any φ ∈ 𝔥^{∗}, we know that (φ, 0) ∈ B_{λ} by assumption. Therefore there is a h ∈ 𝔥 and ξ ∈ 𝔤^{∗} such that Λ_{λ}^{#}(h, ξ) = (φ, 0), i.e., (i^{∗}ξ, −h + r(λ)^{#}ξ) = (φ, 0) according to Equation (5). This implies that φ = i^{∗}ξ and r(λ)^{#}ξ = h. Hence φ ∈ i^{∗}(r(λ)^{#−1}𝔥). Therefore, we conclude that 𝔥^{∗} ⊂ i^{∗}(r(λ)^{#−1}𝔥).

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Remark In the proof above, the decomposition 𝔤 = 𝔥⊕𝔪 and the choice of the basis {e_{1}, ⋯, e_{m}} in (iv) depend on the particular point λ. It is not clear whether it is possible to find a decomposition so that Equation (7) holds uniformly for all points in 𝔥^{∗}. On the other hand, if there exists such a decomposition 𝔤 = 𝔥 ⊕ 𝔪 so that a triangular dynamical r-matrix is of the form as in Equation (7), it is always splittable.

An immediate consequence of Proposition 2.7 is the following:

Corollary 2.9 If r : 𝔥^{∗} ⟶ ∧^{2}𝔤 is a splittable triangular dynamical r-matrix, then

(i). 𝔤_{λ} is independent of λ, i.e., 𝔤_{λ} = 𝔤_{μ}, ∀λ, μ ∈ 𝔥^{∗}. We will denote 𝔤_{λ} by 𝔤_{1}.

(ii). r can be considered as a non-degenerate triangular dynamical r-matrix valued in ∧^{2}𝔤_{1}.

Proof. By Proposition 2.7, T 𝔥^{∗} ×{0} is a Lie subalgebroid of B. Hence for any X ∈ X (𝔥^{∗}), (X, 0) ∈ Γ(B). Let φ_{t} be the (local) flow on 𝔥^{∗} generated by X. The bisection exp t(X, 0) on the groupoid Γ = 𝔥^{∗} × 𝔥^{∗} × G generated by the section (X, 0) ∈ Γ(A) is {(λ, φ_{t}(λ), 1)|λ ∈ 𝔥^{∗}}. Hence its induced isomorphism between Γ_{λ} and Γ_{φ_{t}(λ)} is the identity map, when both groups are naturally identified with G. Here Γ_{λ} and Γ_{φ_{t}(λ)} denote the isotropic groups of Γ at the points λ and φ_{t}(λ), respectively. Therefore, Ad_{exp t(X,0)} is an identity map between their corresponding isotropic Lie algebras. On the other hand, since (X, 0) ∈ Γ(B), hence Ad_{exp t(X,0)}, when being restricted to B, is exactly the map which establishes the isomorphism between 𝔤_{λ} and 𝔤_{φ_{t}(λ)}. Hence, 𝔤_{λ} and 𝔤_{φ_{t}(λ)} are equal as Lie subalgebras of 𝔤.

For the second part, since r is splittable, we have r(λ)^{#}𝔤^{∗} ⊂ 𝔤_{1} according to Proposition 2.7. Hence ∀λ ∈ 𝔥^{∗}, r(λ) ∈ ∧^{2}(r(λ)^{#}𝔤^{∗}) ⊂ ∧^{2}𝔤_{1}. By dimension counting, one easily sees that r is non-degenerate when being considered as a dynamical r-matrix valued in ∧^{2}𝔤_{1}.

□

Let g : 𝔥^{∗} ⟶ G^{H} be a smooth map, where G^{H} denotes the centralizer of H in G with its Lie algebra being denoted by 𝔤^{H}. Then g can be naturally considered as a bisection of the groupoid Γ = 𝔥^{∗} × 𝔥^{∗} × G, and hence we can talk about the induced automorphism Ad_{g} of the corresponding Lie algebroid. In particular, we have an automorphism Ad_{g} : Γ(∧^{∗}A) ⟶ Γ(∧^{∗}A) of the Gerstenhaber algebra ⊕Γ(∧^{∗}A) [41].

Given a smooth function r : 𝔥^{∗} ⟶ ∧^{2}𝔤, let Λ_{r} = ∑_{i} h_{i} ∧ \frac{∂}{∂λ^{i}} + r(λ) ∈ Γ(∧^{2}A) as in Proposition 2.1. Then

Ad_{g}Λ_{r} = Ad_{g}(∑_{i} h_{i} ∧ \frac{∂}{∂λ^{i}} + r)
= ∑_{i} Ad_{g}h_{i} ∧ (\frac{∂}{∂λ^{i}} − \frac{∂g}{∂λ^{i}} g^{−1}) + Ad_{g}r

= ∑_{i} h_{i} ∧ (\frac{∂}{∂λ^{i}} − \frac{∂g}{∂λ^{i}} g^{−1}) + Ad_{g}r
= ∑_{i} h_{i} ∧ \frac{∂}{∂λ^{i}} + (Ad_{g}r − ∑_{i} h_{i} ∧ \frac{∂g}{∂λ^{i}} g^{−1}).

Here in the second from the last equality, we used Ad_{g}h_{i} = h_{i} since g ∈ G^{H}. Let

r_{g} = Ad_{g}r − ∑_{i} h_{i} ∧ \frac{∂g}{∂λ^{i}} g^{−1}.

Combining with Proposition 2.1, we thus have proved the following:

Proposition 2.10 Assume that g : 𝔥^{∗} ⟶ G^{H} is a smooth map. Then

(i). Λ_{r_{g}} = Ad_{g}Λ_{r};

(ii). r is a triangular dynamical r-matrix iff r_{g} is a triangular dynamical r-matrix.

(iii). rank r_{g} =rank r; in particular, if r is non-degenerate, so is r_{g}.

This proposition naturally leads us to the notion of gauge transformation on dynamical rmatrices, first introduced by Etingof and Varchenko [15]. Two triangular dynamical r-matrices r_{1} and r_{2} are said to be gauge equivalent if there exists a smooth function g : 𝔥^{∗} ⟶ G^{H} such that r_{2} = (r_{1})_{g}.

Remark Although non-degenerate triangular dynamical r-matrices are preserved by gauge transformations, splittable dynamical r-matrices in general are not. For example, the trivial triangular dynamical r-matrix r = 0 is always splittable. However r_{g} = − ∑ h_{i} ∧ \frac{∂g}{∂λ^{i}} g^{−1} is in general not splitable unless G^{H} = H.

By M(𝔤, 𝔥), we denote the quotient space of the space of all triangular dynamical r-matrices r : 𝔥^{∗} ⟶ ∧^{2}𝔤 modular gauge transformations. We will call M(𝔤, 𝔥) the modular space of triangular dynamical r-matrices.

Next we will introduce the dynamical r-matrix cohomology H_{r}^{∗}(𝔤, 𝔥), whose second cohomology group describes the tangent space of the modular space M(𝔤, 𝔥). As we will see in Section 6, the second cohomology group H_{r}^{2}(𝔤, 𝔥) is connected with the classification of quantization of r when r is non-degenerate.

Consider C^{k} = C^{∞}(𝔥^{∗}, (∧^{k}𝔤)^{H}) (or equivalently denoted as C^{∞}(𝔥^{∗}, (∧^{k}𝔤)^{𝔥})), and define the differential δ_{r} : C^{k} ⟶ C^{k+1} by

δ_{r}τ = ∑_{i} h_{i} ∧ \frac{∂τ}{∂λ^{i}} + [r, τ], ∀τ ∈ C^{k}.  (8)

Proposition 2.11 δ_{r} : C^{k} ⟶ C^{k+1} is well-defined and δ_{r}^{2} = 0.

Proof. It is clear that δ_{r}τ is in C^{∞}(𝔥^{∗}, (∧^{k+1}𝔤)^{H}) provided that τ ∈ C^{∞}(𝔥^{∗}, (∧^{k}𝔤)^{H}). For any τ ∈ C^{k} = C^{∞}(𝔥^{∗}, (∧^{k}𝔤)^{H}), τ can be naturally considered as a section in ∧^{k}A, and

[Λ, τ]
= [∑_{i} h_{i} ∧ \frac{∂}{∂λ^{i}} + r, τ]
= ∑_{i} h_{i} ∧ \frac{∂τ}{∂λ^{i}} + [r, τ]
= δ_{r}τ.

Since [Λ, Λ] = 0, it thus follows that δ_{r}^{2} = 0.

□

Hence the cochain complex δ_{r} : C^{k} ⟶ C^{k+1} defines a cohomology, called the dynamical rmatrix cohomology and denoted by H_{r}^{∗}(𝔤, 𝔥). Two remarks are in order.

Remark (1). The cochain complex δ_{r} : C^{k} ⟶ C^{k+1} is in fact a subcomplex of the Lie algebroid cohomology cochain complex d_{∗} : Γ(∧^{k}A) ⟶ Γ(∧^{k+1}A), d_{∗}X = [Λ, X]. Therefore it is easy to see that such a cochain complex is always defined for an arbitrary dynamical r-matrix, which is not necessary triangular.

(2). When r is triangular, H_{r}^{∗}(𝔤, 𝔥) can be naturally identified with a “special” G × H-invariant Poisson cohomology of the Poisson manifold (M, π), i.e., the Poisson cohomology obtained by restricting to G × H-invariant multi-vector fields which are tangent to the fibers of the fibration 𝔥^{∗} × G ⟶ 𝔥^{∗}.

Proposition 2.12 If g : 𝔥^{∗} ⟶ G^{H} is a smooth map, then

(i). δ_{r_{g}} ∘Ad_{g} = Ad_{g} ∘δ_{r};

(ii). Ad_{g} : (C^{∗}, δ_{r}) ⟶ (C^{∗}, δ_{r_{g}}) induces an isomorphism H_{r}^{∗}(𝔤, 𝔥) ≅ H_{r_{g}}^{∗}(𝔤, 𝔥).

Proof. For any τ ∈ C^{∞}(𝔥^{∗}, (∧^{k}𝔤)^{H}),

(Ad_{g} ∘δ_{r})τ = Ad_{g}[Λ, τ]
= [Ad_{g}Λ, Ad_{g}τ]
= [Λ_{r_{g}}, Ad_{g}τ]
= (δ_{r_{g}} ∘Ad_{g})τ.

The conclusion thus follows immediately.

□

As a consequence, we deduce that H_{r}^{∗}(𝔤, 𝔥) only depends on the gauge equivalence class of the dynamical r-matrix. So sometimes, we will denote this group as H_{[r]}^{∗}(𝔤, 𝔥) in order to emphasize that it only depends on the class of r.

Proposition 2.13 For any triangular dynamical r-matrix r : 𝔥^{∗} ⟶ ∧^{2}𝔤, T_{[r]}M(𝔤, 𝔥) ≅ H_{[r]}^{2}(𝔤, 𝔥).

Proof. In Equation (2), replace r by r + tτ and take the derivative at t = 0, one obtains the linearization equation: ∑_{i} h_{i} ∧ \frac{∂τ}{∂λ^{i}} + [r, τ] = 0, i.e., δ_{r}τ = 0. It is clear that τ is of zero weight since r + tτ is of zero weight.

To find the tangent space to the gauge orbit at r, one needs to compute \frac{d}{dt}|_{t=0}r_{exp tf}, for f ∈ C^{∞}(𝔥^{∗}, 𝔤^{H}). Now r_{exp tf} = Ad_{exp tf}r − ∑_{i} h_{i} ∧ \frac{∂ exp tf}{∂λ^{i}} (exp tf)^{−1}. It is thus simple to see that \frac{d}{dt}|_{t=0}(r_{exp tf}) = [f, r] − ∑_{i} h_{i} ∧ \frac{∂f}{∂λ^{i}} = −δ_{r}f. The conclusion thus follows immediately.

□

Given a Lie algebra 𝔤, one may also consider triangular classical dynamical r-matrices valued in 𝔤[[ℏ]], r_{ℏ} : 𝔥^{∗} ⟶ (∧^{2}𝔤)[[ℏ]] such that r_{ℏ}(λ) = r(λ) + ℏr_{1}(λ) + · · ·. The gauge transformation can be defined formally in an obvious way. Thus one can form the modular space M(𝔤[[ℏ]], 𝔥). Assume that r : 𝔥^{∗} ⟶ ∧^{2}𝔤 is a triangular classical dynamical r-matrix. From Proposition 2.13, it follows that T_{[r]}M(𝔤[[ℏ]], 𝔥) ≅ H_{[r]}^{2}(𝔤, 𝔥)[[ℏ]]. By a formal neighbourhood of r in M(𝔤[[ℏ]], 𝔥), denoted by M_{r}(𝔤[[ℏ]], 𝔥), we mean the subset in M(𝔤[[ℏ]], 𝔥) consisting of the classes of those elements r +O(ℏ). Then H_{[r]}^{2}(𝔤, 𝔥)[[ℏ]] can be considered as a kind of linearization of M_{r}(𝔤[[ℏ]], 𝔥). In general, these two spaces are different. However, when r is a non-degenerate triangular classical dynamical r-matrix, they expect to be isomorphic, which should follow from Moser lemma. Content-Length: 24030 X-Lines: 573 Status: RO

In fact, as we will see in the next theorem, when r is non-degenerate, H_{[r]}^{∗}(𝔤, 𝔥) is isomorphic to the relative Lie algebra cohomology.

Theorem 2.14 If r : 𝔥^{∗} ⟶ ∧^{2}𝔤 is a non-degenerate dynamical r-matrix, then H_{[r]}^{∗}(𝔤, 𝔥) is isomorphic to H^{∗}(𝔤, 𝔥), the relative Lie algebra cohomology of 𝔤 with respect to the Lie subalgebra 𝔥.

Proof. Since r is non-degenerate, (M, π) is a symplectic manifold. As it is well known, then π^{#} : Ω^{∗}(M) ⟶ X^{∗}(M) induces an isomorphism between the de Rham cohomology cochain complex and the Poisson cohomology cochain complex. Now a k-muti-vector field P ∈ X^{k}(M) is in C^{k} iff (i) P is left G-invariant and right H-invariant; and (ii)dλ^{i} P = 0, ∀i = 1, ⋯ l. This, however, is equivalent to that (i) (π^{#})^{−1}P is left G-invariant and right H-invariant; and (ii) \overrightarrow{h_{i}} (π^{#})^{−1}P = 0, since the Poisson tensor π is G × H-invariant and π^{#}(dλ^{i}) = \overrightarrow{h_{i}}, ∀i = 1, ⋯ l. Note that a k-form ω ∈ Ω^{k}(M) is H-invariant and \overrightarrow{h_{i}} ω = 0, ∀i = 1, ⋯, l iff ω is the pull back of a k-form on the quotient space M/H, i.e, ω = p^{∗}ω′, where ω′ ∈ Ω^{k}(M/H). Moreover, ω is left G-invariant iff ω′ is left G-invariant since the left G-action on M commutes with the right H-action. In summary, we have proved that the space (π^{#})^{−1}(C^{k}) can be naturally identified with the space of left G-invariant k-form on M/H ≅ 𝔥^{∗} × G/H. Under such an identification, the differential δ_{r} goes to the de-Rham differential. Hence H_{[r]}^{k}(𝔤, 𝔥) is isomorphic to the invariant de-Rham cohomology H^{k}(𝔥^{∗} × G/H)^{G}. Since G does not act on the first factor 𝔥^{∗}, the latter is isomorphic to H^{k}(G/H)^{G}, which is in turn isomorphic to the relative Lie algebra cohomology H^{k}(𝔤, 𝔥) [10].

□

3 Quantization and star products

This section is devoted to the discussion on the relation between quantization of a triangular dynamical r-matrix and star products on its associated Poisson manifold (M, π). The main theme is to show that quantizing r is equivalent to finding a certain special type of star products on M. Let us first introduce the precise definition of a quantization.

Definition 3.1 Let r : 𝔥^{∗} ⟶ ∧^{2}𝔤 be a triangular dynamical r-matrix. A quantization of r is an element F (λ) = 1 + ℏF_{1}(λ) + O(ℏ^{2}) ∈ C^{∞}(𝔥^{∗}, U𝔤 ⊗ U𝔤)[[ℏ]] satisfying

(i). the zero weight condition: [1⊗h + h⊗1, F (λ)] = 0, ∀h ∈ 𝔥;

(ii). the shifted cocycle condition:

(Δ⊗id)F (λ)F^{12}(λ − \frac{1}{2}ℏh^{(3)}) = (id⊗Δ)F (λ)F^{23}(λ + \frac{1}{2}ℏh^{(1)});  (9)

(iii). the normal condition:

(ϵ⊗id)F (λ) = 1; (id⊗ϵ)F (λ) = 1,  (10)

(iv). the quantization condition: F_{1}(λ) − F_{1}^{21}(λ) = r(λ),

where Δ : U𝔤 ⟶ U𝔤⊗U𝔤 is the standard comultiplication, ϵ : U𝔤 ⟶ ℂ is the counit map, and F^{12}(λ − \frac{1}{2}ℏh^{(3)}), F^{23}(λ + \frac{1}{2}ℏh^{(1)}) are U𝔤⊗U𝔤⊗U𝔤-valued functions on 𝔥^{∗} defined by

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

+ ⋯ + \frac{1}{k!}(−\frac{ℏ}{2})^{k} ∑ \frac{∂^{k}F}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} ⊗h_{i_{1}} ⋯ h_{i_{k}} + ⋯,  (11)

and similarly for F^{23}(λ + \frac{1}{2}ℏh^{(1)}).

The relation between this definition of quantization and the well known quantum dynamical Yang-Baxter equation (QDYBE) is indicated by the following proposition, which can be proved by a straightforward verification.

Proposition 3.2 If F (λ) is a quantization of a triangular dynamical r-matrix r(λ) : 𝔥^{∗} ⟶ ∧^{2}𝔤, then R(λ) = F^{21}(λ)^{−1}F^{12}(λ) can be written as R(λ) = 1 + ℏr(λ) + O(ℏ^{2}) and satisfies the quantum dynamical Yang-Baxter equation (QDYBE):

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)}).

(12)

Remark. This is a symmetrized version of QDYBE, which is known [18] to be equivalent to the non-symmetrized QDYBE:

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

The reason for us to choose to work with the symmetrized QDYBE in the present paper is because it is related to the Weyl quantization, while non-symmetrized QDYBE is related to the normal order quantization, as indicated in [42]. Since we will use Fedosov method in our quantization procedure later on, the Weyl quantization is of an obvious advantage.

To proceed, we need some preparation on notations. Let A = D⊗U𝔤[[ℏ]], where D is the algebra of smooth differential operators on 𝔥^{∗}. Then D⊗U𝔤 can be naturally identified with the algebra of left G-invariant differential operators on M. Hence A becomes a Hopf algebroid [42] with base algebra R = C^{∞}(𝔥^{∗})[[ℏ]]. The comultipicaltion

Δ : A ⟶ A⊗_{R}A ≅ D⊗_{C^{∞}(𝔥^{∗})}D⊗U𝔤⊗U𝔤[[ℏ]]

is the natural extension of the comultiplications on D and on U𝔤:

Δ(D⊗u) = ΔD⊗Δu, ∀D ∈ D, and u ∈ U𝔤,

where ΔD is the bidifferential operator on 𝔥^{∗} given by (ΔD)(f, g) = D(fg), f, g ∈ C^{∞}(𝔥^{∗}) and Δu ∈ U𝔤⊗U𝔤 is the usual comultiplication on U𝔤. Let us fix a basis in 𝔥, say {h_{1}, ⋯, h_{l}}, and let {ξ_{1}, ⋯, ξ_{l}} be its dual basis, which in turn defines a coordinate system (λ^{1}, ⋯, λ^{l}) on 𝔥^{∗}.

Set

θ = ∑_{i=1}^{l} \frac{1}{2}(h_{i}⊗\frac{∂}{∂λ^{i}} − \frac{∂}{∂λ^{i}} ⊗h_{i}) ∈ A⊗A, and Θ = exp ℏθ ∈ A⊗A.  (13)

Note that θ, and hence Θ, is independent of the choice of the basis in 𝔥.

For each D ∈ D⊗U𝔤, we denote by \overrightarrow{D} its corresponding left G-invariant differential operator on M = 𝔥^{∗} × G. We also use a similar notation to denote multi-differential operators on M as well. Now let r(λ) : 𝔥^{∗} ⟶ ∧^{2}𝔤 be a triangular dynamical r-matrix, and M = 𝔥^{∗} × G its associated (regular) Poisson manifold with Poisson tensor π = ∑_{i} \overrightarrow{h_{i}} ∧ \frac{∂}{∂λ^{i}} + \overrightarrow{r(λ)}. It is simple to see that the Poisson brackets on C^{∞}(M) can be described as follows.

(i). for any f, g ∈ C^{∞}(𝔥^{∗}), {f, g} = 0;

(ii). for any f ∈ C^{∞}(𝔥^{∗}) and g ∈ C^{∞}(G), \{f, g\} = − ∑_{i} \frac{∂f}{∂λ^{i}} \overrightarrow{h_{i}}g;

(iii). for any f, g ∈ C^{∞}(G), \{f, g\} = \overrightarrow{r(λ)}(f, g).

This Poisson bracket relation naturally motivates the following theorem, which is indeed the main theorem of this section.

Theorem 3.3 Let (M, π) be the Poisson manifold associated to a triangular dynamical r-matrix as in Proposition 2.2. Assume that ∗_{ℏ} is a G × H-invariant star product on (M, π) satisfying the properties:

(i). for any f, g ∈ C^{∞}(𝔥^{∗}),

f(λ) ∗_{ℏ} g(λ) = f(λ)g(λ);

(ii). for any f(λ) ∈ C^{∞}(𝔥^{∗}) and g(x) ∈ C^{∞}(G),

f(λ) ∗_{ℏ} g(x) = \overrightarrow{Θ}(f, g) = ∑_{k=0}^{∞}(−\frac{ℏ}{2})^{k} \frac{1}{k!} \frac{∂^{k}f}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} \overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}g,

g(x) ∗_{ℏ} f(λ) = \overrightarrow{Θ}(g, f) = ∑_{k=0}^{∞}(\frac{ℏ}{2})^{k} \frac{1}{k!}\overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}g \frac{∂^{k}f}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}};

(iii). there is a smooth map F : 𝔥^{∗} ⟶ U𝔤⊗U𝔤[[ℏ]] such that for any f(x), g(x) ∈ C^{∞}(G),

f ∗_{ℏ} g = \overrightarrow{F (λ)}(f, g).  (14)

Then F (λ) is a quantization of the dynamical r-matrix r(λ). Conversely, any quantization of r(λ) corresponds to a G × H-invariant star product on M satisfying the properties (i)-(iii).

Any G × H-invariant star product on M with properties (i)-(iii) is called a compatible star product. The above theorem can then be rephrased that a quantization of r(λ) is equivalent to a compatible star-product on M.

To prove Theorem 3.3, we need several lemmas.

Lemma 3.4 Θ satisfies

(Δ⊗id)ΘΘ^{12} = (id⊗Δ)ΘΘ^{23} in A⊗A⊗A.  (15)

Proof. Normally, both sides of Equation (15) are only elements in A⊗_{R}A⊗_{R}A. In our situation, however, they can be considered as elements in A⊗A⊗A in an obvious manner.

Now

(Δ⊗id)ΘΘ^{12}
= ((Δ⊗id) exp ℏθ) exp ℏθ^{12}
= exp ℏ((Δ⊗id)θ + θ^{12})
= exp \frac{1}{2}ℏ ∑_{i=1}^{k}(h_{i}⊗1⊗\frac{∂}{∂λ^{i}} + 1⊗h_{i}⊗\frac{∂}{∂λ^{i}} + h_{i}⊗\frac{∂}{∂λ^{i}} ⊗1 − \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.

A similar computation leads to the same expression for (id⊗Δ)ΘΘ^{23}. This proves Equation (15).

□

Lemma 3.5 ∀D_{1}, D_{2}, D_{3} ∈ A, and ∀f_{1}(λ) ∈ C^{∞}(𝔥^{∗}), f_{2}(x) ∈ C^{∞}(G), and g(λ, x) ∈ C^{∞}(𝔥^{∗} × G), \overrightarrow{[(Δ⊗id)F (λ)(D_{1}⊗D_{2}⊗D_{3})]}(f_{1}(λ), f_{2}(x), g(λ, x)) = \overrightarrow{[F^{23}(λ)(D_{1}⊗D_{2}⊗D_{3})]}(f_{1}(λ), f_{2}(x), g(λ, x)). Proof. Write F (λ) = ∑ a_{αβ}(λ)u_{α}⊗u_{β}, with u_{α}, u_{β} ∈ U𝔤 and a_{αβ}(λ) ∈ C^{∞}(𝔥^{∗})[[ℏ]]. Then

((Δ⊗id)F (λ))(D_{1}⊗D_{2}⊗D_{3}) = ∑ a_{αβ}(λ)Δu_{α}(D_{1}⊗D_{2})⊗u_{β}D_{3}.

Hence

\overrightarrow{[(Δ⊗id)F (λ)(D_{1}⊗D_{2}⊗D_{3})]}(f_{1}(λ), f_{2}(x), g(λ, x))
= ∑ a_{αβ}(λ)\overrightarrow{Δu_{α}(D_{1}⊗D_{2})}(f_{1}(λ), f_{2}(x))(\overrightarrow{u_{β}D_{3}}g)(λ, x)
= ∑ a_{αβ}(λ)\overrightarrow{u_{α}}[(\overrightarrow{D_{1}}f_{1})(λ)(\overrightarrow{D_{2}}f_{2})(x)](\overrightarrow{u_{β}D_{3}}g)(λ, x)
= ∑ a_{αβ}(λ)(\overrightarrow{D_{1}}f_{1})(λ)((\overrightarrow{u_{α}}\overrightarrow{D_{2}})f_{2})(x)(\overrightarrow{u_{β}D_{3}}g)(λ, x)
= \overrightarrow{D_{1}⊗F (λ)(D_{2}⊗D_{3})}(f_{1}(λ), f_{2}(x), g(λ, x))
= \overrightarrow{F^{23}(λ)(D_{1}⊗D_{2}⊗D_{3})}(f_{1}(λ), f_{2}(x), g(λ, x)).
□

Corollary 3.6 ∀f_{1}(λ) ∈ C^{∞}(𝔥^{∗}), f_{2}(x) ∈ C^{∞}(G) and g(λ, x) ∈ C^{∞}(𝔥^{∗} × G),

\overrightarrow{F (λ)Θ}(f_{1}(λ) ∗_{ℏ} f_{2}(x), g(λ, x)) = \overrightarrow{Θ}(f_{1}(λ), \overrightarrow{F (λ)Θ}(f_{2}(x), g(λ, x))).

Proof.

\overrightarrow{F (λ)Θ}(f_{1}(λ) ∗_{ℏ} f_{2}(x), g(λ, x))
= \overrightarrow{F (λ)Θ}(\overrightarrow{Θ}(f_{1}(λ), f_{2}(x)), g(λ, x))
= \overrightarrow{(Δ⊗id)(F (λ)Θ)Θ^{12}}(f_{1}(λ), f_{2}(x), g(λ, x))
= \overrightarrow{(Δ⊗id)F (λ)(Δ⊗id)ΘΘ^{12}}(f_{1}(λ), f_{2}(x), g(λ, x)) (by Lemma 3.4)
= \overrightarrow{(Δ⊗id)F (λ)(id⊗Δ)ΘΘ^{23}}(f_{1}(λ), f_{2}(x), g(λ, x)) (by Lemma 3.5)
= \overrightarrow{F^{23}(λ)(id⊗Δ)ΘΘ^{23}}(f_{1}(λ), f_{2}(x), g(λ, x)).

Let us write Θ = ∑ D_{α}⊗D_{β}. Then (id⊗Δ)Θ = ∑ D_{α}⊗ΔD_{β}, and

\overrightarrow{F^{23}(λ)(id⊗Δ)ΘΘ^{23}}(f_{1}(λ), f_{2}(x), g(λ, x))

= ∑ \overrightarrow{[D_{α}⊗F (λ)ΔD_{β}Θ]}(f_{1}(λ), f_{2}(x), g(λ, x))
= ∑(\overrightarrow{D_{α}}f_{1})(λ)\overrightarrow{F (λ)ΔD_{β}Θ}(f_{2}(x), g(λ, x)).

Using the expansion Θ = ∑_{k=0}^{∞}(\frac{ℏ}{2})^{k} \frac{1}{k!}(∑_{i=1}^{l}(h_{i}⊗\frac{∂}{∂λ_{i}} − \frac{∂}{∂λ_{i}} ⊗h_{i}))^{k}, one obtains that

\overrightarrow{F (λ)Θ}(f_{1}(λ) ∗_{ℏ} f_{2}(x), g(λ, x))
= ∑_{k=0}^{∞}(−\frac{ℏ}{2})^{k} \frac{1}{k!} \frac{∂^{k}f_{1}(λ)}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} \overrightarrow{F (λ)Δ(h_{i_{1}} ⋯ h_{i_{k}})Θ}(f_{2}(x), g(λ, x))

= ∑_{k=0}^{∞}(−\frac{ℏ}{2})^{k} \frac{1}{k!} \frac{∂^{k}f_{1}(λ)}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} \overrightarrow{Δ(h_{i_{1}} ⋯ h_{i_{k}})F (λ)Θ}(f_{2}(x), g(λ, x))

= ∑_{k=0}^{∞}(−\frac{ℏ}{2})^{k} \frac{1}{k!} \frac{∂^{k}f_{1}(λ)}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} \overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}[\overrightarrow{F (λ)Θ}(f_{2}(x), g(λ, x))]
= \overrightarrow{Θ}(f_{1}(λ), \overrightarrow{F (λ)Θ}(f_{2}(x), g(λ, x))).

Here the second equality follows from the fact that F (λ) is of zero weight, i.e., F (λ)Δh = (Δh)F (λ), ∀h ∈ 𝔥. This concludes the proof.

□

Proposition 3.7 Under the same hypothesis as in Theorem 3.3, we have (1). for any f(λ) ∈ C^{∞}(𝔥^{∗}) and g(λ, x) ∈ C^{∞}(𝔥^{∗} × G),

f(λ) ∗_{ℏ} g(λ, x) = \overrightarrow{Θ}(f, g) = ∑_{k=0}^{∞}(−\frac{ℏ}{2})^{k} \frac{1}{k!} \frac{∂^{k}f}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} \overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}g,  (16)

g(λ, x) ∗_{ℏ} f(λ) = \overrightarrow{Θ}(g, f) = ∑_{k=0}^{∞}(\frac{ℏ}{2})^{k} \frac{1}{k!}\overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}g \frac{∂^{k}f}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}};  (17)

(2). for any f(λ, x) ∈ C^{∞}(𝔥^{∗} × G) and g(x) ∈ C^{∞}(G),

f(λ, x) ∗_{ℏ} g(x) = (\overrightarrow{F (λ)Θ})(f, g) = ∑_{k=0}^{∞}(−\frac{ℏ}{2})^{k} \frac{1}{k!}\overrightarrow{F (λ)}(\frac{∂^{k}f}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} , \overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}g), (18)

g(x) ∗_{ℏ} f(λ, x) = (\overrightarrow{F (λ)Θ})(g, f) = ∑_{k=0}^{∞}(\frac{ℏ}{2})^{k} \frac{1}{k!}\overrightarrow{F (λ)}(\overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}g, \frac{∂^{k}f}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}}).  (19)

Proof. We will prove Equation (16) first. It suffices to show this equation for g(λ, x) = g_{1}(λ)∗_{ℏ}g_{2}(x), ∀g_{1}(λ) ∈ C^{∞}(𝔥^{∗}) and g_{2}(x) ∈ C^{∞}(G), since, at each point, the C^{∞}-jet space of C^{∞}(𝔥^{∗} × G)[[ℏ]] is spanned by the C^{∞}-jets of this type of functions. Now

f(λ) ∗_{ℏ} g(λ, x) = f(λ) ∗_{ℏ} (g_{1}(λ) ∗_{ℏ} g_{2}(x))
= (f(λ) ∗_{ℏ} g_{1}(λ)) ∗_{ℏ} g_{2}(x)
= (f(λ)g_{1}(λ)) ∗_{ℏ} g_{2}(x)
= \overrightarrow{Θ}(f(λ)g_{1}(λ), g_{2}(x))
= \overrightarrow{Θ}(\overrightarrow{Θ}(f(λ), g_{1}(λ)), g_{2}(x))
= \overrightarrow{(Δ⊗id)ΘΘ^{12}}(f(λ), g_{1}(λ), g_{2}(x)) (by Lemma 3.4)

= \overrightarrow{(id⊗Δ)ΘΘ^{23}}(f(λ), g_{1}(λ), g_{2}(x))
= \overrightarrow{Θ}(f(λ), \overrightarrow{Θ}(g_{1}(λ), g_{2}(x)))
= \overrightarrow{Θ}(f(λ), g_{1}(λ) ∗_{ℏ} g_{2}(x))
= \overrightarrow{Θ}(f(λ), g(λ, x)).

Equation (17) can be proved similarly.

To prove Equation (18), similarly we may assume that f(λ, x) = f_{1}(λ) ∗_{ℏ} f_{2}(x), for f_{1}(λ) ∈ C^{∞}(𝔥^{∗}) and f_{2}(x) ∈ C^{∞}(G). Then

f(λ, x) ∗_{ℏ} g(x) = (f_{1}(λ) ∗_{ℏ} f_{2}(x)) ∗_{ℏ} g(x)
= f_{1}(λ) ∗_{ℏ} (f_{2}(x) ∗_{ℏ} g(x)) (using Equation (16) )
= \overrightarrow{Θ}(f_{1}(λ), f_{2}(x) ∗_{ℏ} g(x))
= \overrightarrow{Θ}(f_{1}(λ), \overrightarrow{F (λ)}(f_{2}(x), g(x)))
= \overrightarrow{Θ}(f_{1}(λ), \overrightarrow{F (λ)Θ}(f_{2}(x), g(x))) (by Corollary 3.6)
= \overrightarrow{F (λ)Θ}(f_{1}(λ) ∗_{ℏ} f_{2}(x), g(x))
= \overrightarrow{F (λ)Θ}(f(λ, x), g(x)).

Equation (19) can also be proved similarly.

□

We are now ready to prove

Theorem 3.8 Under the same hypothesis as in Theorem 3.3, \overrightarrow{F (λ)Θ} is the formal bidifferential operator defining the star product ∗_{ℏ}, i.e., for any f(λ, x), g(λ, x) ∈ C^{∞}(𝔥^{∗} × G),

f(λ, x) ∗_{ℏ} g(λ, x) = \overrightarrow{F (λ)Θ}(f, g).

Proof. We may assume that f(λ, x) = f_{1}(λ) ∗_{ℏ} f_{2}(x), for f_{1}(λ) ∈ C^{∞}(𝔥^{∗}) and f_{2}(x) ∈ C^{∞}(G). Then

f(λ, x) ∗_{ℏ} g(λ, x)
= (f_{1}(λ) ∗_{ℏ} f_{2}(x)) ∗_{ℏ} g(λ, x)
= f_{1}(λ) ∗_{ℏ} (f_{2}(x) ∗_{ℏ} g(λ, x)) (by Proposition 3.7)
= \overrightarrow{Θ}(f_{1}(λ), \overrightarrow{F (λ)Θ}(f_{2}(x), g(λ, x))) (by Corollary 3.6)
= \overrightarrow{F (λ)Θ}(f_{1}(λ) ∗_{ℏ} f_{2}(x), g(λ, x))
= \overrightarrow{F (λ)Θ}(f(λ, x), g(λ, x)).

This concludes the proof.

□

Finally, before proving Theorem 3.3, we need the following result, which reveals the connection between the shifted cocycle condition and the associativity condition of a star-product.

Proposition 3.9 Under the same hypothesis as in Theorem 3.3, then for any f_{1}(x), f_{2}(x), f_{3}(x) ∈ C^{∞}(G),

(i). \overrightarrow{(Δ⊗id)F (λ)F^{12}(λ − \frac{1}{2}ℏh^{(3)})}(f_{1}(x), f_{2}(x), f_{3}(x)) = (f_{1}(x) ∗_{ℏ} f_{2}(x)) ∗_{ℏ} f_{3}(x);

(ii). \overrightarrow{(id⊗Δ)F (λ)F^{23}(λ + \frac{1}{2}ℏh^{(1)})}(f_{1}(x), f_{2}(x), f_{3}(x)) = f_{1}(x) ∗_{ℏ} (f_{2}(x) ∗_{ℏ} f_{3}(x)).

Proof. From Equation (11), it follows that

(Δ⊗id)F (λ)F^{12}(λ − \frac{1}{2}ℏh^{(3)}) = ∑ \frac{1}{k!}(−\frac{ℏ}{2})^{k}[(Δ⊗id)F (λ)]\frac{∂^{k}F}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} ⊗h_{i_{1}} ⋯ h_{i_{k}}.

Hence

\overrightarrow{(Δ⊗id)F (λ)F^{12}(λ − \frac{1}{2}ℏh^{(3)})}(f_{1}(x), f_{2}(x), f_{3}(x))

= ∑ \frac{1}{k!}(−\frac{ℏ}{2})^{k}\overrightarrow{F (λ)}[\overrightarrow{\frac{∂^{k}F (λ)}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}}}(f_{1}(x), f_{2}(x)), (\overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}f_{3})(x)]
= ∑ \frac{1}{k!}(−\frac{ℏ}{2})^{k}\overrightarrow{F (λ)}[\frac{∂^{k}(f_{1} ∗_{ℏ} f_{2})}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} , (\overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}f_{3})(x)] (using Equation (18))
= (f_{1}(x) ∗_{ℏ} f_{2}(x)) ∗_{ℏ} f_{3}(x).

The second identity can be proved similarly.

□

Proof of Theorem 3.3 Since ∗_{ℏ} is invariant under the right H-action, \overrightarrow{F (λ)} is right H-invariant. This implies that F (λ) is Ad_{H}-invariant, and therefore is of zero weight. The normal condition follows from the fact that 1 is the unit of the star algebra, i.e., 1 ∗_{ℏ} f = f ∗_{ℏ} 1 = f. And the shifted cocycle condition follows from the associativity of the star algebra and Proposition 3.9. Finally, let us write F (λ) = 1 + ℏF_{1}(λ) + O(ℏ^{2}). Since ∗_{ℏ} is a star product quantizing π, it follows that \overrightarrow{(F_{1}(λ) − F_{1}^{21}(λ))}(f, g) = \{f, g\} = \overrightarrow{r(λ)}(f, g), ∀f, g ∈ C^{∞}(G). Hence it follows that F_{1}(λ) − F_{1}^{21}(λ) = r(λ).

Conversely, if F (λ) is a quantization of r(λ), according to Theorem 7.5 in [42], \overrightarrow{F (λ)Θ} is indeed an associator and therefore it defines a star product on M = 𝔥^{∗} × G. It is simple to see that this is a quantization of π and satisfies Properties (i)-(iii) in Theorem 3.3.

□

We end this section by the following

Remark Bordemann et. al. has found a nice elementary star-product on ℝ × SU(2) [9] using a quantum analogue of Marsden-Weinstein reduction. It would be interesting to see if this is a compatible star-product.

4 Symplectic connections

From now on, we will confine ourselves mostly to non-degenerate triangular dynamical r-matrices. In this case, the corresponding Poisson manifolds are in fact symplectic, and therefore can be quantized by Fedosov method [20] [21]. As is well known, Fedosov quantization relies on the choice of a symplectic connection. Serving as a preliminary for the study on Fedosov quantization in the next section, this section is devoted to the discussion on symplectic connections. We will start with some general notations and constructions. Content-Length: 42427 X-Lines: 1033 Status: RO

Let ∇ be a torsion-free symplectic connection on a symplectic manifold (M, ω). Define the symplectic curvature [20] by

R(X, Y, Z, W) = ω(X, R(Z, W)Y), ∀X, Y, Z, W ∈ X (M),  (20)

where R(Z, W)Y = ∇_{Z}∇_{W} Y − ∇_{W} ∇_{Z}Y − ∇_{[Z,W]}Y is the usual curvature tensor of ∇.

Proposition 4.1 (i). R(X, Y, Z, W) is skew symmetric with respect to Z and W, and symmetric with respect to X and Y, i.e.,

R(X, Y, Z, W) = −R(X, Y, W, Z), R(X, Y, Z, W) = R(Y, X, Z, W).  (21)

(ii). The following Bianchi’s identity holds:

R(X, Y, Z, W) + R(X, Z, W, Y) + R(X, W, Y, Z) = 0.  (22)

Proof. It is clear by definition that R(X, Y, Z, W) is skew symmetric with respect to Z and W. Now since ∇ is a symplectic connection, then

ω(X, ∇_{Z}∇_{W} Y)
= Z(ω(X, ∇_{W} Y)) − ω(∇_{Z}X, ∇_{W} Y)
= Z(W ω(X, Y)) − Zω(∇_{W} X, Y) − W ω(∇_{Z}X, Y) + ω(∇_{W} ∇_{Z}X, Y).

Similarly,

ω(X, ∇_{W} ∇_{Z}Y) = W (Zω(X, Y)) − W ω(∇_{Z}X, Y) − Zω(∇_{W} X, Y) + ω(∇_{Z}∇_{W} X, Y),

and

ω(X, ∇_{[Z,W]}Y) = [Z, W](ω(X, Y)) − ω(∇_{[Z,W]}X, Y).

Thus

R(X, Y, Z, W) = ω(X, R(Z, W)Y)
= ω(X, ∇_{Z}∇_{W} Y − ∇_{W} ∇_{Z}Y − ∇_{[Z,W]}Y)
= −ω(∇_{Z}∇_{W} X − ∇_{W} ∇_{Z}X − ∇_{[Z,W]}X, Y)
= ω(Y, R(Z, W)X)
= R(Y, X, Z, W).

This concludes the proof of (i). Finally, (ii) follows from the usual Bianchi’s identity for a torsionfree connection.

□

Symplectic connections always exist on any symplectic manifold. In fact, there is a standard procedure to construct a torsion free symplectic connection out of an arbitrary torsion free linear connection [28] [20]. Since this construction is essential to our discussion here, let us recall it briefly below.

Assume that ∇^{0} is a torsion-free linear connection on a symplectic manifold M. Then any linear connection on M can be written as

∇_{X}Y = ∇_{X}^{0}Y + S(X, Y), ∀X, Y ∈ X (M),  (23)

where S is a (2, 1)-tensor on M. Clearly, ∇ is torsion-free iff S is symmetric, i.e., S(X, Y) = S(Y, X), ∀X, Y ∈ X (M). ∇ is symplectic iff ∇_{X}ω = 0. The latter is equivalent to

ω(S(X, Y), Z) − ω(S(X, Z), Y) = (∇_{X}^{0}ω)(Y, Z) ∀X, Y, Z ∈ X (M).  (24)

Lemma 4.2 If ∇^{0} is a torsion-free linear connection, and S a (2, 1)-tensor defined by the equation:

ω(S(X, Y), Z) = \frac{1}{3}[(∇_{X}^{0}ω)(Y, Z) + (∇_{Y}^{0} ω)(X, Z)],  (25)

then ∇_{X}Y = ∇_{X}^{0}Y +S(X, Y) is a torsion-free symplectic connection. Moreover, if M is a symplectic G-space and ∇^{0} is a G-invariant connection, then the connection ∇ is also G-invariant.

Proof. 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}^{0}ω)(Y, Z) + (∇_{Y}^{0} ω)(X, Z)] − \frac{1}{3}[(∇_{X}^{0}ω)(Z, Y) + (∇_{Z}^{0}ω)(X, Y)]
= \frac{1}{3}[(∇_{X}^{0}ω)(Y, Z) + (∇_{Y}^{0} ω)(X, Z) + (∇_{X}^{0}ω)(Y, Z) + (∇_{Z}^{0}ω)(Y, X)]
= (∇_{X}^{0}ω)(Y, Z),

where the last step follows from the identity:

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

This means that ∇ is a torsion-free symplectic connection. The second statement is obvious according to Equation (25).

□

Now we retain to the case that M = 𝔥^{∗} × G, the symplectic manifold associated with a nondegenerate triangular dynamical r-matrix r, which is our main subject of interest in the present paper. The main result is the following Theorem 4.3 Assume that r : 𝔥^{∗} ⟶ ∧^{2}𝔤 is a non-degenerate triangular dynamical r-matrix. Let M = 𝔥^{∗} × G be equipped with the symplectic structure as in Corollary 2.5. Then M admits a G×H-invariant torsion free symplectic connecting ∇ satisfying the condition that ∇_{X} \overrightarrow{h} = 0, ∀X ∈ X (M), h ∈ 𝔥.

We need a couple of lemmas.

Lemma 4.4 Assume that 𝔤 admits a reductive decomposition 𝔤 = 𝔥 ⊕ 𝔪, i.e., [𝔥, 𝔪] ⊂ 𝔪. Then, the following equations define a biinvariant torsion free linear connection on M with respect to both the left and the right G-actions on M.

\begin{matrix}∇_{X}^{0} \frac{∂}{∂λ^{i}} = 0, & ∇_{X}^{0} \overrightarrow{h} = 0, & ∇_{X}^{0} \overrightarrow{e} = 0 \\ ∇\overrightarrow{h}^{0} \frac{∂}{∂λ^{i}} = 0, & ∇\overrightarrow{h_{1}}^{0}\overrightarrow{h_{2}} = 0, & ∇\overrightarrow{h}^{0}\overrightarrow{e} = \overrightarrow{[h, e]} \\ ∇\overrightarrow{e}^{0} \frac{∂}{∂λ^{i}} = 0, & ∇\overrightarrow{e}^{0}\overrightarrow{h} = 0, & ∇\overrightarrow{e_{1}}^{0}\overrightarrow{e_{2}} = \frac{1}{2} \overrightarrow{[e_{1}, e_{2}]}, \end{matrix}  (26)

where X ∈ X (𝔥^{∗}), h ∈ 𝔥 and e, e_{1}, e_{2} ∈ 𝔪.

Proof. This follows from a straightforward verification.

□

Lemma 4.5 Given a Lie algebra 𝔤, if there exists a non-degenerate triangular dynamical r-matrix r : 𝔥^{∗} ⟶ ∧^{2}𝔤, then 𝔤 admits a reductive decomposition 𝔤 = 𝔥 ⊕ 𝔪 so that [𝔥, 𝔪] ⊂ 𝔪.

Proof. Fixing any λ ∈ 𝔥^{∗}, we take 𝔪 = r(λ)^{#}𝔥^{⊥}. Since r(λ) is non-degenerate, by definition, we have 𝔤 = 𝔥 + 𝔪. On the other hand, it is clear that dim𝔪 ≤ dim𝔥^{⊥} = dim𝔤 − dim𝔥. Hence, dim𝔥 + dim𝔪 ≤ dim𝔤. Therefore 𝔤 = 𝔥 + 𝔪 must be a direct sum. For any h ∈ 𝔥 and ξ ∈ 𝔤^{∗}, since r(λ) is of zero weight, [h, r(λ)^{#}ξ] = r(λ)^{#}ad_{h}^{∗}ξ. Since ad_{h}^{∗}ξ ∈ 𝔥^{⊥} for any ξ ∈ 𝔤^{∗}, it follows that 𝔪 = r(λ)^{#}𝔥^{⊥} is stable under the adjoint action of 𝔥.

□

Remark. Note that, in our proof above, the decomposition 𝔤 = 𝔥 ⊕ 𝔪 depends on the choice of a particular point λ ∈ 𝔥^{∗}, because it is not clear if 𝔪 = r(λ)^{#}𝔥^{⊥} is independent of λ.

Proof of Theorem 4.3 According to Lemma 4.5, we may find a reductive decomposition 𝔤 = 𝔥⊕𝔪 such that [𝔥, 𝔪] ⊂ 𝔪. Let ∇^{0} be the G-biinvariant torsion free connection on M as given by Lemma 4.4. Using Lemma 4.2, we obtain a torsion free symplectic connection ∇ on M. Since the symplectic structure is G × H-invariant, the resulting symplectic connection ∇ is G × H-invariant. It remains to show that ∇ still satisfies the condition that ∇_{X} \overrightarrow{h} = 0, ∀X ∈ X (M) and h ∈ 𝔥. The latter is equivalent to that S(\overrightarrow{h}, X) = 0. To show this identity, first note that ∀X ∈ X (M), ∇\overrightarrow{h}^{0}X = L\overrightarrow{h}X, since ∇^{0} is torsion free and ∇_{X}^{0} \overrightarrow{h} = 0. Hence ∇\overrightarrow{h}^{0}ω = L\overrightarrow{h}ω. However, L\overrightarrow{h}ω = 0

since ω is invariant under the right H-action. It follows that ∇\overrightarrow{h}^{0}ω = 0. According to Equation

(25), ∀Y ∈ X (M), ω(S(\overrightarrow{h}, X), Y) = \frac{1}{3}[(∇\overrightarrow{h}^{0}ω)(X, Y) + (∇_{X}^{0}ω)(\overrightarrow{h}, Y)] = \frac{1}{3}(∇_{X}^{0}ω)(\overrightarrow{h}, Y). This

implies that ω^{b}(S(\overrightarrow{h}, X)) = \frac{1}{3}(\overrightarrow{h} ∇_{X}^{0}ω) = \frac{1}{3}∇_{X}^{0}(\overrightarrow{h} ω) since ∇_{X}^{0} \overrightarrow{h} = 0. Finally, for any i, \overrightarrow{h_{i}} ω = dλ^{i} and from the table in Lemma 4.4, it is easy to check that ∇_{X}^{0}(dλ^{i}) = 0, ∀i = 1, ⋯, l. It thus follows that S(\overrightarrow{h_{i}}, X) = 0, ∀i = 1, ⋯, l. This concludes the proof.

□

In the case that r(λ) ∈ ∧^{2}𝔪, the symplectic connection can be described more explicitly.

Proposition 4.6 Suppose that 𝔤 = 𝔥 ⊕ 𝔪 is a reductive decomposition, {h_{1}, ⋯, h_{l}} is a basis of 𝔥, and {e_{1}, ⋯, e_{m}} is a basis of 𝔪. Suppose that r(λ) = ∑_{ij} r^{ij}(λ)e_{i} ∧ e_{j} is a non-degenerate triangular dynamical r-matrix. Then the symplectic connection on M obtained from ∇^{0}, using the standard construction, as in Lemma 4.4, has the following form:

\begin{matrix}∇_{\frac{∂}{∂λ^{i}}} \frac{∂}{∂λ^{j}} = 0, & ∇_{\frac{∂}{∂λ^{i}}} \overrightarrow{h_{j}} = 0, & ∇_{\frac{∂}{∂λ^{i}}} \overrightarrow{e_{j}} = ∑_{k} d_{ij}^{k}(λ)\overrightarrow{e_{k}} \\ ∇\overrightarrow{h_{i}} \frac{∂}{∂λ^{j}} = 0, & ∇\overrightarrow{h_{i}}\overrightarrow{h_{j}} = 0, & ∇\overrightarrow{h_{i}}\overrightarrow{e_{j}} = \overrightarrow{[h_{i}, e_{j}]} \\ ∇\overrightarrow{e_{i}} \frac{∂}{∂λ^{j}} = ∑_{k} d_{ij}^{k}(λ)\overrightarrow{e_{k}}, & ∇\overrightarrow{e_{i}}\overrightarrow{h_{j}} = 0, & ∇\overrightarrow{e_{i}}\overrightarrow{e_{j}} = \frac{1}{2} \overrightarrow{[e_{i}, e_{j}]} + ∑_{k} f_{ij}^{k}(λ)\overrightarrow{e_{k}}, \end{matrix}  (27)

where d_{ij}^{k}(λ) and f_{ij}^{k}(λ) are smooth functions on 𝔥^{∗}.

Proof. The proof is essentially a straightforward computation. We omit it here.

□

Corollary 4.7 Under the same hypothesis as in Proposition 4.6, if {h_{∗}^{1}, ⋯, h_{∗}^{l}, e_{∗}^{1}, ⋯, e_{∗}^{m}} denotes the dual basis of {h_{1}, ⋯, h_{l}, e_{1}, ⋯, e_{m}}, then

\begin{matrix}∇_{\frac{∂}{∂λ^{i}}} dλ^{j} = 0, & ∇_{\frac{∂}{∂λ^{i}}} \overrightarrow{h_{∗}^{j}} = 0, & ∇_{\frac{∂}{∂λ^{i}}} \overrightarrow{e_{∗}^{j}} = ∑_{k} d_{ik}^{j}(λ)\overrightarrow{e_{∗}^{k}} \\ ∇\overrightarrow{h_{i}}dλ^{j} = 0, & ∇\overrightarrow{h_{i}}\overrightarrow{h_{∗}^{j}} = 0, & ∇\overrightarrow{h_{i}}\overrightarrow{e_{∗}^{j}} = \overrightarrow{ad_{h_{i}}^{∗}e_{∗}^{j}} \\ ∇\overrightarrow{e_{i}}dλ^{j} = 0, & ∇\overrightarrow{e_{i}}\overrightarrow{h_{∗}^{j}} = −\frac{1}{2} ∑_{k} a_{ik}^{j}\overrightarrow{e_{∗}^{k}}, & ∇\overrightarrow{e_{i}}\overrightarrow{e_{∗}^{j}} = − ∑_{k} d_{ik}^{j}(λ)dλ^{k} − (\frac{1}{2}a_{ik}^{j} + f_{ik}^{j}(λ))\overrightarrow{e_{∗}^{k}}, \end{matrix} (28)

where the coadjoint action is defined by < ad_{u}^{∗}ξ, v >= − < ξ, [u, v] >, ∀u, v ∈ 𝔤 and ξ ∈ 𝔤^{∗}, and the constants a_{ij}^{k} are defined by the equation [e_{i}, e_{j}] = ∑_{k} a_{ij}^{k}h_{k} (mod 𝔪).

We end this section by generalizing Theorem 4.3 to the case of splittable triangular dynamical r-matrix. According to Corollary 2.9, one may reduce a splittable triangular dynamical r-matrix to a non-degenerate one by considering a Lie subalgebra 𝔤_{1} ⊂ 𝔤. Thus immediately we obtain the following

Corollary 4.8 Assume that r : 𝔥^{∗} ⟶ ∧^{2}𝔤 is a splittable triangular dynamical r-matrix. Let M = 𝔥^{∗} × G be its associated Poisson manifold as in Proposition 2.2, which admits a (regular) symplectic foliation. Then there exists a G × H-invariant torsion free leafwise Poisson connecting ∇ satisfying the condition that ∇_{X} \overrightarrow{h} = 0, ∀h ∈ 𝔥, and X ∈ X (M) a vector field tangent to the symplectic foliation.

However, when a triangular dynamical r-matrix r is not splittable, such a Poisson connection may not exist. We give a counterexample below.

Example 4.9. Consider a two dimensional Lie algebra 𝔤 with basis {h, e} satisfying the bracket relation [h, e] = ah, where a is a fixed constant. Let 𝔥 = ℝh and r(λ) = f(λ)h ∧ e, where f(λ) is a smooth function. It is simple to see that r(λ) is a triangular dynamical r-matrix of rank zero, and is not splittable unless a = 0. Nevertheless, r(λ) defines a regular rank 2 Poisson structure on the three dimensional space M = ℝ × G with the Poisson tensor π = \overrightarrow{h} ∧ \frac{d}{dλ} + f(λ)\overrightarrow{h} ∧ \overrightarrow{e}, where G is a 2-dimensional Lie group integrating the Lie algebra 𝔤. It is simple to see that the symplectic foliation is spanned by the vector fields \overrightarrow{h} and \frac{d}{dλ} + f(λ)\overrightarrow{e}. Let us denote X = \frac{d}{dλ} + f(λ)\overrightarrow{e}. Then, obviously we have [\overrightarrow{h}, X] = af(λ)\overrightarrow{h}. Now suppose that ∇ is a G × H-invariant torsion free leafwise Poisson connection on M satisfying the condition that ∇\overrightarrow{h}\overrightarrow{h} = 0 and ∇_{X} \overrightarrow{h} = 0.

Since ∇ is torsion free, it thus follows that ∇\overrightarrow{h}X = [\overrightarrow{h}, X] = af(λ)\overrightarrow{h}. Now assume that

∇_{X}X = b(λ, x)\overrightarrow{h} + c(λ, x)X, where b(λ, x) and c(λ, x) are smooth functions on M. Then, ∇_{X}π = ∇_{X}(\overrightarrow{h} ∧ X) = \overrightarrow{h} ∧ ∇_{X}X = c(λ, x)\overrightarrow{h} ∧ X. Since ∇ is a Poisson connection, it follows that c(λ, x) = 0. Finally, we still need to check that ∇ is G × H-invariant. It is clear that ∇ is G-invariant iff the function b(λ, x) is independent of x ∈ G. For it to be invariant under the right H-action, one needs the following condition:

∇_{[\overrightarrow{h},X]}X + ∇_{X}[\overrightarrow{h}, X] = [\overrightarrow{h}, ∇_{X}X] = [\overrightarrow{h}, b(λ)\overrightarrow{h}] = 0.

It thus follows that ∇_{af(λ)\overrightarrow{h}}X + ∇_{X}(af(λ)\overrightarrow{h}) = 0, which implies that f^{2}(λ)a^{2}\overrightarrow{h} + a(−\frac{df}{dλ})\overrightarrow{h} = 0.

Therefore, we arrive at the following equation (under the assumption that a ≠ 0)

\frac{df}{dλ} = af^{2}(λ).  (29)

In conclusion, we have proved that such a connection does not exist unless f(λ) is a solution of the above equation. It would be interesting to find out what is the geometric meaning of this equation.

This indicates that the quantization method we are using in the present paper fails in this particular example. On the other hand, since M is a very simple Poisson manifold, it would be interesting to see directly if there exists a compatible ∗-product on M. In fact, since 𝔤 is a 2dimensional Lie algebra and the premier obstruction to a quantization problem normally lies in some third cohomology, we believe that this dynamical r-matrix is quantizable for any f(λ).

5 Compatible Fedosov star products

In this section, we consider Fedosov star products on a symplectic manifold M admitting an (right) action of an Abelian group H. For the reader’s convenience, we will give a brief account of the general construction of Fedosov star products in Appendix. Readers may refer to that section for various notations and formulas that will be used here. What is eventually reveralent to our situation is the case that M = 𝔥^{∗} × G, the symplectic manifold corresponding to a nondegenerate dynamical r-matrix. However, we believe that our presentation in a general framework would be of its own interest. The main result of this section is the following Theorem 5.1 Suppose that M is a symplectic manifold, on which an Abelian group H acts freely in a Hamiltonian way with a momentum map J : M ⟶ 𝔥^{∗}, which is assumed to be a submersion. Assume that there exists a H-invariant symplectic connection ∇, under which \overrightarrow{h} is parallel for any h ∈ 𝔥, i.e., ∇_{X} \overrightarrow{h} = 0, ∀X ∈ X (X). Let ∗_{ℏ} be the corresponding Fedosov star product on M with Weyl curvature Ω = ω + ℏω_{1} + ⋯ + ℏ^{i}ω_{i} + ⋯ ∈ Z^{2}(M)[[ℏ]], which satisfies the condition that i\overrightarrow{h}ω_{i} = 0, (i = 1, ⋯), ∀h ∈ 𝔥. Then for any f(λ) ∈ C^{∞}(𝔥^{∗}) and g(x) ∈ C^{∞}(M),

(J^{∗}f) ∗_{ℏ} g(x) = = ∑_{k=0}^{∞}(−\frac{ℏ}{2})^{k} \frac{1}{k!}J^{∗}(\frac{∂^{k}f}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}})\overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}g;

g(x) ∗_{ℏ} (J^{∗}f) = ∑_{k=0}^{∞}(\frac{ℏ}{2})^{k} \frac{1}{k!}\overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}gJ^{∗}(\frac{∂^{k}f}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}}).

Here, for any h ∈ 𝔥, \overrightarrow{h} denotes its corresponding Hamiltonian vector field on M.

Remark. It follows from Theorem 5.1 that J^{∗} : C^{∞}(𝔥^{∗})[[ℏ]] ⟶ C^{∞}(M)[[ℏ]] is an algebra homomorphism, where C^{∞}(𝔥^{∗})[[ℏ]] is equipped with pointwise multiplication. In other words, J^{∗} is a quantum momentum map [38]. It would be interesting to see how to generalize this result to the case that H is not Abelian.

Applying Theorem 5.1 to the symplectic manifold M = 𝔥^{∗} × G associated to a nondegenerate triangular dynamical r-matrix, and using Theorem 4.3, we obtain the following

Corollary 5.2 Let r : 𝔥^{∗} ⟶ ∧^{2}𝔤 be a nondegenerate triangular dynamical r-matrix, and M = 𝔥^{∗} × G its associated symplectic manifold. Let ∇ be the symplectic connection on M as in Theorem 4.3. Suppose that Ω = ω + ℏω_{1} + ⋯ + ℏ^{i}ω_{i} + ⋯ ∈ Z^{2}(M)[[ℏ]] satisfies the condition that i\overrightarrow{h}ω_{i} =

0, (i = 1, ⋯), ∀h ∈ 𝔥. Then the Fedosov star product on M corresponding to (∇, Ω) is a compatible star product.

Using Theorem 3.3, we are lead to the following main result of the paper.

Theorem 5.3 Any non-degenerate triangular dynamical r-matrix is quantizable.

More generally, if r is a splittable triangular dynamical r-matrix, the corresponding Poisson manifold M = 𝔥^{∗} × G admits a G × H-invariant leafwise (w.r.t. the symplectic foliation) Poisson connection such that ∇_{X} \overrightarrow{h} = 0, ∀h ∈ 𝔥 according to Corollary 4.8. Applying 5.1 leafwisely, we obtain the following

Theorem 5.4 Any splittable triangular dynamical r-matrix is quantizable.

The rest of the section is devoted to the proof of Theorem 5.1. We will start with the following Proposition 5.5 Under the same hypothesis as in Theorem 5.1, we have

(i). For any (r, s)-type tensor S ∈ T^{(r,s)}M, and any h ∈ 𝔥, ∇\overrightarrow{h}S = L\overrightarrow{h}S.

(ii). For any i = 1, ⋯, l, ∇_{X}(J^{∗}dλ^{i}) = 0.

(iii). For θ ∈ Ω^{1}(M), if \overrightarrow{h} θ = 0, then \overrightarrow{h} ∇_{X}θ = 0, ∀X ∈ X (M).

(iv). R(X, Y, Z, W) = 0, as long as any one of the vectors X, Y, Z, W is tangent to the H-orbits.

(v). ∇\overrightarrow{h}R = 0, for any h ∈ 𝔥.

Proof. (i). Since ∇ is torsion-free, for any vector field X ∈ X (M), we have ∇\overrightarrow{h}X = ∇_{X} \overrightarrow{h} +

[\overrightarrow{h}, X] = [\overrightarrow{h}, X] = L\overrightarrow{h}X. This implies that ∇\overrightarrow{h}θ = L\overrightarrow{h}θ for any one form θ ∈ Ω^{1}(M). Therefore

∇\overrightarrow{h}S = L\overrightarrow{h}S for any (r, s)-type tensor S ∈ T^{(r,s)}M.

(ii). Since J : M ⟶ 𝔥^{∗} is a momentum map, it follows that J^{∗}dλ^{i} = ω^{b}\overrightarrow{h_{i}}, where ω^{b} : X (M) ⟶ Ω^{1}(M) is the isomorphism induced by the symplectic structure ω. Hence ∇_{X}(J^{∗}dλ^{i}) = ∇_{X}(ω^{b}\overrightarrow{h_{i}}) = ω^{b}(∇_{X} \overrightarrow{h_{i}}) = 0, since ∇ is a symplectic connection.

(iii). ∇_{X}(\overrightarrow{h} θ) = (∇_{X} \overrightarrow{h}) θ + \overrightarrow{h} ∇_{X}θ = \overrightarrow{h} ∇_{X}θ. If \overrightarrow{h} θ is zero, so is \overrightarrow{h} ∇_{X}θ.

(iv). Let Φ denote the H-action on M. For any h ∈ 𝔥, since ∇ is H-invariant, it follows that ∀W, Y ∈ X (M), ∇_{Φ_{exp th∗}W} Φ_{exp th∗}Y = Φ_{exp th∗}(∇_{W} Y). Taking the derivative at t = 0, one obtains that

∇_{[\overrightarrow{h},W]}Y + ∇_{W} [\overrightarrow{h}, Y] = [\overrightarrow{h}, ∇_{W} Y].

Hence,

R(\overrightarrow{h}, W)Y = ∇\overrightarrow{h}∇_{W} Y − ∇_{W} ∇\overrightarrow{h}Y − ∇_{[\overrightarrow{h},W]}Y
= [\overrightarrow{h}, ∇_{W} Y] − ∇_{W} [\overrightarrow{h}, Y] − ∇_{[\overrightarrow{h},W]}Y
= 0.

On the other hand, we have R(Z, W)\overrightarrow{h} = 0, since \overrightarrow{h} is parallel by assumption. This means that R(X, Y, Z, W) = 0 if Y = \overrightarrow{h} or Z = \overrightarrow{h}. Since R(X, Y, Z, W) is antisymmetric with respect to W, Z, and symmetric with respect to X, Y according to Proposition 4.1, the conclusion thus follows.

(v). Since both the connection ∇ and the symplectic structure ω are H-invariant, the symplectic curvature R, as defined by Equation (20), is also H-invariant. Hence, for any h ∈ 𝔥, according to (i), ∇\overrightarrow{h}R = L\overrightarrow{h}R = 0.

This concludes the proof of the proposition.

□

By K ⊂ T M, we denote the integrable distribution on M corresponding to the H-orbits, and K^{⊥} its conormal subbundle. That is, a covector θ is in K^{⊥} iff < θ, \overrightarrow{h} >= 0, ∀h ∈ 𝔥. For any x ∈ M, by pol(K_{x}^{⊥}), we denote the polynomials on T_{x}M generated by covectors in K_{x}^{⊥} being considered as linear functions on T_{x}M. By W_{x}^{⊥}, we denote the formal power series in ℏ with coefficients in pol(K_{x}^{⊥}). Clearly W_{x}^{⊥} is a subalgebra of the Weyl algebra W_{x}. Let W^{⊥} = ∪_{x∈M}W_{x}^{⊥} be the subbundle of W. We may also consider W^{⊥}⊗ ∧^{q} K^{⊥}, a subbundle of W ⊗ ∧^{q} T^{∗}M, whose space of sections is denoted by ΓW^{⊥}⊗(Λ^{⊥})^{q}. As before, let us fix a basis {h_{1}, ⋯, h_{l}} of 𝔥, and denote by (λ^{1}, ⋯, λ^{l}) its induced coordinate system on 𝔥^{∗}. Since J : M ⟶ 𝔥^{∗} is a momentum map, X_{J^{∗}λ^{i}} = \overrightarrow{h_{i}}, ∀i = 1, ⋯, l. It thus follows that J_{∗}\overrightarrow{h_{i}} = J_{∗}X_{J^{∗}λ^{i}} = 0, ∀i = 1, ⋯, l, since 𝔥 is Abelian. Now we extend \{\overrightarrow{h_{1}}, ⋯, \overrightarrow{h_{l}}\} to a local basis of the tangent bundle T M around a fixed point x ∈ M. For this purpose, we will first choose local vector fields {u_{1}, ⋯, u_{m}}, which are all tangent to the J-fibers such that \{\overrightarrow{h_{1}}, ⋯, \overrightarrow{h_{l}}, u_{1}, ⋯, u_{m}\} is a basis of the tangent space of the J-fibers at each point of the manifold. Then we choose local vector fields {v_{1}, ⋯, v_{l}} such that J_{∗}v_{i} = \frac{∂}{∂λ^{i}} , ∀i = 1, ⋯, l, which is always possible since J is a submersion. It is easy to see that \{\overrightarrow{h_{1}}, ⋯, \overrightarrow{h_{l}}, v_{1}, ⋯, v_{l}, u_{1}, ⋯, u_{m}\} constitutes a local basis of the tangent bundle T M. Assume that \{\overrightarrow{h_{∗}^{1}}, ⋯, \overrightarrow{h_{∗}^{l}}, v_{∗}^{1}, ⋯ v_{∗}^{l}, u_{∗}^{1}, ⋯, u_{∗}^{m}\} is its dual basis. Then any section of W ⊗Λ can be written as

a = ∑ ℏ^{k}a_{k,i_{1}···i_{p},j_{1}···j_{q}}y_{∗}^{i_{1}} ⋯ y_{∗}^{i_{p}}x_{∗}^{j_{1}} ∧ ⋯ ∧ x_{∗}^{j_{q}},  (30)

where all y_{∗}^{i}’s and x_{∗}^{i}’s are any of \overrightarrow{h_{∗}^{i}}, v_{∗}^{i} or u_{∗}^{i}, and the coefficient a_{k,i_{1}···i_{p},j_{1}···j_{q}} is a covariant tensor symmetric with respect to i_{1} ⋯ i_{p} and antisymmetric in j_{1} ⋯ j_{q}. Then a ∈ ΓW^{⊥}⊗(Λ^{⊥})^{q} iff there are no terms involving explicit h_{∗}^{i}’s in the above expression.

Lemma 5.6 (i). For any i = 1, ⋯, l, J^{∗}dλ^{i} = v_{∗}^{i};

(ii). for any i, j, ∇\overrightarrow{h_{i}}v_{∗}^{j} = 0, and ∇\overrightarrow{h_{i}}h_{∗}^{j} and ∇\overrightarrow{h_{i}}u_{∗}^{j} belong to ΓK^{⊥};

(iii). for any i, j, π(v_{∗}^{i}, h_{∗}^{j}) = δ_{ij}, π(v_{∗}^{i}, v_{∗}^{j}) = 0, π(v_{∗}^{i}, u_{∗}^{j}) = 0;

(iv). the ∗-commutatant of {v_{∗}^{1}, ⋯, v_{∗}^{l}} in ΓW is ΓW^{⊥}.

Proof. (i) < J^{∗}dλ^{i}, v_{j} >=< dλ^{i}, J_{∗}v_{j} >=< dλ^{j}, \frac{∂}{∂λ^{i}} >= δ_{ij}. Similarly, we have < J^{∗}dλ^{i}, u_{j} >= 0 and < J^{∗}dλ^{i}, h_{j} >= 0. Therefore, J^{∗}dλ^{i} = v_{∗}^{i}.

(ii) According to Proposition 5.5, ∇\overrightarrow{h_{i}}v_{∗}^{j} = ∇\overrightarrow{h_{i}}(J^{∗}dλ^{j}) = 0. Also, ∀k, < ∇\overrightarrow{h_{i}}h_{∗}^{j}, \overrightarrow{h_{k}} >= ∇\overrightarrow{h_{i}} <

h_{∗}^{j}, \overrightarrow{h_{k}} > − < h_{∗}^{j}, ∇\overrightarrow{h_{i}}\overrightarrow{h_{k}} >= 0. Hence it follows that ∇\overrightarrow{h_{i}}h_{∗}^{j} ∈ ΓK^{⊥}. Similarly, we can prove that

∇\overrightarrow{h_{i}}u_{∗}^{j} ∈ ΓK^{⊥}.

(iii) π(v_{∗}^{i}, h_{∗}^{j}) =< π^{#}(J^{∗}dλ^{i}), h_{∗}^{j} >=< \overrightarrow{h_{i}}, h_{∗}^{j} >= δ_{ij}. Similarly, we have π(v_{∗}^{i}, v_{∗}^{j}) = 0 and π(v_{∗}^{i}, u_{∗}^{j}) = 0.

(iv) Assume that a ∈ ΓW such that [a, v_{∗}^{i}] = 0, ∀i = 1, ⋯, l. It thus follows that {a, v_{∗}^{i}} = 0, where the Poisson bracket refers to the one corresponding to the fiberwise symplectic structure on T M. Thus a ∈ ΓW^{⊥} according to (iii).

□

Lemma 5.7 (i). ΓW^{⊥}⊗Λ^{⊥} is closed under the multiplication ∘ as defined by Equation (45).

(ii). ΓW^{⊥}⊗Λ^{⊥} is closed under both operators δ and δ^{−1}.

(iii). ΓW^{⊥}⊗Λ^{⊥} is invariant under the covariant derivative ∇_{X}, ∀X ∈ X (M).

Proof. (i) and (ii) are obvious. For (iii), note that Γ(K^{⊥}) is invariant under the covariant derivative ∇_{X} according to Proposition 5.5 (iii). Hence ΓW^{⊥}⊗Λ^{⊥} is also invariant.

□

As an immediate consequense, we have the following

Corollary 5.8 If a ∈ ΓW^{⊥}⊗Λ^{⊥} and ∇\overrightarrow{h}a = 0, ∀h ∈ 𝔥, then ∂a ∈ ΓW^{⊥}⊗Λ^{⊥}.

To prove Theorem 5.1, we start with the following

Lemma 5.9 Under the same hypthesis as in Theorem 5.1, we have γ_{0} = δ^{−1}Ω̃ ∈ ΓW^{⊥}⊗Λ^{⊥} and ∇\overrightarrow{h}γ_{0} = 0, ∀h ∈ 𝔥.

Proof. According to Equation (56), we know that Ω̃ = Ω − ω + R = R + ℏω_{1} + ℏ^{2}ω_{2} + · · ·. By assumption, we have ω_{i} ∈ ΓW^{⊥}⊗Λ^{⊥}, ∀i = 1, · · ·. On the other hand, according to Proposition 5.5 (iv), we know that R ∈ ΓW^{⊥}⊗Λ^{⊥}. Therefore, Ω̃ ∈ ΓW^{⊥}⊗Λ^{⊥}. Hence γ_{0} ∈ ΓW^{⊥}⊗Λ^{⊥} by Lemma 5.7.

Finally, note that for any h ∈ 𝔥, L\overrightarrow{h}ω_{i} = i\overrightarrow{h}(dω_{i}) + d(i\overrightarrow{h}ω_{i}) = d(i\overrightarrow{h}ω_{i}) = 0. According

to Proposition 5.5, we obtain that L\overrightarrow{h}R = ∇\overrightarrow{h}R = 0. Hence L\overrightarrow{h}Ω̃ = 0. It thus follows that

∇\overrightarrow{h}γ_{0} = L\overrightarrow{h}γ_{0} = L\overrightarrow{h}δ^{−1}Ω̃ = δ^{−1}L\overrightarrow{h}Ω̃ = 0.

□

Proposition 5.10 Under the same hypthesis as in Theorem 5.1, the element γ, defined as in Theorem 7.2, belongs to ΓW^{⊥}⊗Λ^{⊥} and satisfies ∇\overrightarrow{h}γ = 0, ∀h ∈ 𝔥.

Proof. We prove it by induction. Assume that γ_{n} satisfies γ_{n} ∈ ΓW^{⊥}⊗Λ^{⊥} and ∇\overrightarrow{h}γ_{n} = 0, ∀h ∈ 𝔥.

It suffices to show that γ_{n+1} satisfies the same conditions. By Equation (58), γ_{n+1} and γ_{n} are related by the following equation:

γ_{n+1} = γ_{0} + δ^{−1}(∂γ_{n} + \frac{i}{ℏ}γ_{n}^{2}), ∀n ≥ 0.  (31)

According to Corollary 5.8, we have ∂γ_{n} ∈ ΓW^{⊥}⊗Λ^{⊥}. On the other hand, by Lemma 5.7, γ_{n}^{2} ∈ ΓW^{⊥}⊗Λ^{⊥}. Hence γ_{n+1} ∈ ΓW^{⊥}⊗Λ^{⊥} according to Lemma 5.7 and Lemma 5.9.

Now

∇\overrightarrow{h}(γ_{n+1})
= ∇\overrightarrow{h}γ_{0} + ∇\overrightarrow{h}δ^{−1}(∂γ_{n} + \frac{i}{ℏ}γ_{n}^{2})
= L\overrightarrow{h}δ^{−1}(∂γ_{n} + \frac{i}{ℏ}γ_{n}^{2})
= δ^{−1}(L\overrightarrow{h}∂γ_{n} + \frac{i}{ℏ}L\overrightarrow{h}γ_{n}^{2})
= 0.

Here, in the last step, we used the relation L\overrightarrow{h}∂ = ∂L\overrightarrow{h}, which follows from the fact that the symplectic connection is H-invariant. This concludes the proof.

□

As in Appendix, for any a ∈ C^{∞}(M), we denote by ã ∈ W_{D} its parallel lift, i.e., Dã = 0 and ã|_{y=0} = a. Theorem 5.1 is in fact an immediate consequence of the following

Proposition 5.11 Under the same hypothesis as in Theorem 5.1,

(i). If a = J^{∗}f with f ∈ C^{∞}(𝔥^{∗}), then

ã = ∑_{k=0}^{∞} \frac{1}{k!}J^{∗}(\frac{∂^{k}f}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}})v_{∗}^{i_{1}} ⋯ v_{∗}^{i_{k}};  (32)

(ii). For any a ∈ C^{∞}(M),

ã = ∑_{k=0}^{∞} \frac{1}{k!}(\overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}a)h_{∗}^{i_{1}} ⋯ h_{∗}^{i_{k}} + T,

where the reminder T does not contain any terms which are pure polynomials of h_{∗}^{i}’s.

Proof. For (i), it suffices to prove that ã given by Equation (32) is a parallel section. According to Proposition 5.10 and Lemma 5.6, we have [γ, ã] = 0. Thus it follows that Dã = −δã + ∂ã, which clearly vanishes since ∂v_{∗}^{i} = 0 by Proposition 5.5 (ii) and Lemma 5.6 (ii).

For (ii), recall that ã is given by the iteration formula

a_{n+1} = a + δ^{−1}(∂a_{n} + [\frac{i}{ℏ}γ, a_{n}]).  (33)

So it suffices to prove that

a_{n} = ∑_{k=0}^{n} \frac{1}{k!}(\overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}a_{0})h_{∗}^{i_{1}} ⋯ h_{∗}^{i_{k}} + T_{n},

where each term in the reminder T_{n} is not a pure polynomial of h_{∗}^{i}’s. This can be proved by induction again.

Assume that this assertion holds for a_{n}. To show that it still holds for a_{n+1}, we need to analyze which term in a_{n} would produce pure polynomials of h_{∗}^{i}’s out of Equation (33). Since γ ∈ ΓW^{⊥}⊗Λ^{⊥}, we may ignore δ^{−1}[\frac{i}{ℏ}γ, a_{n}] and only consider δ^{−1}∂a_{n} = δ^{−1}(∑_{i} ∇\overrightarrow{h_{i}}a_{n} ∧ h_{∗}^{i} +

∑_{i} ∇_{v_{i}}a_{n} ∧ v_{∗}^{i} + ∑_{i} ∇_{u_{i}}a_{n} ∧ u_{∗}^{i}). It is clear that those terms containing pure polynomials of h_{∗}^{i}’s arise only from δ^{−1}(∑_{i} ∇\overrightarrow{h_{i}}a_{n} ∧ h_{∗}^{i}). Now a general term in a_{n} has the form ℏ^{k}a_{αβγ}(x)v_{∗}^{α}h_{∗}^{β}u_{∗}^{γ}, where α, β and γ are multi-indexes. However,

∇\overrightarrow{h_{i}}(a_{αβγ}(x)v_{∗}^{α}h_{∗}^{β}u_{∗}^{γ})
= (\overrightarrow{h_{i}}a_{αβγ}(x))v_{∗}^{α}h_{∗}^{β}u_{∗}^{γ} + a_{αβγ}(x)(∇\overrightarrow{h_{i}}v_{∗}^{α})h_{∗}^{β}u_{∗}^{γ} + a_{αβγ}(x)v_{∗}^{α}(∇\overrightarrow{h_{i}}h_{∗}^{β})u_{∗}^{γ} + a_{αβγ}(x)v_{∗}^{α}h_{∗}^{β}(∇\overrightarrow{h_{i}}u_{∗}^{γ})
= (\overrightarrow{h_{i}}a_{αβγ}(x))v_{∗}^{α}h_{∗}^{β}u_{∗}^{γ} + a_{αβγ}(x)v_{∗}^{α}(∇\overrightarrow{h_{i}}h_{∗}^{β})u_{∗}^{γ} + a_{αβγ}(x)v_{∗}^{α}h_{∗}^{β}(∇\overrightarrow{h_{i}}u_{∗}^{γ}).

According to Lemma 5.6, neither ∇\overrightarrow{h_{i}}h_{∗}^{β} nor ∇\overrightarrow{h_{i}}u_{∗}^{γ} will be a pure polynomial of h_{∗}^{i}’s. Hence to

produce a pure h_{∗}^{i}-polynomial term, one needs that α = γ = 0. And in this case, the resulting pure h_{∗}^{i}-polynomial term is ℏ^{k}(\overrightarrow{h_{i}}a_{0β0}(x))h_{∗}^{β}. In conclusion, only pure h_{∗}^{i}-polynomial terms in a_{n} can give rise to pure h_{∗}^{i}-polynomial terms in δ^{−1}∂a_{n}. Hence the pure h_{∗}^{i}-polynomial terms in a_{n+1} is a_{0} +∑_{k=0}^{n} \frac{1}{k!} \frac{1}{k+1} \overrightarrow{h_{i}}(\overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}a_{0})h_{∗}^{i}h_{∗}^{i_{1}} ⋯ h_{∗}^{i_{k}}, which is clear equal to ∑_{k=0}^{n+1} \frac{1}{k!}(\overrightarrow{h_{i_{1}}} ⋯ \overrightarrow{h_{i_{k}}}a_{0})h_{∗}^{i_{1}} ⋯ h_{∗}^{i_{k}}. This concludes the proof.

□

6 Classification

This section is devoted to the classification of quantization of a non-degenerate triangular dynamical r-matrix. Our method relies heavily on the classification result of star products on a symplectic manifold. First, let us introduce the following:

Definition 6.1 Two quantizations F (λ) and G(λ) of a triangular dynamical r-matrix are said to be equivalent if there is a T (λ) : 𝔥^{∗} ⟶ (U𝔤)^{𝔥}[[ℏ]] satisfying the condition that T (λ) = 1(mod ℏ) and ϵ(T (λ)) = 1 such that

G(λ) = ΔT (λ)^{−1}F (λ)T_{1}(λ − \frac{1}{2}ℏh^{(2)})T_{2}(λ + \frac{1}{2}ℏh^{(1)}).  (34)

To justify our definition, we need the following result, which interprets this equivalence in terms of star products.

Theorem 6.2 Given a compatible star product ∗_{ℏ} on the Poisson manifold (M, π) associated to a triangular dynamical r-matrix r(λ), assume that T (λ) : 𝔥^{∗} ⟶ (U𝔤)^{𝔥}[[ℏ]] satisfies the condition that T (λ) = 1(mod ℏ) and ϵ(T (λ)) = 1. Then the ∗-product:

f ˜∗_{ℏ}g = \overrightarrow{T}^{−1}(\overrightarrow{T}f ∗_{ℏ} \overrightarrow{T}g), ∀f, g ∈ C^{∞}(M)  (35)

is still a compatible ∗-product. Moreover, if f, g ∈ C^{∞}(G),

f ˜∗_{ℏ}g = \overrightarrow{G(λ)}(f, g),  (36)

where G(λ) is given by Equation (34).

Thus we are lead to the following

Definition 6.3 Compatible ∗-products ∗_{ℏ} and ˜∗_{ℏ} are said to be strongly equivalent iff they are related by Equation (35) for some T (λ) : 𝔥^{∗} ⟶ (U𝔤)^{𝔥}[[ℏ]] satisfying the condition that T (λ) = 1(mod ℏ) and ϵ(T (λ)) = 1.

An immediate consequence of Theorem 6.2 is the following

Corollary 6.4 If F (λ) is a quantization of a triangular dynamical r-matrix r : 𝔥^{∗} ⟶ ∧^{2}𝔤 and T (λ) : 𝔥^{∗} ⟶ (U𝔤)^{𝔥}[[ℏ]] satisfies the condition that T (λ) = 1(mod ℏ) and ϵ(T (λ)) = 1, then

G(λ) = ΔT (λ)^{−1}F (λ)T_{1}(λ − \frac{1}{2}ℏh^{(2)})T_{2}(λ + \frac{1}{2}ℏh^{(1)})

is also a quantization of r(λ).

Because of this fact, Definition (6.1) is well justified. In fact, Theorem 6.2 allows us to reduce the classification problem of quantization of a triangular dynamical r-matrix to that of strongly equivalent star products on M.

Remark If we denote R_{G}(λ) = G^{21}(λ)^{−1}G^{12}(λ) and R_{F} (λ) = F^{21}(λ)^{−1}F^{12}(λ), then it is easy to see that they are related by

R_{G}(λ) = T_{2}(λ − \frac{1}{2}ℏh^{(1)})^{−1}T_{1}(λ + \frac{1}{2}ℏh^{(2)})^{−1}R_{F} (λ)T_{1}(λ − \frac{1}{2}ℏh^{(2)})T_{2}(λ + \frac{1}{2}ℏh^{(1)}).  (37)

Alternatively, we may define a quantization of a triangular dynamical r-matrix r(λ) to be an element R(λ) ∈ U𝔤⊗U𝔤[[ℏ]] such that R(λ) = 1 + ℏr(λ) + ⋯, and satisfies the QDYBE, and say two such quantizations are equivalent if they are related by Equation (37). Obviously, this definition is weaker than our original definition. We, however, do not know at this moment whether these two definitions are equivalent. It would be interesting to have this clarified.

To prove Theorem 6.2, we need a lemma.

Lemma 6.5 Assume that T (λ) : 𝔥^{∗} ⟶ (U𝔤)^{𝔥}[[ℏ]] is as in Theorem 6.2, then

(i). (ad_{θ})^{n}(T ⊗1) = (−\frac{1}{2})^{n} ∑_{i_{1}···i_{n}} \frac{∂^{n}T}{∂λ^{i_{1}}···∂λ^{in}} ⊗h_{i_{1}} ⋯ h_{i_{n}};

(ii). Θ(T ⊗1)Θ^{−1} = T_{1}(λ − \frac{1}{2}ℏh^{(2)});

(iii). Θ(1⊗T)Θ^{−1} = T_{2}(λ + \frac{1}{2}ℏh^{(1)}); (iv). Θ(T ⊗T)Θ^{−1} = T_{1}(λ − \frac{1}{2}ℏh^{(2)})T_{2}(λ + \frac{1}{2}ℏh^{(1)}). Proof. (i) We prove this by induction. Obviously, it holds for n = 0. Assume that it holds for n = k. Now

(ad_{θ})^{k+1}(T ⊗1)

= ad_{θ}[(−\frac{1}{2})^{k} ∑_{i_{1}···i_{k}} \frac{∂^{k}T}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} ⊗h_{i_{1}} ⋯ h_{i_{k}}]
= (−\frac{1}{2})^{k} \frac{1}{2} ∑_{i_{1}···i_{k}} ∑_{i}([h_{i}⊗\frac{∂}{∂λ^{i}} , \frac{∂^{k}T}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} ⊗h_{i_{1}} ⋯ h_{i_{k}}]

−[\frac{∂}{∂λ^{i}} ⊗h_{i}, \frac{∂^{k}T}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} ⊗h_{i_{1}} ⋯ h_{i_{k}}])

= (−\frac{1}{2})^{k+1} ∑_{i_{1}···i_{k+1}} \frac{∂^{k+1}T}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k+1}}} ⊗h_{i_{1}} ⋯ h_{i_{k+1}}.

(ii)

Θ(T ⊗1)Θ^{−1}
= exp (ℏad_{θ})(T ⊗1)
= ∑_{k=0}^{∞} \frac{1}{k!}(ℏad_{θ})^{k}(T ⊗1)
= ∑_{k=0}^{∞} \frac{1}{k!}(−\frac{ℏ}{2})^{k} \frac{∂^{k}T}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}} ⊗h_{i_{1}} ⋯ h_{i_{k}}
= T_{1}(λ − \frac{1}{2}ℏh^{(2)}).

(iii) is proved similarly, and (iv) follows from (ii) and (iii).

□

Proof of Theorem 6.2 If f, g ∈ C^{∞}(𝔥^{∗}), then \overrightarrow{T}f = f and \overrightarrow{T}g = g since ϵ(T) = 1. Hence

\overrightarrow{T}^{−1}(\overrightarrow{T}f ∗_{ℏ} \overrightarrow{T}g) = fg.

Now if f ∈ C^{∞}(𝔥^{∗}) and g ∈ C^{∞}(G),

\overrightarrow{T}f ∗_{ℏ} \overrightarrow{T}g

= f ∗_{ℏ} \overrightarrow{T}g
= \overrightarrow{Θ}(f, \overrightarrow{T}g)
= \overrightarrow{Θ(1⊗T)}(f, g) (by Lemma 6.5)
= \overrightarrow{T_{2}(λ + \frac{1}{2}ℏh^{(1)})Θ}(f, g)

= ∑_{k=0}^{∞} \frac{1}{k!}(\frac{ℏ}{2})^{k}\overrightarrow{(h_{i_{1}} ⋯ h_{i_{k}}⊗\frac{∂^{k}T}{∂λ^{i_{1}} ⋯ ∂λ^{i_{k}}})Θ}(f, g)
= \overrightarrow{(1⊗T)Θ}(f, g)
= \overrightarrow{T}(f ∗_{ℏ} g).

Here in the last equality, we used the fact that \overrightarrow{T} does not involve any derivative \frac{∂}{∂λ^{i}}. So we have

\overrightarrow{T}^{−1}(\overrightarrow{T}f ∗_{ℏ} \overrightarrow{T}g) = \overrightarrow{Θ}(f, g).

Finally, assume that f, g ∈ C^{∞}(G). According to Theorem 3.8,

\overrightarrow{T}f ∗_{ℏ} \overrightarrow{T}g
= \overrightarrow{(F (λ)Θ})(\overrightarrow{T}f, \overrightarrow{T}g)
= \overrightarrow{F (λ)Θ(T ⊗T)}(f, g) (by Lemma 6.5)
= \overrightarrow{F (λ)T_{1}(λ − \frac{1}{2}ℏh^{(2)})T_{2}(λ + \frac{1}{2}ℏh^{(1)})Θ}(f, g).

It thus follows that

\overrightarrow{T}^{−1}(\overrightarrow{T}f ∗_{ℏ} \overrightarrow{T}g)
= \overrightarrow{ΔT (λ)^{−1}F (λ)T_{1}(λ − \frac{1}{2}ℏh^{(2)})T_{2}(λ + \frac{1}{2}ℏh^{(1)})Θ}(f, g)
= \overrightarrow{G(λ)Θ}(f, g).

This concludes the proof.

□

.

The rest of the section is devoted to the classification of strongly equivalent compatible star products on M = 𝔥^{∗} × G. The classification of star products on a general symplectic manifold was studied by many authors, for example, see [8] [11] [34] [37] [38]. Here we follow the elementary approach due to Bertelson, Bieliavsky and Gutt [7] concerning invariant star products.

First we prove

Theorem 6.6 Let M = 𝔥^{∗} × G be the symplectic manifold corresponding to a non-degenerate dynamical r-matrix r(λ). Two compatible Fedosov ∗-products are strongly equivalent iff their Weyl curvatures Ω_{∗} and Ω are strongly cohomologous, i.e., Ω_{∗} − Ω = dθ, where θ ∈ Ω^{1}(M)[[ℏ]] is G × Hinvariant and satisfies i\overrightarrow{h}θ = 0, ∀h ∈ 𝔥.

From now on, in this section, by M we always mean the symplectic manifold 𝔥^{∗} × G associated with a non-degenerate dynamical r-matrix. Let 𝔤 = 𝔥 ⊕ 𝔪 be a reductive decomposition as in Lemma 4.5, and \{h_{1}, ⋯, h_{l}\} a basis in 𝔥, and \{e_{1}, ⋯, e_{m}\} a basis of 𝔪. If we choose v_{i} = \frac{∂}{∂λ^{i}} and u_{i} = \overrightarrow{e_{i}}, then \{\overrightarrow{h_{1}}, ⋯, \overrightarrow{h_{l}}, v_{1}, ⋯, v_{l}, u_{1} ⋯, u_{m}\} constitutes a local (in fact global in this case) basis of T M, which satisfies all the required properties as in the construction preceding Lemma 5.6. In what follows, we will fix such a choice, and denote by \{\overrightarrow{h_{∗}^{1}}, ⋯, \overrightarrow{h_{∗}^{l}}, v_{∗}^{1}, ⋯ v_{∗}^{l}, u_{∗}^{1}, ⋯, u_{∗}^{m}\} its dual basis. Content-Length: 2909 X-Lines: 69 Status: RO

Lemma 6.7 Suppose that D is an Abelian connection defining a compatible ∗-product on M as in Corollary 5.2. For any a ∈ C^{∞}(M), let

ã = ∑ ℏ^{k}D_{k,αβγ}(a)v_{∗}^{α}h_{∗}^{β}u_{∗}^{γ}  (38)

be its parallel lift, where α, β and γ are multi-indexes, and D_{k,αβγ} are certain differential operators on M. If the operator D_{k,αβγ} in any given term of Equation (38) involves a derivative with respect to λ ∈ 𝔥^{∗}, then the corresponding term satisfies |α| > 0.

Proof. As we know, ã is given by the iteration formula

a_{n+1} = a_{0} + δ^{−1}(∂a_{n} + [\frac{i}{ℏ}γ, a_{n}]),

so it suffices to show that a_{n} possesses such a property for any n, which we shall prove by induction.

Assume that all terms in a_{n} possess such a property, and we need to show that so do all the terms in a_{n+1}. Let ℏ^{k}D_{k,αβγ}(a)v_{∗}^{α}h_{∗}^{β}u_{∗}^{γ} be an arbitrary term in a_{n+1} that D_{k,αβγ} involves a derivative with respect to λ. There are two possible sources that this term may come from. One is from δ^{−1}[\frac{i}{ℏ}γ, a_{n}]. Since this operation does not affect the part involving derivatives on a, so it must come from some term having the form:

δ^{−1}[\frac{i}{ℏ}γ, ℏ^{k′}D_{k,αβγ}(a)v_{∗}^{α′}h_{∗}^{β′}u_{∗}^{γ′}],  (39)

where ℏ^{k′}D_{k,αβγ}(a)v_{∗}^{α′}h_{∗}^{β′}u_{∗}^{γ′} is one of the terms in a_{n}. By assumption, we have that |α′| > 0. Since γ ∈ ΓW^{⊥}⊗Λ^{⊥}, it follows from Lemma 5.6 that any resulting term in Equation (39) has at least a factor v_{∗}^{α′}.

Another possible source is from δ^{−1}(∂a_{n}). Now

δ^{−1}∂a_{n} = ∑_{i}(∇_{\frac{∂}{∂λ^{i}}} a_{n})v_{∗}^{i} + ∑_{i}(∇\overrightarrow{h_{i}}a_{n})h_{∗}^{i} + ∑_{i}(∇\overrightarrow{e_{i}}a_{n})u_{∗}^{i}.

If it arises from the first term, we are done. Assume that it comes from the second term: (∇\overrightarrow{h_{i}}a_{n})h_{∗}^{i}.

Let ℏ^{k}D_{k,αβγ}(a)v_{∗}^{α}h_{∗}^{β}u_{∗}^{γ} be a general term in a_{n}, then

∇\overrightarrow{h_{i}}(ℏ^{k}D_{k,αβγ}(a)v_{∗}^{α}h_{∗}^{β}u_{∗}^{γ}) = ℏ^{k}(\overrightarrow{h_{i}}D_{k,αβγ})(a)v_{∗}^{α}h_{∗}^{β}u_{∗}^{γ}+ℏ^{k}D_{k,αβγ}(a)v_{∗}^{α}(∇\overrightarrow{h_{i}}h_{∗}^{β})u_{∗}^{γ}+ℏ^{k}D_{α,β,γ}(a)v_{∗}^{α}h_{∗}^{β}(∇\overrightarrow{h_{i}}u_{∗}^{γ}).

From this, it is obvious to see that D_{k,αβγ} must already involve some derivative to λ ∈ 𝔥^{∗}. Then the conclusion follows from the inductive assumption. A similar argument can be used if it arises from the last term (∇\overrightarrow{e_{i}}a_{n})u_{∗}^{i}. This concludes the proof.

□

Proof of Theorem 6.6 Our proof here is essentially a modification of the proof of Corollary 5.5.4 in [21]. Content-Length: 28086 X-Lines: 787 Status: RO

“Necessity.” Assume that A : W_{D} ⟶ W_{D_{∗}} is the isomorphism of algebras corresponding to the equivalence of the compatible Fedosov ∗-products. It is standard that A lifts to an automorphism of the Weyl bundle W, which will be denoted by the same symbol A : W ⟶ W. Then A is G × H-equivariant. As in [21], we may assume that Aa = U ∘a∘U^{−1} for some U ∈ ΓW_{+}. We may also assume that U is G × H-invariant since A is G × H-equivariant. By assumption, we also know that Aa = a if a = ∑_{0}^{∞} \frac{1}{k!} \frac{∂^{k}a_{0}}{∂λ^{i_{1}}···∂λ^{i_{k}}} v_{∗}^{i_{1}} ⋯ v_{∗}^{i_{k}}, which is the parallel lift of a function a_{0} ∈ C^{∞}(𝔥^{∗}) according to Proposition 5.11. This implies that U commutes with v_{∗}^{i}, i = 1, ⋯, l, and therefore U ∈ ΓW_{+}^{⊥} according to Lemma 5.6. Consider another Abelian connection: D̃a = (A∘D∘A^{−1})(a) = U ∘D(U^{−1}aU)∘U^{−1} = Da − [DU ∘U^{−1}, a]. Then D̃ has the same Weyl curvature as D (see Theorem 5.5.3 and Corollary 5.5.4 in [21]), which is assumed to be Ω. On the other hand,

D_{∗}a − D̃a = \frac{i}{ℏ}[γ_{∗} − γ − iℏ(DU ∘U^{−1}), a]
= \frac{i}{ℏ}[Δγ, a].  (40)

Since U ∈ ΓW_{+}^{⊥} and γ_{∗}, γ ∈ ΓW^{⊥}⊗Λ^{⊥}, it follows that Δγ ∈ ΓW^{⊥}⊗Λ^{⊥}. It is also clear that Δγ is G × H-invariant. Moreover, from Equation (40), it follows that [Δγ, a] = 0, if a ∈ W_{D_{∗}}. Hence Δγ is a scalar form. Thus Ω_{∗} − Ω = dΔγ. Clearly, Δγ is G × H-invariant and i\overrightarrow{h}Δγ = 0, ∀h ∈ 𝔥.

“Sufficientity”. Let

D = −δ + ∂ + \frac{i}{ℏ}[γ, ·],
D_{∗} = −δ + ∂ + \frac{i}{ℏ}[γ_{∗}, ·]

be Abelian connections with Weyl curvatures Ω and Ω_{∗}, respectively. Assume that Ω_{∗}−Ω = dθ, θ ∈ Ω^{1}(M)[[ℏ]] is G×H-invariant and i\overrightarrow{h}θ = 0, ∀h ∈ 𝔥. Let Ω(t) = Ω+tdθ and D_{t} = −δ +∂ + \frac{i}{ℏ}[γ(t), ·]

be the Abelian connection with Weyl curvature Ω(t), where γ(t) is as in Theorem 7.2 satisfying δ^{−1}γ(t) = 0.

Let H(t) ∈ ΓW be the solution of the equation D_{t}H(t) = −θ + γ̇(t) satisfying H(t)|_{y=0} = 0. Then H(t) is G × H-invariant since D_{t}, θ, γ(t) are all G × H-invariant. On the other hand, since γ(t) ∈ ΓW^{⊥}⊗Λ^{⊥} according to Proposition 5.10, and θ ∈ ΓΛ^{⊥} by assumption, it follows that H(t) ∈ ΓW^{⊥}.

According to Theorem 5.5.3 [21], the solution of the Heisenberg equation:

\frac{dã}{dt} + [H(t), ã] = 0  (41)

establishes an isomorphism W_{D} ⟶ W_{D_{∗}} given by ã(0) ⟶ ã(1). In fact, D_{t}ã(t) = 0 if Dã(0) = 0.

Clearly, this correspondence is G × H-equivariant since H(t) is G × H-invariant. So its corresponding formal differential operator T : C^{∞}(M)[[ℏ]] ⟶ C^{∞}(M)[[ℏ]] is G × H-invariant. It remains to show that T, as a formal differential operator, does not involve any derivative with respect to λ ∈ 𝔥^{∗}.

To show this, for any a ∈ C^{∞}(M), let ã ∈ W_{D} be its parallel lift, and ã(t) be the solution of Equation (41) satisfying the initial condition ã(0) = ã. Then D_{t}ã(t) = 0. Also, let a(t) = ã(t)|_{y=0}. Write

ã(t) = ∑ ℏ^{k}D_{t,k,αβγ}(a(t))v_{∗}^{α}h_{∗}^{β}u_{∗}^{γ}.

If D_{t,k,αβγ} involves a derivative to λ ∈ 𝔥^{∗}, we know that α ≠ 0 according to Lemma 6.7. Since H(t) ∈ ΓW^{⊥}, it thus follows that [H(t), D_{t,k,αβγ}(a(t))v_{∗}^{α}h_{∗}^{β}u_{∗}^{γ}]|_{y=0} = 0. This implies that [H(t), ã(t)]|_{y=0} = D_{t}a(t), where D_{t} is a formal differential operator on M involving no derivatives to λ ∈ 𝔥^{∗}. Now Equation (41) implies that

\frac{da(t)}{dt} + D_{t}(a(t)) = 0.

Therefore the equivalence operator T : C^{∞}(M)[[ℏ]] ⟶ C^{∞}(M)[[ℏ]], which sends a(0) to a(1), does not involve any derivative to λ ∈ 𝔥^{∗}. This concludes the proof.

□

As in [7], by C_{diff,0}^{k}(M), we denote the space of differential Hochschild k-cochains on C^{∞}(M) (i.e. k-differential operators on M) vanishing on constants, and denote by b : C_{diff,0}^{k}(M) ⟶ C_{diff,0}^{k+1}(M) the Hochschild coboundary operator.

Proposition 6.8 Suppose that ∗_{ℏ} and ∗′_{ℏ} are two compatible ∗-products on M:

u ∗_{ℏ} v = ∑_{k=0}^{∞} ℏ^{k}C_{k}(u, v), u ∗′_{ℏ} v = ∑_{k=0}^{∞} ℏ^{k}C′_{k}(u, v), ∀u, v ∈ C^{∞}(M).

Assume that ∗_{ℏ} and ∗′_{ℏ} coincide up to order n, i.e., C_{k} = C′_{k}, 0 ≤ k ≤ n. Then

(i). (C_{n+1} − C′_{n+1})(u, v) = \overrightarrow{B}(u, v) + (b\overrightarrow{E})(u, v), where B ∈ C^{2}(𝔥^{∗}, (∧^{2}𝔤)^{𝔥}) is a δ 2-cocycle (i.e., δB = 0), and E : 𝔥^{∗} ⟶ (U𝔤)^{𝔥}.

(ii). C_{1} = \frac{1}{2}\{·, ·\} + b\overrightarrow{c_{1}} for some c_{1} ∈ C^{∞}(𝔥^{∗}, (U𝔤)^{𝔥}).

(iii). If B = δX, X ∈ C^{∞}(𝔥^{∗}, 𝔤^{𝔥}), then the formal operator T = 1 + ℏ^{n}\overrightarrow{X} + ℏ^{n+1}\overrightarrow{E_{1}} transforms ∗_{ℏ} to another star-product, which coincides with ∗′_{ℏ} up to order n + 1. Here E_{1} = E(u) − [X, c_{1}].

Proof. We use a similar argument as in [7].

(i). By definition, if either u or v is in C^{∞}(𝔥^{∗}), u ∗_{ℏ} v = u ∗′_{ℏ} v = \overrightarrow{Θ}(u, v). This implies that (C_{n+1} − C′_{n+1})(u, v) = 0 if either u or v belongs to C^{∞}(𝔥^{∗}).

It is standard that C_{n+1} − C′_{n+1} is a Hochschild 2-cocycle [7] [38]. Hence we may write

C_{n+1} − C′_{n+1} = S + bT,

where S ∈ Γ(∧^{2}T M) and T is a Hochschild 1-cochain. Since S and bT are, respectively, the skewsymmetric part and the symmetric part of C_{n+1} − C′_{n+1}, they share many common properties as C_{n+1}−C′_{n+1}. In particular, both of them are G×H-invariant and vanish when either u or v belongs to C^{∞}(𝔥^{∗}). This implies that S = \overrightarrow{B}, for some B ∈ C^{∞}(𝔥^{∗}, (∧^{2}𝔤)^{𝔥}). It is also standard [7] [38] that S satisfies the equation [π, S] = 0, which is equivalent to δB = 0 according to Proposition 2.11.

Now M = 𝔥^{∗}×G clearly admits a G×H-invariant (in fact a G bi-invariant) connection. Since bT is G × H-invariant, according to Proposition 2.1 in [7], we may assume that T is a G × H-invariant 1-cochain. For any u, v ∈ C^{∞}(𝔥^{∗}), it follows from (bT)(u, v) = 0 that uT v − T (uv) + (T u)v = 0. On the other hand, since T u is G-invariant, it is a function of λ ∈ 𝔥^{∗} only, i.e., T u ∈ C^{∞}(𝔥^{∗}). Hence the restriction of the operator T to C^{∞}(𝔥^{∗}) defines a vector field Y on 𝔥^{∗}. Now since (bT)(u, v) = 0, ∀u ∈ C^{∞}(𝔥^{∗}), it follows that

(T − Y)(uv) = u(T − Y)(v), ∀u ∈ C^{∞}(𝔥^{∗}), v ∈ C^{∞}(M).

Hence T −Y does not involve any derivative with respect to λ ∈ 𝔥^{∗}. Since T −Y is G×H-invariant, this implies that T − Y = \overrightarrow{E}, for some E : 𝔥^{∗} ⟶ (U𝔤)^{𝔥}. Therefore, bT = b\overrightarrow{E}.

(ii) It is standard that C_{1} = \frac{1}{2}\{·, ·\} + bc′_{1}, where c′_{1} is a Hochschild 1-cochain. By repeating a similar argument as in (ii), we can prove that c′_{1} can be chosen so that c′_{1} = \overrightarrow{c_{1}} for some c_{1} ∈ C^{∞}(𝔥^{∗}, (U𝔤)^{𝔥}).

(iii) If B = δX, then \overrightarrow{B} = [π, \overrightarrow{X}] according to Proposition 2.11. Then it is easy to check that T = 1 + ℏ^{n}\overrightarrow{X} + ℏ^{n+1}\overrightarrow{E_{1}} transforms ∗_{ℏ} to another star-product, which coincides with ∗′_{ℏ} up to order n + 1.

□

As a consequence, we have

Corollary 6.9 If r is a non-degenerate triangular dynamical r-matrix and M = 𝔥^{∗} × G its associated symplectic manifold, then every compatible ∗-product on M is strongly equivalent to a Fedosov ∗-product as constructed in Corollary 5.2.

Proof. This follows essentially from the same argument as in the proof of Proposition 4.1 in [7]. We will omit it here.

□

According to Theorem 6.6, we thus have proved the following result.

Theorem 6.10 Let M = 𝔥^{∗} × G be the symplectic manifold associated with a non-degenerate triangular dynamical r-matrix r : 𝔥^{∗} ⟶ ∧^{2}𝔤. Then compatible ∗-products on M are classified by the relative Lie algebra cohomology (with coefficients being formal power series of ℏ) H^{2}(𝔤, 𝔥)[[ℏ]].

Using Theorem 6.2, we are thus lead to the following

Theorem 6.11 The equivalence classes of quantization of a non-degenerate triangular dynamical r-matrix r : 𝔥^{∗} ⟶ ∧^{2}𝔤 are classified by the relative Lie algebra cohomology (with coefficients being formal power series of ℏ) H^{2}(𝔤, 𝔥)[[ℏ]].

Remark It would be interesting to see if one can prove this classification theorem by directly applying the usual classification theorem of star products on a symplectic manifold. One of the difficulties is that the characteristic class of a star product is difficult to computer in practise. Recently, Tsygan comes up a nice way of redefining the characteristic class using the jet bundle [36]. This may shed some new light on our problem.

Inspired by Kontesvich’s formality theorem, we end this section with the following

Conjecture For an arbitrary triangular classical dynamical r-matrix r : 𝔥^{∗} ⟶ ∧^{2}𝔤, the quantization is classified by M_{r}(𝔤[[ℏ]], 𝔥), the formal neighbourhood of r in the modular space M(𝔤[[ℏ]], 𝔥).

7 Appendix

In this section, we recall some basic ingredients of the Fedosov construction of ∗-products on a symplectic manifold, as well as some useful notations, which are used throughout the paper. For details, readers should consult [20] [21].

Let (M, ω) be a symplectic manifold of dimension 2n. Then, each tangent space T_{x}M is equipped with a linear symplectic structure, which can be quantized using the standard Moyal-Weyl product. The resulting space is denoted by W_{x}. More precisely,

Definition 7.1 A formal Weyl algebra W_{x} associated to T_{x}M is an associative algebra with a unit over ℂ, whose elements consist of formal power series in the formal parameter ℏ with coefficients being formal polynomials in T_{x}M. In other words, each element has the form

a(y, ℏ) = ∑ ℏ^{k}a_{k,α}y^{α}  (42)

where y = (y^{1}, ⋯, y^{2n}) is a linear coordinate system on T_{x}M, α = (α_{1}, ⋯, α_{2n}) is a multi-index, y^{α} = (y^{1})^{α_{1}} ⋯ (y^{2n})^{α_{2n}}, and a_{k,α} are constants. The product is defined according to the Moyal-Weyl rule:

a ∗ b = ∑_{k=0}^{∞}(\frac{ℏ}{2})^{k} \frac{1}{k!}π^{i_{1}j_{1}} ⋯ π^{i_{k}j_{k}} \frac{∂^{k}a}{∂y^{i_{1}} ⋯ ∂y^{i_{k}}} \frac{∂^{k}b}{∂y^{j_{1}} ⋯ ∂y^{j_{k}}} .  (43)

Let W = ∪_{x∈M}W_{x}. Then W is a bundle of algebras over M, called the Weyl bundle. Its space of sections ΓW forms an associative algebra with unit under fiberwise multiplication. One may think of W as a “quantum tangent bundle” of M, whose space of sections ΓW gives rise to a deformation quantization for the tangent bundle T M, considered as a Poisson manifold with fiberwise linear symplectic structures. Also as in [20], by W^{+}, we denote the extension of the algebra W consisting of the elements described as follows:

(i). elements a ∈ W^{+} are given by series (42), but the powers of ℏ can be both positive and negative;

(ii). the total degree 2k + |α| of any term of the series is nonnegative; (iii). there exists a finite number of terms with a given nonnegative total degree.

The center Z(W) of ΓW consists of sections not containing y′s, thus can be naturally identified with C^{∞}(M)[[ℏ]].

By assigning degrees to y′s and ℏ with degy^{i} = 1 and degℏ = 2, there is a natural filtration

C^{∞}(M) ⊂ Γ(W_{1}) ⊂ ⋯ Γ(W_{i}) ⊂ Γ(W_{i+1}) ⋯ ⊂ Γ(W)

with respect to the total degree (eg. any individual term in the summation of the RHS of Equation (42) has degree 2k + |α|.)

A differential form with values in W is a section of the bundle W ⊗ ∧^{q}T^{∗}M, which can be expressed locally as

a(x, y, ℏ, dx) = ∑ ℏ^{k}a_{k,i_{1}···i_{p},j_{1}···j_{q}}y^{i_{1}} ⋯ y^{i_{p}}dx^{j_{1}} ∧ ⋯ ∧ dx^{j_{q}}.  (44)

Here the coefficient a_{k,i_{1}···i_{p},j_{1}···j_{q}} is a covariant tensor symmetric with respect to i_{1} ⋯ i_{p} and antisymmetric in j_{1} ⋯ j_{q}. For short, we denote the space of these sections by ΓW ⊗ Λ^{q}. There is an associative product ∘ on ΓW ⊗ Λ^{∗}, which naturally extends the multiplication ∗ on ΓW and the wedge product on Λ^{∗}:

(a⊗θ)∘(b⊗ω) = (a ∗ b)⊗(θ ∧ ω), ∀a, b ∈ ΓW, and θ, ω ∈ Λ^{∗}.  (45)

The usual exterior derivative on differential forms extends, in a straightforward way, to an operator δ on W -valued differential forms:

δa = ∑_{i} dx^{i} ∧ \frac{∂a}{∂y^{i}} , ∀a ∈ ΓW ⊗ Λ^{∗}.  (46)

By δ^{−1}, we denote its “inverse” operator as defined by:

δ^{−1}a = ∑_{i} \frac{1}{p + q} y^{i}(\frac{∂}{∂x^{i}} a)  (47)

when p + q > 0, and δ^{−1}a = 0 when p + q = 0, where a ∈ ΓW ⊗ Λ^{q} is homogeneous of degree p in y.

There is a “Hodge”- decomposition:

a = δδ^{−1}a + δ^{−1}δa + a_{00}, ∀a ∈ ΓW ⊗ Λ^{∗},  (48)

where a_{00}(x) is the constant term of a, i.e, the 0-form term of a|_{y=0} or a_{00}(x) = a(x, 0, 0, 0). The operator δ possesses most of the basic properties of the usual exterior derivatives. For example,

δ^{2} = 0 and (δ^{−1})^{2} = 0.

It is also clear that both δ and δ^{−1} commute with the Lie derivative, i.e., ∀X ∈ X (M),

L_{X} ∘δ = δ∘L_{X}, and L_{X} ∘δ^{−1} = δ^{−1}∘L_{X}.  (49)

Let ∇ be a torsion-free symplectic connection on M and

∂ : ΓW ⟶ ΓW ⊗ Λ^{1}

its induced covariant derivative.

Consider a connection on W of the form:

D = −δ + ∂ + \frac{i}{ℏ}[γ, · ],  (50)

with γ ∈ ΓW ⊗ Λ^{1}.

Clearly, D is a derivation with respect to the Moyal-Weyl product, i.e.,

D(a ∗ b) = a ∗ Db + Da ∗ b, ∀a, b ∈ ΓW.  (51)

A simple calculation yields that

D^{2}a = −[\frac{i}{ℏ}Ω, a], ∀a ∈ ΓW,  (52)

where

Ω = ω − R + δγ − ∂γ − \frac{i}{ℏ}γ^{2}.  (53)

Here R = \frac{1}{4}R_{ijkl}y^{i}y^{j}dx^{k} ∧ dx^{l} and R_{ijkl} = ω_{im}R_{jkl}^{m} is the curvature tensor of the symplectic connection as defined in Equation (20).

A connection of the form (50) is called Abelian if Ω is a scalar 2-form, i.e., Ω ∈ Ω^{2}(M)[[ℏ]]. It is called a Fedosov connection if it is Abelian and in addition γ ∈ ΓW_{3} ⊗ Λ^{1}. For an Abelian connection, the Bianchi identity implies that dΩ = DΩ = 0, i.e., Ω ∈ Z^{2}(M)[[ℏ]]. In this case, Ω is called the Weyl curvature.

Theorem 7.2 (Fedosov [21]) Let ∇ be any torsion free symplectic connection, and Ω = ω + ℏω_{1} + ⋯ ∈ Z^{2}(M)[[ℏ]] a perturbation of the symplectic form in the space Z^{2}(M)[[ℏ]]. There exists a unique γ ∈ ΓW_{3} ⊗ Λ^{1} such that D, given by Equation (50), is a Fedosov connection, which has Weyl curvature Ω and satisfies

δ^{−1}γ = 0.

Proof. It suffices to solve the equation:

ω − R + δγ − ∂γ − \frac{i}{ℏ}γ^{2} = Ω.  (54)

It is equivalent to

δγ = Ω̃ + ∂γ + \frac{i}{ℏ}γ^{2},  (55)

where

Ω̃ = Ω − ω + R.  (56)

Applying the operator δ^{−1} to Equation (55) and using the Hodge decomposition: Equation (48), we obtain

γ = δ^{−1}Ω̃ + δ^{−1}(∂γ + \frac{i}{ℏ}γ^{2}).  (57)

Here we note that γ_{00} = 0 since γ is a 1-form.

Take γ_{0} = δ^{−1}Ω̃, and consider the following iteration equation:

γ_{n+1} = γ_{0} + δ^{−1}(∂γ_{n} + \frac{i}{ℏ}γ_{n}^{2}), ∀n ≥ 0.  (58)

Since the operator ∂ preserves the filtration and δ^{−1} raises it by 1, γ_{n} defined by Equation (58) converges to a unique γ ∈ ΓW ⊗ Λ^{1}, which is clearly a solution to Equation (57). Moreover since γ_{0} is at least of degree 3, γ is indeed in ΓW_{3} ⊗ Λ^{1}.

□

This theorem indicates that a Fedosov connection D is uniquely determined by a torsion free symplectic connection ∇ and a Weyl curvature Ω = ∑_{i=0}^{∞} ℏ^{i}ω_{i} ∈ Z^{2}(M)[[ℏ]]. For this reason, we will say that D is a Fedosov connection corresponding to the pair (∇, Ω).

If D is a Fedosov connection, the space of all parallel sections W_{D} automatically becomes an associative algebra. Let σ denote the projection from W_{D} to its center C^{∞}(M)[[ℏ]] defined by σ(a) = a|_{y=0}.

Theorem 7.3 (Fedosov [21]) For any a_{0}(x, ℏ) ∈ C^{∞}(M)[[ℏ]] there is a unique section a ∈ W_{D} such that σ(a) = a_{0}. Therefore, σ establishes an isomorphism between W_{D} and C^{∞}(M)[[ℏ]] as vector spaces.

Proof. The equation Da = 0 can be written as

δa = ∂a + [\frac{i}{ℏ}γ, a].

Applying the operator δ^{−1}, it follows from the Hodge decomposition (Equation (48)) that

a = a_{0} + δ^{−1}(∂a + [\frac{i}{ℏ}γ, a]).  (59)

In analogue to the proof of Theorem 7.2, we can solve this equation by the iteration formula:

a_{n+1} = a + δ^{−1}(∂a_{n} + [\frac{i}{ℏ}γ, a_{n}]).  (60)
□

References

[1] Arnaudon, D., Buffenoir, E., Ragoucy, E., and Roche, Ph., Universal solutions of quantum dymanical Yang-Baxter equation, Lett. Math. Phys. 44 (1998), 201-214.

[2] Avan. J, Babelon, O., Billey, E., The Gervais-Neveu-Felder equation and the quantum Calogero-Moser systems, Comm. Math. Phys. 178 (1996), 281-299.

[3] Babelon, O., Universal exchange algebra for Bloch waves and Liouville theory, Comm. Math. Phys. 139 (1991), 619-643.

[4] Bangoura, M. and Kosmann-Schwarzbach, Y., Equation de Yang-Baxter dynamique classique et algebroides de Lie, C. R. Acad. Sci. Paris Serie I, 327 (1998), 541–546.

[5] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D., Deformation theory and quantization, I and II, Ann. Phys. 111 (1977), 61–151.

[6] Berezin, F.A., Some remarks about the associated envelope of a Lie algebra, Funct. Anal. Appl. 1 (1967), 91-102.

[7] Bertelson, M., Bieliavsky, P., and Gutt, S., Parametrizing equivalence classes of invariant star products, Lett. Math. Phys. 46 (1998), 339-345.

[8] Bertelson, M., Cahen, M., and Gutt, S., Equivalence of star products, Classical and Quantum Gravity 14 (1997), A93–A107.

[9] Bordemann, M., Brischle, M., Emmrich, C., and Waldmann, S., Phase space reduction for star-products: an explicit construction for CP^{n} Lett. Math. Phys. 36 (1996), 357-371.

[10] Borel, A., and Wallach, N., Continuous cohomology, discrete subgroups, and representations of reductive groups, Second edition, Mathematical Surveys and Monographs, 67, American Mathematical Society, Providence, RI, 2000.

[11] Deligne, P., Déformations de l’algèbre des fonctions d’une variété symplectique: Comparaison entre Fedosov et De Wilde, Lecomte, Selecta Mathematica, New Series 1 (1995), 667–697.

[12] Drinfel’d, V.G., Quantum groups, Proc. ICM, Berkeley, (1986), 1, 789-820.

[13] Drinfel’d, V.G., On constant quasiclassical solutions of the Yang-Baxter quantum equation, Soviet Math. Dokl. 28, (1983), 667-671.

[14] Etingof, P. and Kazhdan, D., Quantization of Lie bialgebras I, Selecta Mathematica, New series 2 (1996), 1-41.

[15] Etingof, P. and Varchenko, A., Geometry and classification of solutions of the classical dynamical Yang-Baxter equation, Comm. Math. Phys., 192 (1998), 77-120.

[16] Etingof, P. and Varchenko, A., Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups, q-alg/9708015.

[17] Etingof, P. and Varchenko, A., Exchange dynamical quantum groups, math.QA/9801135.

[18] Etingof, P., Schedler, T., and Schiffmann, O., Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras, math.QA/9912009.

[19] Etingof, P., and Schiffmann, O., Lectures on the dynamical Yang-Baxter equations, math.QA/9908064.

[20] Fedosov, B., A simple geometrical construction of deformation quantization, J. Diff. Geom. 40 (1994), 213-238.

[21] Fedosov, B., Deformation quantization and index theory, Mathematical Topics 9, Akademie Verlag, Berlin, 1996.

[22] Felder, G., Conformal field theory and integrable systems associated to elliptic curves, Proc. ICM Zürich, Birkhäuser, Basel, (1994), 1247-1255.

[23] Fronsdal, C., Quasi-Hopf deformation of quantum groups, Lett. Math. Phys. 40 (1997), 117-134.

[24] Gervais, J.-L., and Neveu, A., Novel triangle relation and absense of tachyons in Liouville string field theory, Nucl. Phys. B 238, (1984), 125-141.

[25] Jimbo, M., Konno, H., Odake, and Shiraishi, J., Quasi-Hopf twistors for elliptic quantum groups, Transform. Groups 4, (1999), 303–327.

[26] Kontsevich, M, Deformation quantization of Poisson manifolds I, q-alg/9709040.

[27] Kosmann-Schwarzbach, Y., Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math. 41 (1995), 153-165.

[28] Lichnerowicz, A., Déformations d’algébres associées à une variété symplectique (les ∗_{ν}- produits), Ann. Inst. Fourier, Grenoble, 32 No.1 (1982), 157-209.

[29] Liu, Z.-J. and Xu, P., Exact Lie bialgebroids and Poisson groupoids, Geom. and Funct. Anal 6 (1996), 138-145.

[30] Liu, Z.-J. and Xu, P., Dirac structures and dynamical r-matrices, math.DG/9903119.

[31] Mackenzie, K., Lie Groupoids and Lie Algebroids in Differential Geometry, LMS Lecture Notes Series, 124, Cambridge Univ. Press, 1987.

[32] Mackenzie, K. and Xu, P., Lie bialgebroids and Poisson groupoids, Duke Math. J. 18 (1994), 415-452.

[33] Mackenzie, K. and Xu, P., Integration of Lie bialgebroids, Topology 39, (2000), 445-467.

[34] Nest, R. and Tsygan, B., Algebraic index theorem for families, Advances in Math. 113 (1995), 151-205.

[35] Schiffmann, O., On classification of dynamical r-matrices, q-alg/9706017.

[36] Tsygan, B., Private communication.

[37] Weinstein, A. and Xu, P., Hochschild cohomology and characteristic classes for star-products, Geometry of differential equations (Festschrift for V.I. Arnol’d’s 60th birthday), Amer. Math. Soc. Transl. Ser. 2, 186, (1998), 177–194,

[38] Xu, P, Fedosov ∗-products and quantum momentum maps, Comm. Math. Phys., 197 (1998), 167-197.

[39] Xu, P, Quantum groupoids and deformation quantization, C. R. Acad. Sci. Paris, Serie I. 326 (1998), 289-294.

[40] Xu, P., Quantum groupoids associated to universal dynamical R-matrices, C. R. Acad. Sci. Paris, Serie I, 328 (1999), 327-332.

[41] Xu, P., Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys., 200 (1999), 545-560.

[42] Xu, P., Quantum groupoids/math.QA9905192.