{'fraction_non_alphanumeric': 0.055998723675813655, 'fraction_numerical': 0.014571367794086364, 'mean_word_length': 3.834190231362468, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A universal deformation formula for Connes-Moscovici's Hopf algebra without any projective structure

We construct a universal deformation formula for Connes-Moscovici's Hopf algebra without any projective structure using Fedosov's quantization of symplectic diffeomorphisms.


Introduction
In the study of index theory of a transverse elliptic differential operator of a codimension one foliation, Connes and Moscovici discovered a Hopf algebra H 1 which governs the local symmetry in computing the Chern character. In this paper, we study deformation theory of this Hopf algebra. In particular, we prove that the Hopf algebra H 1 has a universal deformation formula.
In [4], inspired from Rankin-Cohen brackets on modular forms, Connes and Moscovici constructed a universal deformation formula of the Hopf algebra H 1 with a projective structure. By a universal deformation formula of a Hopf algebra A, we mean an element R ∈ In [1], we together with Bieliavsky provided a geometric interpretation of a projective structure in the case of a codimension one foliation. And as a result, we (with Bieliavsky) obtained a geometric way to reconstruct Connes-Moscovici's universal deformation formula. Furthermore, a new and interesting result we proved in [1][Prop. 6.1] is that even without a projective structure, the first Rankin-Cohen bracket is a noncommutative Poisson structure, i.e. RC 1 is a Hochschild cocycle and (1 ⊗ ∆)RC 1 (1 ⊗ RC 1 ) − (∆ ⊗ 1)RC 1 (RC 1 ⊗ 1) is a Hochschild coboundary. This leads to a question whether H 1 has a universal deformation formula without a projective structure.
In this article, we give a positive answer to the above question and introduce a geometric construction of a universal deformation formula of H 1 without a projective structure. The idea of this construction goes back to Fedosov [6] in his study of deformation quantization of a symplectic diffeomorphism. Fedosov developed in [6] a systematic way to quantize a symplectic diffeomorphism to an endomorphism of the quantum algebra no matter whether it preserves or not the chosen symplectic connection. Fedosov also observed that composition of quantized symplectic diffeomorphisms does not preserve associativity. Instead, it satisfies a weaker associativity property-associative up to an inner endomorphism. This picture can be explained using the language of "gerbes and stacks" as [2]. In any case, Fedosov's construction does give rise to a deformation quantization of the groupoid algebra associated to a pseudogroup action on a symplectic manifold.
In this paper, we apply this idea to the special case that the symplectic manifold is R × R + and the Poisson structure is ∂ x ∧ ∂ y , with x the coordinate on R and y the coordinate on R + .
We consider symplectic diffeomorphisms on R × R + of the form where γ is a local diffeomorphism on R. In this case, Fedosov's construction of quantization of symplectic diffeomorphism can be computed explicitly. In particular, we are able to prove that the resulting star product on the groupoid algebra C ∞ c (R × R + ) ⋊ Γ can be expressed by f α ⋆ gβ = m(R(f α ⊗ gβ)), where m is the multiplication map on C ∞ c (R × R + ) ⋊ Γ, and R is an element in H 1 [[ ]] ⊗ H 1 [[ ]]. An important property is that the H 1 action on the collection of all C ∞ c (R × R + ) ⋊ Γ for all pseudogroup Γ is fully injective because this action is equivalent to the action used by Connes and Moscovici to define H 1 . With this observation, we can derive all the property of R as a universal deformation formula from the corresponding properties about the star product on The notion of universal deformation formula of a Hopf algebra is closely related to the solution of quantum Yang-Baxter equation. We hope that our construction will shed a light on the study of deformation theory of the Hopf algebra H 1 and also codimension one foliations.
This article is organized as follows. We review in section 2 Fedosov's theory of deformation quantization of symplectic diffeomorphisms. We provide a detailed proof of the fact that this defines a deformation of the groupoid algebra C ∞ c (R × R + ) ⋊ Γ. In section 3, we prove the main theorem of this paper that H 1 has a universal deformation formula using Fedosov's theory reviewed in section 2. In section 4, we compute explicitly our universal deformation formula up to 2 . We observe that when the H 1 action is projective, the universal deformation formula obtained in this paper does not agree with the one introduced by Connes and Moscovici in [4]. Instead, our lower order term computation suggests that in the case of a projective action, these two universal deformation formulas should be related by an isomorphism expressed by elements in H 1 [[ ]] and the projective structure Ω. In the appendix we discuss the associativity of the Eholzer product on modular forms, which was used by Connes and Moscovici in constructing their Rankin-Cohen deformation.
Acknowledgements: We would like to thank A. Connes and H. Moscovici for explaining the Hopf algebra H 1 and Rankin-Cohen brackets. The first author would like to thank A. Gorokhovsky, R. Nest, and B. Tsygan for explaining Fedosov's quantization of symplectic diffeomorphisms and their ideas of deformation of groupoid algebras. The research of the first author is partially supported by NSF Grant 0703775.

Quantization of symplectic diffeomorphisms
In this section, we briefly recall Fedosov's construction of quantization of a symplectic diffeomorphism. And we use this idea to define a deformation of a groupoid algebra coming from a pseudogroup action on a symplectic manifold. We learned this construction from A. Gorokhovsky, R. Nest, and B. Tsygan.
In Fedosov's approach to deformation quantization of A symplectic manifold (M, ω), a flat connection D (also called Fedosov connection) on the Weyl algebra bundle W plays an essential role. The quantum algebra is identified with the space of flat sections W D of W. A question arises when one wants to quantize a symplectic diffeomorphism. Because a symplectic diffeomorphism may not preserve D, the canonical lifting of a symplectic diffeomorphism to the Weyl algebra bundle W may not act on the quantum algebra W D . How can we quantize a symplectic diffeomorphism in this case? Fedosov in [6] studied this question. The answer he came up with fits well with the language of "stack of algebras". In the following we briefly review Fedosov's results [6][Section 4].
A symplectic diffeomorphism γ : M → M naturally acts on the cotangent bundle γ : T * M → T * M . Therefore γ lifts to an endomorphism on the Weyl algebra bundle γ : W → W. It is easy to check that if γ(D) := γ • D • γ −1 = D, then γ defines an algebra endomorphism on the quantum algebra W D = ker(D), which is called a quantization of the symplectic diffeomorphism γ. We with Bieliavsky in [1] used this idea to construct a universal deformation formula of H 1 with a projective structure.
The quantization of γ when γ(D) = D is more involved. Fedosov in [6][Sec. 4] purposed the following construction of quantization. We start with extending the standard Weyl algebra W to W + , (1) an element u of W + can be written as where (y 1 , · · · , y 2n ) are coordinates on the standard symplectic vector space (V, ω). In the above sum, we allow k to be negative. (2) There are a finite number of terms with a given total degree 2k + l ≥ 0. We remark that the Moyal product extends to a well define product on W + . And we consider the corresponding extension W + of the Weyl algebra bundle W associated to W + . Write γ(D) = D + i/ [∆ γ , ·] with ∆ γ a smooth section of the Weyl algebra bundle W. We remark that in principle, we can change ∆ γ by any section in the center of W. But when we fix the expression of D = d + i/ [r, ·], ∆ γ has a canonical choice γ(r) − r. In the following of this paper, we will always work with this choice of ∆ γ . We consider the following equation where U γ is an invertible section of W + . Fedosov [6][Thm 4.3] proved that equation (1) always has solutions. In general, solutions to equation (1) are not unique. But the following induction procedure U n+1 = 1 + δ −1 {(D + δ)U n + (i/ )∆ γ • U n }, U 0 = 1 uniquely determines an invertible solution to Equation (1). By this induction, we see that U is a solution to the following equation which actually has a unique solution because δ −1 {(D + δ)U + i/ ∆ γ • U } raises the total degree of U by 1. We will always work with this solution in this paper. By Equation (1), U −1 α satisfies the following equation, Using Equation (2), it is not difficult to check that U −1 α is the unique solution to the following equation, which can be constructed by same induction as above.
We have the following lemma on ∆ α .
We will need the following properties of U α later in our construction.
Proposition 2.2. The assignment α → U α satisfies the following properties, Proof.
(1)We can use Equation (2) to prove a stronger statement. We compute By applying Lemma 2.1 to the last line, we conclude that α(U β ) is the unique solution to the following equation We remark that solution to Equation (4) is unique because • ∆ α } raises the total degree of α(U β ) by 1. Taking δ on both sides of the above equation, we obtain the first identity of this Proposition.
(2) Setting β = α −1 in Equation (4), we have that α( which is same as the defining equation for U −1 α . By the uniqueness of the solution to the above equation, we have that α(U α −1 ) = U −1 α .
Fedosov [6] defined quantization of a symplectic diffeomorphism γ on (M, ω) aŝ which is an algebra endomorphism of the quantum algebra W D . The "defect" of this quantization is that the associativity of composition fails, i.e. αβ = αβ.
Instead, Fedosov proves the following property of the associator v α,β : Proposition 2.3. The associator v α,β is a flat section of W and We remark that in the above formula we have used the property D(α(U −1 The property of the associator is a straight forward computation. We use f →f to represent the isomorphism between C ∞ (M ) and the quantum algebra W D . We know that α(ĝ) is a flat section of the connection α(D). Therefore, α(ĝ) satisfies the following equation Then U −1 α • α(ĝ) • U α satisfies the following equation In the following, we apply the above idea to quantize the groupoid algebra of a pseudogroup Γ on a symplectic manifold M . We define the following product on where y's are coordinates along the fiber direction of T * M . We remark that becausef , U −1 α • α(ĝ) • U α , and v α,β are all flat with respect to the connection D, the productf is also flat with respect to the connection D.
We check the associativity of ⋆ on

A universal deformation formula
In this section, we apply the construction described in the previous section to construct a universal deformation formula of Connes-Moscovici's Hopf algebra H 1 . We start with briefly recalling the definition of H 1 .
We consider the defining representation of .
We remark that the above expression of γ action does not agree with the formulas in [3], but the two actions are isomorphic under the transformation y → 1/y.
On R × R + , we consider the Poisson structure ∂ x ∧ ∂ y , which can be expressed by −X ⊗ Y + Y ⊗ X. Our main goal is to use the method reviewed in the previous section to construct a star product on . We prove that this star product as a bilinear operator actually can be expressed by an element R of . The associativity of the star product is equivalent to the property that R is a universal deformation formula. We start with fixing a symplectic connection ∇ on the cotangent bundle of R × R + , which was introduced in [ Using X and Y , we can express the above connection by We compute α(∇) by α∇α −1 .
We observe that both ∇ and α(∇) are flat and torsion free. We consider the lifting of ∇ and α(∇) onto the Weyl algebra bundle. Use u, v to denote the generators along the fiber direction of the Weyl algebra bundle W. We have for any section a of W, Therefore, using the notation of the previous section, we have that ∆ α = y 3 δ ′ 2 (α)u 2 dx. In the following, we solve the expression forf , α(ĝ), Then the above equation is expressed as Therefore, we have thatf is the unique solution to the following family of equations with f 00 = f . Solving Equation (5), we have that 3.2. α(ĝ). We know that α(ĝ) is the unique solution to the following equation with α(g) 00 = α(g). Therefore, α(ĝ) is the unique solution to the following family of equations with α(g) 00 = α(g). By the second equation of Equations (6), we have Setting n = 0 in the first equation of Equations (6), we obtain By induction, we can solve the above equation as where A m ∈ H 1 is defined inductively by 3.3. U −1 α . We compute U −1 α using the equation . And u α mn satisfies the following family of equations with u 0,0 = 1. The second equation of Equations (7) implies that We use the n = 0 version of the first equation of Equation (7) to solve u m0 .
By induction, we have the following expression of u, where A m is defined by with A 0 = 1. ). This shows that once m + n > 0, the smallest degree term contained in u α mn u m v n has degree greater than or equal to 1. Therefore the degree 0 term of the solutionŨ −1 α = u α mn u m v n is equal to 1. Accordingly, we compute using that DŨ −1 This remark applies also to the following solutions α(U −1 β ) and αβ(U −1 (αβ) −1 ).

Theorem 3.2. There exists an element R ∈ H 1 ⊗ H 1 [[ ]] such that the star product on
Proof. We start by rewriting u α mn , u α,β mn , v αβ mn , and α(ĝ). (1) As X and Y vanishes on 1, u α mn can be written as y m−n times a sum of terms like powers of X and Y acting on powers of δ ′ 2 (α). If we rewrite δ ′ 2 (α) as δ 2 − 1/2δ 2 1 , we can write a term of powers of X and Y acting on powers of δ ′ 2 (α) into a sum of products like δ j 1 i 1 (α) · · · δ jp ip (α). We point out that as there are 1/ in the induction formula of A m , u α mn may contain terms with negative powers of . However, by induction on the negative power of , we can easily prove that the negative power of is no more than [m/3]. Therefore, we can write u α mn as −[ m 3 ] y m−n µ mn (δ 1 (α), δ 2 (α), · · · ), where µ mn is a polynomial of variables , δ 1 , δ 2 , · · · independent of α.
When g is 1 and β is identity, we have that α(1) = 1, U −1 β = 1 and U −1 From the computation in the next section, we know that

Formulas of lower order terms
We must say that the formula for R constructed in the previous section Theorem 3.2 could be very complicated. We do not know an easy way to write it down explicitly. We will provide in this section the computation of R up to the second order of .
We do not know the explicit relation between these two universal deformation formulas R 2 | δ ′ 2 =[Ω,·] and RC 2 , but only a heuristic and geometric explanation to their difference. The geometric constructions in [1] and in this article are not exactly same. In [1], we used a projective structure to redefine a symplectic connection and a Fedosov connection D ′ on the Weyl algebra bundle W, and therefore a symplectic diffeomorphism preserving this new Fedosov connection naturally lifts to an endomorphism of the quantum algebra W D ′ . In this paper, we do not change the symplectic connection because of lack of data, but change the quantization process of a symplectic diffeomorphism by introducing sections like U α , U β , · · · . Furthermore, we notice that the difference R 2 | δ ′ 2 =[Ω,·] − RC 2 is actually a Hochschild coboundary of a 1-Hochschild cochain −1/3Ω(Y + 1)(2Y + 1)Y . This suggests if we define an isomorphism , then I −1 (m(R(I(a) ⊗ I(b)))) = m + RC 1 + 2 RC 2 + o( 2 ). In general, we expect that if H 1 acts on A with a projective structure Ω, there is an isomorphism I on A[[ ]] which can be expressed using elements in H 1 [[ ]] and the projective structure Ω such that I −1 (m(R(I(a) ⊗ I(b)))) = m (RC(a ⊗ b)).

Appendix : Associativity of the Eholzer Product
In this appendix we study associativity of the Eholzer product, which was used in Connes and Moscovici's approach [4] to obtain the general associativity at the Hopf algebra level. This associativity theorem was first proved by Cohen, Manin, and Zagier in [5]. In the first part of this appendix, we give a new proof of the associativity using the method developed by the second author [8]. In the second part, we study an important combinatorial identity used by Cohen, Manin, and Zagier in [5]. This interesting identity was obtained by Zagier [11], but its complete proof is missing in literature. We prove this identity in the special case corresponding to the Eholzer product.
5.1. Proof of Associativity. First we follow the argument developed in [8] according to which the associativity of the product is equivalent to prove the identity (with the notation X n = n−1 i=0 (X + i)) r=0 n − r p A n−r (2k + 2l + 2r, 2m)A r (2k, 2l) (2k + 2l + 2r) n−p−r (2m) p (2k) r = s=0 n − s n − p A n−s (2k, 2l + 2m + 2s)A s (2l, 2m) (2k) n−p (2l + 2m + 2s) p−s (2m) s , for p = 0, 1, . . . , n and for Our proof is based on manipulation of combinatorial identities. We have, for the left hand side, Once n, p is fixed, what to be verified is an identity about polynomials in 2k, 2l, 2m. When 2l is a negative integer, we can use the following two combinatorial relations: Then (10) We then simplify the quantity inside the above brackets by where δ x,y is the Kronecker symbol (δ x,y = 1 if x = y, = 0 if not). Finally we get, On the right hand side of (9), we have By the same method as before, we compute the above quantity as follows, which gives out the same quantity. We obtain then , what we proved above is that for every triple of indices (l 1 , l 2 , l 3 ), the following combinatorial identity : where E is the Euler operator. and a polynomial Q n (y, z, a) on the right hand side of (11), (we replace x by n − y − z − 1) Q n (y, z, a) := [ n 2 ] j=0 ((n − 2j)!) 2 (j!) 6 In summary, identity (11) is equivalent to the identity "P n (y, z, a) = Q n (y, z, a)" for n ≥ 0. In order to have the associativity of the Eholzer product, we shall prove this identity when a = 1 2 . Explicitly, we want to prove the following identity, 2z − n + r r (2(n − r))! n! = 2z n 2z − n + r n − r + 2 2z − n + r n − r − 1 (n − r)! r! .

5.2.2.
The sums S 0 (n; A, B) and S(n; X). We consider the sum S 0 (n; A, B) = n k=0 k + A n − k n − k + B k .
On the other hand, we define now S(n; X) = It is clear that S 0 (0; X) = 1 and S(n + 1; X) = S(n + 1; X − 1) + S(n; X) for the same reason as above.
We conclude that the identity holds for all n and all pairs (A, B) ∈ N 2 . But as what we want to prove is a polynomial identity in A and B(for fixed n), the identities for all natural numbers implies its correctness for arbitrary (A, B). 2