Hopf Algebroid Twists for Deformation Quantization of Linear Poisson Structures

In our earlier article [Lett. Math. Phys. 107 (2017), 475-503, arXiv:1409.8188], we explicitly described a topological Hopf algebroid playing the role of the noncommutative phase space of Lie algebra type. Ping Xu has shown that every deformation quantization leads to a Drinfeld twist of the associative bialgebroid of h-adic series of differential operators on a fixed Poisson manifold. In the case of linear Poisson structures, the twisted bialgebroid essentially coincides with our construction. Using our explicit description of the Hopf algebroid, we compute the corresponding Drinfeld twist explicitly as a product of two exponential expressions.


Introduction
Given a possibly noncommutative associative k-algebra A, a (left) A-bialgebroid H is an associative algebra with an additional A-bimodule structure A H A , coproduct ∆ : H → H ⊗ A H and counit : H → A generalizing appropriately the structures of a k-vector space, coproduct and counit in the definition of a k-bialgebra. We refer to A as the base algebra and H as the total algebra of the bialgebroid. For details see [4,7] and Section 2. Lu [13] introduced a class of A-bialgebroids meant to be noncommutative analogues of transformation groupoids, hence a novel symmetry useful in geometry and mathematical physics. These bialgebroids are smash products of a Hopf algebra and a braided commutative algebra in its category of Yetter-Drinfeld modules and nowadays they are often called scalar extension bialgebroids. A particular case is the smash product of a Hopf algebra and its dual (with its canonical Yetter-Drinfeld module structure); this smash product is known under the name of Heisenberg double and may require some completions in the infinite dimensional case. Our present focus is on the Lie algebra noncommutative phase spaces whose Hopf algebroid structure is presented in detail in [18], crudely shown to be a Heisenberg double of the universal enveloping algebra U (g) in [22]. An improved variant of this Heisenberg double and an entire class of generalizations are categorically treated in [24], namely the Heisenberg doubles (and more generally, scalar extension Hopf algebroids) in the internalized context of the symmetric monoidal category of (countably cofinite strict) filtered-cofiltered vector spaces, see [24]. They are shown to be internal Hopf algebroids in the sense of Böhm [3]. It is compelling to interpret the Heisenberg double H g of the universal enveloping algebra U (g) of a Lie algebra g as a noncommutative phase space of Lie algebra type where the universal enveloping algebra U (g) is interpreted as its coordinate sector (configuration space) and its (topological) Hopf dual as the momentum sector. In the case of κ-deformed Minkowski space, one can extend the Hopf algebroid adding full κ-deformed Poincaré symmetry into the Heisenberg double [14,15]. The physical discussion of the coproduct for the momentum sector has been studied in many references including [1,11].
Based on a work of Xu [25], this example can also be understood as a deformation quantization of the phase space with the corresponding linear Poisson structure. Xu defines an analogue of Drinfeld twist F ∈ H ⊗ A H twisting the coproduct ∆ to a new coproduct ∆ F , by a recipe ∆ F (h) = F −1 ∆(h)F, for all h ∈ H, where however the Hopf algebra H from Drinfeld's theory is replaced by an A-bialgebroid H. On the other hand, if we consider the formal power series in one variable t with coefficients in the Poisson algebra of functions, then the deformation quantization gives a recipe for an associative star product f g which is a formal power series in t whose zero term is the usual product of functions and the first correction gives the Poisson bracket. One usually restricts to the case when there exist a series F in t whose coefficients are bidifferential operators such that the star product is of the form f g = (µ • F)(f ⊗ g). Xu shows that such formal bidifferential operator F is in fact a twist for the topological bialgebroid whose total algebra consists of formal power series with coefficients in the original Poisson algebra of functions.
Excluding the bialgebroid twists induced by bialgebra twists [6], very few examples where both the deformation twist and the twisted bialgebroid structures are given by explicit formulas are known. The linear Poisson structures are clearly among the most important classes to study and here we find two explicit expressions F l and F r giving the same twist for the left bialgebroid with completed Heisenberg-Weyl algebraÂ n to H g as the total algebra. Though F l = F r as elements in H g ⊗ U (g) H g the corresponding formulas come by projecting from two different elementsF l ,F r in the usual tensor square H g ⊗ H g . In [17] we exhibited another formula F c for this twist which is however far less explicit and involves a series where each term involves both the original and twisted coproduct, see (4.8).
Our main motivation is to find explicit examples of Xu's construction, and is reinforced by the recent interest in field theories on Lie algebra type noncommutative spaces, including κ-deformations.
A Hopf algebroid is defined in Section 6 as a left bialgebroid with an antihomomorphism S : H → H called the antipode and satisfying some axioms. Unlike for the bialgebra twists, it is not known in general if the bialgebroid twists of Hopf algebroids are sufficient to formulate a general recipe for twisting the antipode as well. In our case, both the bialgebroid before and the bialgebroid after the twist have an antipode. We give a conjectural formula S(h) = V −1 F S 0 (h)V F for the new antipode S in terms of old antipode S 0 along with a partial argument for it where V F = µ(S 0 ⊗ id)(F). We can show that S(h) = V −1 S 0 (h)V for some V , but we do not have the complete proof that V F = V . Both can be expressed as exponentials in formal power series of only the momentum variables ∂ µ , starting with free term 1 and agreeing in the first few terms. Note that Xu [25] says Hopf algebroid for the notion which does not involve an antipode and which is in [7] shown to be equivalent to the left bialgebroid of other authors.
The representation of H g via a concrete twist, besides its conceptual appeal, is useful to twist systematically many other constructions (for example, basic constructions in differential geometry) from the undeformed Heisenberg-Weyl algebra case to the case of phase space of Lie algebra type. This is an additional tool for physical applications, e.g., development of field theories on the noncommutative spaces of Lie algebra type. Such applications are under investigation.
To orient the reader within the subject, we note that some other physically important examples of Hopf algebroids are built from the data of weak Hopf algebras [4] (as those coming from the symmetries in low dimensional QFTs [16]) and the study of the dynamical quantum Yang-Baxter equation [8,25].
Conventions. In this article, all algebras are over a field k of characteristic 0, and the unadorned tensor product ⊗ is over the ground field (in the deformation quantization and quantum gravity examples, the field of real or complex numbers). We freely use Sweedler notation ∆(h) = h (1) ⊗ h (2) for the coproducts with or without explicit summation sign. If A is an algebra, A op denotes the algebra with opposite multiplication. We use the Einstein summation convention.

Bialgebroids
In this section, we define bialgebroids and some auxiliary constructions. 4,7,13]). Given an associative algebra (A, µ A ), which is in this context called the base algebra, a left A-bialgebroid (H, µ, α, β, ∆, ) consists of • an associative algebra (H, µ); • two algebra maps, the source map α : A → H and the target map β : A op → H such that [α(a), β(a )] = 0 for all a, a ∈ A; assume in the following that on H we fix the A-bimodule structure given by a.h.a = α(a)β(a )h, a, a ∈ A, h ∈ H; • A-bimodule map ∆ : H → H ⊗ A H, called coproduct, satisfying the coassociativity The following compatibilities are required for these data: contains the image of ∆ and the corestriction ∆| : H → H × A H is an algebra map with respect to the factorwise multiplication.
To see the meaning of (ii), notice that, unlike for the Takeuchi product H × A H, the factorwise multiplication on H ⊗ H does not factor to a well defined map on the vector space H ⊗ A H in general. Accordingly, it does not make sense to say that ∆ : H → H ⊗ A H is an algebra map and the multiplicativity of ∆ should be interpreted differently, say within H × A H. Indeed, the kernel of the canonical projection of the A-bimodules is the right ideal in the algebra H ⊗ H generated by β(a) ⊗ 1 − 1 ⊗ α(a) for all a ∈ A, but this kernel is not a two-sided ideal in general. The compatibility (ii) in the definition of a bialgebroid holds iff the map H ⊗ (H ⊗ A H) → H, (g, i h i ⊗ k i ) → ∆(g)( i h i ⊗ k i ) (multiplied in each tensor factor) is well defined (does not depend on the choice of the sum within the equivalence class in H ⊗ A H = H ⊗ H/I A ). In other words, ∆(g) · I A ⊂ I A for all g ∈ H. This last characterization will be useful in the proof of Proposition 5.9.
Lemma 2.2. The composition By similar factorization, we get analogous maps from the multiple tensor products (H ⊗ A · · · ⊗ A H) ⊗ (A ⊗ · · · ⊗ A) → A. For elements we extend the operation syntax h a := (h ⊗ a) to products ⊗ (and multiple analogues), e.g., if F ∈ H ⊗ A H and a, b ∈ A we write Proof . We need to show that the composition (2.2) gives zero on the subspace We show this in terms of the generators. For a, b, c ∈ A, h, h ∈ H we need By the definition of (see (i) above), it is hence enough to show Since is a map of A-bimodules, both sides yield (hα(a)) · c · (h α(b)) ∈ A.

Twists for bialgebroids
In this section, we explain how Ping Xu [25] generalized the Drinfeld twists to bialgebroids (see also [8]), give some examples and focus on the case relevant for deformation quantization. This involves formal completions and completed tensor products and somewhat different formal completions in the later part of the article. For that reason, in this section, we also include an extensive review of completions adapted to our formalism (see also Remark 4.5). Unlike in some other publications, we here use Xu's convention for twists (elsewhere we use F for his F −1 ).
and the counitality ( We use the Sweedler-like notation for the twist F = F (1) ⊗ F (2) (upper labels, while we use lower for the coproduct) and the notation from Lemma 2.2. 25]). If H is a left A-bialgebroid and F ∈ H ⊗ A H a Drinfeld twist, then the formula defines an associative k-algebra A = (A, ) structure on A with the same unit; the formulas α F (a) = α F (1) a F (2) and β F (a) = β F (2) a F (1) define respectively an algebra homomorphism α F : A → H and antihomomorphism β F : A → H with [α F (a), β F (a)] = 0 for all a ∈ A . Given the A -bimodule structure on H by a.h.b := α F (a)β F (b)h and denoting by I A the kernel of the canonical projection H ⊗H → H ⊗ A H, there is an inclusion F ·I A ⊂ I A ⊂ H ⊗H (cf. (2.1)). In other words, F induces a linear map F : H ⊗ A H → H ⊗ A H given by the left multiplication of classes in H ⊗ H by F, that is by defines a map ∆ F : H → H ⊗ A H which is coassociative and counital with the same counit.
In that case, F −1 = F −1 . This will be the situation throughout the article, hence we do not need to distinguish F −1 from the left multiplication with F −1 . In terms of F −1 and A , the cocycle condition (3.1) is equivalent to the condition Thus the cocycle condition (3.1) implies the coassociativity. The converse does not hold for arbitrary bialgebroids. Indeed, let µ 2 = µ(µ ⊗ id) = µ(id ⊗ µ) be the second iterate of the multiplication and F 3a and F 3b the left and the right-hand sides of the cocycle condition (3.1); we can see only that the coassociativity implies is nondegenerate in the first argument, which is not true for an arbitrary bialgebroid. The nondegeneracy holds for the undeformed Heisenberg algebra and by Theorem 4.3 more generally for the U (g)-bialgebroid H g studied in this paper -this is the content of Theorem 4.3 with the consequence for the cocycle condition, Corollary 4.4.
Example 3.5 (basic example of a noncommutative bialgebroid over a commutative base). Let A = C ∞ (M ), where M is a smooth real manifold and let H = D be the algebra of global differential operators with smooth coefficients. The formula ∆(D)(f, g) = D(f · g) defines a bidifferential operator ∆(D) ∈ D ⊗ C ∞ (M ) D; this defines a cocommutative coproduct ∆. The base C ∞ (M ) is commutative and α = β is the canonical embedding of the algebra of functions into the algebra of differential operators.
This example generalizes. The algebra of differential operators D is canonically isomorphism to the universal enveloping algebra of the Lie algebroid given by the tangent bundle V = T M where the anchor map a is the identity. Universal enveloping algebras of Lie algebroids are bialgebroids in a canonical way. To explain this in few lines let us say that a Lie algebroid L = (V, a, Thesis [12] settled the question when bialgebroid structures on universal enveloping algebras of Lie algebroids, and more generally of Lie-Rinehart algebras, have an antipode (see Section 6), namely iff a flat connection in the appropriate context exists. In [25] an erroneous argument (confusing the role of the antipode with the duality for functional spaces) was given that the bialgebroids of differential operators can not have an antipode. The special case of the Heisenberg-Weyl algebra in this article of course has an antipode.
Remark 3.6 (notation on completions). Given a vector space V an N-cofiltration of V is given by specifying for all n ∈ N the quotient spaces V n called cofiltering components with projections π n : V → V n , connecting quotient maps π nm : V m → V n for all n ≤ m, satisfying π nm • π mk = π nk for all n ≤ m ≤ k, and commuting with the projections, π nm • π m = π n . One considers the completionV as the inverse limit lim ←− n V n which may be realized by threads, that is sequences (v n ) n∈N where v n ∈ V n and π nm (v m ) = v n for all n ≤ m. The canonical map V →V = lim ←− n V n , v → (π n (v)) n is injective iff for every v ∈ V there is n ∈ N such that π n (v) = 0. In that case, we usually identify V with its image withinV . If the canonical map V →V is an isomorphism we say that V is complete. A cofiltration is often given by a decreasing filtration, that is a decreasing sequence of subspaces V n with empty common intersection; the quotients are given by V n = V /V n and the inclusions V n → V n+1 induce the quotient maps V /V n+1 → V /V n . If all V n are of finite codimension in V , then the completion may be understood in the topological sense (consisting of all equivalence classes of Cauchy sequences), where V is understood as a topological vector space with linear topology for which the subspaces V n form a basis of neighborhoods of 0. Given two cofiltered spaces V and W , there is a completed tensor product also adorned with the hat symbol, V⊗W = lim ←− n,m V n ⊗ W m , where the inverse limit is over the directed set N × N; the spaces V n ⊗ V m together with the connecting morphisms form an N × N-cofiltration [24], which can be replaced, in the generality of this paper, by an equivalent N-cofiltration [18, Appendix 2]. Given a cofiltered vector space V , an expression λ∈Λ v λ where v λ ∈ V is a profinite (or formal) sum [24] if for each n there are only finitely many λ ∈ Λ such that π n (v λ ) = 0. Then the finite sums s n = λ,πn(λ) =0 π n (v λ ) ∈ V n define a thread (s n ) n∈N which is therefore an element of the completionV called the value v of the profinite sum λ∈Λ v λ and we write v = λ∈Λ v λ . A linear map f : V → W between cofiltered vector spaces distributes over profinite sums (equivalently: is a cofiltered map) if for each profinite sum λ∈Λ v λ in V the expression λ∈Λ f (v λ ) is profinite in W and the equation f λ∈Λ v λ = λ∈Λ f (v λ ) holds for the values; cofiltered spaces (even if noncomplete) with cofiltered maps form a monoidal category with the usual tensor product ⊗. Similarly, the subcategory of complete cofiltered spaces is a monoidal category with respect to the completed tensor product⊗. Each cofiltered map f : V → W induces the morphisms among the limits of cofiltrations, which can be interpreted as the completed cofiltered mapf :V →Ŵ , the completion of f , which commutes with the canonical maps profinite sums whose summands are in Z their values are also in Z. The cofiltered maps among the complete vector spaces for which the topological interpretation holds, coincide with the continuous linear maps. Algebra in the monoidal category of cofiltered vector spaces with the usual tensor product is said to be a noncomplete cofiltered algebra. The modules and ideals of complete algebras should be completed as well, the fact which we often pass over silently. In particular, if A is a noncomplete cofiltered algebra, M a right A-module and N a left A-module, such that the actions are cofiltered maps, the kernel and its completion is identified with a subspaceÎ A ⊂ M⊗N . If the actions on M and N are cofiltered maps, then the vector space M⊗ A N = M⊗N/Î A is cofiltered, and called the completed tensor product of M and N over A, see [24]. The multiplication of A is cofiltered, hence it extends to a cofiltered mapÂ ⊗Â →Â and even to a unique cofiltered map A⊗A ∼ =Â⊗Â →Â; thusÂ becomes an algebra in the category of complete cofiltered vector spaces with the completed tensor product. In this situation, if M is a right A-module and the action is a cofiltered map A ⊗ M → M , thenM is a right complete cofilteredÂmodule, in other words the action is a cofiltered mapÂ⊗M →M (note the complete tensor product), hence by restriction a fortiori an A-module. It is straightforward to observe that . Therefore, if U is the (faithful) forgetful functor from the category ofÂ-bimodules with the completed tensor product to the category of the usual A-bimodules, then there is a natural isomorphism of functorŝ  Again, ∆(D)(f, g) = D(f · g), the base A is commutative and α = β is the canonical inclusion of elements of A as the operators of left multiplication. Here and in Example 3.5 the counit is taking the constant term and is the usual action of differential operators on functions (below denoted or as evaluation D(f ) = D f ). If M is the affine n-space, H is the Weyl algebra A n and the corresponding coproduct ∆ 0 is primitive on the generators ∂ 1 , . . . , ∂ n and ∆ 0 (x µ ) = x µ on the remaining generators x 1 , . . . , x n of the Weyl algebra. This O(M )-bialgebroid is in fact a Hopf algebroid with the antipode S(x µ ) = x µ and S(∂ ν ) = −∂ ν . We may complete A n by the degree of a differential operator to the algebraÂ n and the Hopf algebroid structure continuously extends ( [18]) to a topological coproduct onÂ n with values in the completed tensor product A n⊗AÂn relative over A = k[x 1 , . . . , x n ], By abuse of notation, we denote by ∆ 0 also the restriction of ∆ 0 to the polynomial algebra of ∂ 1 , . . . , ∂ n (or to its completion k[[∂ 1 , . . . , ∂ n ]], the algebra of formal power series), which therefore becomes a Hopf k-algebra (respectively, a topological Hopf k-algebra).
Example 3.8 (related to deformation quantization). Ping Xu [25] extends the base algebra  We are interested in using the bialgebroid techniques to find explicit formulas for F and also to describe the Xu's bialgebroid in detail in special cases. There are rather few explicit formulas for bialgebroid twists in the literature which are not induced from bialgebra twists by a procedure studied in [6].

Phase spaces of Lie type as bialgebroids
In this section, we present the bialgebroid H g from our earlier article [18]. This bialgebroid is in fact a Hopf algebroid, but the discussion of the antipode is deferred to Section 6.
Throughout, g is a fixed Lie algebra over k with basisx 1 , . . . ,x n , U (g) is the universal enveloping algebra and S(g) the symmetric algebra of g. The generators of U (g) are also denotedx 1 , . . . ,x n (and viewed as noncommutative coordinates), but the corresponding generators of S(g) are x 1 , . . . , x n (and viewed as coordinates on the undeformed commutative space). The structure constants C λ µν are given by Let ∂ 1 , . . . , ∂ n be the dual basis of g * , which are also (commuting) generators of S(g * ). LetŜ(g * ) be the formal completion of S(g * ) (which should be interpreted as the algebra of formal functions on the space of deformed momenta). A basis of neighborhoods of 0 is made out of powers of the ideal J = g ·Ŝ(g * ) generated by g (J-adic topology of the formal power series ring). The ring M n (Ŝ(g * )) of n × n-matrices with entries inŜ(g * ) is rank n 2 freeŜ(g * )-module and as such it inherits the topology of componentwise convergence. By a straightforward check we can see that this topology is equivalent to the M n (J)-adic topology, where M n (J) = M n (g ·Ŝ(g * )) consists of all matrices with entries in g ·Ŝ(g * ). Therefore M n (Ŝ(g * ) is a linearly compact topological ring as well. We introduce an auxiliary matrix C ∈ M n (Ŝ(g * )) with entries with the summation over repeated Greek indices and where, for the purposes of this article, we use the convention that α is the row and β the column index. In this notation, we introduce matrices where B N are the Bernoulli numbers. By a simple comparison of the expressions (4.2) we obtaiñ ByÂ n denote the completion by the degree of differential operators of the n-th Weyl algebra A n with generators x 1 , . . . , x n , ∂ 1 , . . . , ∂ n hence the underlying vector space of A n is S(g) ⊗ S(g * ) andÂ n is some completion of it. The symmetric algebra S(g) is a Hopf algebra which acts onŜ(g * ) by a unique action δ : S(g) → End k (Ŝ(g * )) which is a Hopf action, namely and which satisfies δ(x ν )(∂ ν ) = δ ν µ on the generators. It is a useful point of view for the deformations below that the product in A n is the multiplication of the smash product algebra given by the formula The algebraÂ n acts on the symmetric algebra S(g) of polynomials in x 1 , . . . , x n , extending the Fock action of A n on S(g) by differential operators (P, With our choice of the matrix φ, the right-hand side in the equations (4.4) are dual, as explained and derived in [18], Section 2, to the expressions for the left and right invariant vector fields respectively, corresponding to the Lie algebra generators, in the chart given by the exponential map. The expressions for φ τ ρ ,x φ ρ andŷ φ ρ were derived many times historically, most notably by Berezin in the geometric context [2] and the corresponding expression for the star product studied in C ∞ -context has been introduced by Gutt [10]; the supersymmetric generalization of the formula in the algebraic context and in the dual language of coderivations (rather than derivations and vector fields here) is in [20]; finally the article [9] gives three proofs of the formula valid over any ring containing rationals: a direct calculation, a proof in formal geometry over arbitrary rings and an algebraic proof close in spirit to the approach in [20]. The expression for φ corresponds to the part of Baker-Hausdorff series linear in one of the variables. The same star product is often defined for exponentials via the full Baker-Hausdorff series [1,11,19].
From (4.3) it follows immediately that and one can also prove (see, e.g., [18,Appendix 1]) In the geometric interpretation, equation (4.6) means that all left invariant vector fields on a Lie group commute with all right invariant vector fields. Thenx ρ →x φ ρ extends to a unique algebra map (−) φ : U (g) →Â n . This realization map (−) φ is related to the symmetrization (PBW) isomorphism where Σ(r) is the symmetric group on r letters, in the sense that ξ −1 (u) = u φ (1) (φ-realization of u acting on 1) and more generally u · ξ(f ) = u φ (f ) for all u ∈ U (g), f ∈ S(g), hence the star product f g := ξ −1 (ξ(f ) · U (g) ξ(g)) on g may be written as f g = ξ(f ) φ (g). Thus, in physics literature, our choice of φ is said to correspond to the symmetric ordering. Analogously, other coalgebra isomorphisms S(g) ∼ = U (g) identical on g correspond to different choices of φ or to different orderings, see [17]. Our φ corresponds to the symmetric ordering [9]. The formula φ(x µ )(∂ ν ) = φ ν µ defines a linear map φ(x µ ) : g * →Ŝ(g * ) which extends by the chain rule to a (equally denoted) unique continuous derivation φ(x µ ) :Ŝ(g * ) →Ŝ(g * ). Denote by Derc(Ŝ(g * )) the Lie algebra of all continuous derivations ofŜ(g * ). A key property of φ is that the linear map φ : g → Derc(Ŝ(g * )) extending the rulex µ → φ(x µ ) ∈ Derc(Ŝ(g * )) is a Lie algebra homomorphism, hence it further extends to a unique right Hopf action which we denoted by the same symbol. The property that this action φ is Hopf ensures that we can define the smash product algebra in the usual way: it is the tensor product of vector spaces U (g) ⊗Ŝ(g * ) with the multiplication where, for the emphasis, the expression u f denotes u ⊗ f as an element in the smash product (whenever the meaning is clear the simpler product notation uf may be used as well). One can check that the algebra map H g →Â n , which on elements of U (g) k → H g agrees with the realizationx µ →x φ µ and sends 1 ∂ β ∈ k ⊗Ŝ(g * ) ⊂ H g to ∂ β ∈ A n , is an algebra isomorphism; hence from now on this algebra map H g →Â n is viewed as an identification and sometimes called realization as well. We thus identifyx µ ∈ U (g) andx φ µ ∈Â n etc. The isomorphism of coalgebras ξ : S(g) → U (g) induces the dual isomorphism of algebras ξ T : U (g) * → S(g) * ∼ =Ŝ(g * ) hence the map µ T dual to the multiplication µ : U (g)⊗U (g) → U (g) can be identified with a deformed comultiplication ∆Ŝ (g * ) = ξ T ⊗ξ T •µ T :Ŝ(g * ) →Ŝ(g * )⊗Ŝ(g * ) (the completion on ⊗ comes from (co)filtrations, see [18]). The source map α : U (g) → H g is given by u → u 1. Introducê In the realization H g ∼ =Â n , this expression becomesŷ φ µ . The rulex µ →ŷ µ extends to a unique algebra map β : U (g) op → H g , the target map. Equation (4.6) implies that [α(u), β(v)] = 0 for all u, v ∈ U (g). We also often identify u with its image α(u) and P ∈Ŝ(g * ) with 1 P ∈ H g . Thus, H g becomes a U (g)-bimodule via u.h.v = α(u)β(v)h. It has been shown in [18] that, with the appropriate completions, H g is a formally completed version of a Hopf U (g)-algebroid with the coproduct Notice that in the undeformed case (that is, when g is Abelian) the (topological) bialgebroid H g is some completion of the Weyl algebra A n of regular differential operators on k n where A n is viewed as a bialgebroid over the commutative base algebra consisting of polynomial functions on k n . Hence, the undeformed H g is very close to the bialgebroid of differential operators over a smooth manifold.
Notation. In the case of deformed bialgebroid H = H g , the action will be denoted and in the undeformed case . The action : H g ⊗ S(g) → S(g) is the "Fock action"ofÂ n by differential operators on polynomials precomposed by the realization isomorphism H g ∼ =Â n .
Proposition 4.1. Given u, v ∈ U (g), f, g ∈ S(g) the following formulas hold: ,ŷ µ g u = ux µ and more generally β(v) g u = uv. The last two formulas implyŷ Proof . These properties are at length discussed and proved in our articles [17,18,22]. The actions g and are there denoted and respectively.
Proof . I U (g) is generated by β(u)⊗1−1⊗α(u) where u ∈ U (g) and it is enough to confine to the generators u =x µ of U (g). Then β( of the universal enveloping algebra U (g), considered as a bialgebroid over A = U (g), the right ideal I (k) which is the kernel of the canonical projection H ⊗ · · · ⊗ H → H ⊗ A · · · · · · ⊗ A H (k factors, all tensor products properly completed) coincides with the right ideal I (k) consisting of all elements r in H ⊗ · · · ⊗ H such that µ k−1 (r( g ⊗ · · · ⊗ g )(a 1 ⊗ · · · ⊗ a k )) = 0 for all a 1 , . . . , a k ∈ H.
For an arbitrary bialgebroid one can easily show that I (k) ⊂ I (k) , but the converse inclusion is not present in general. Proof . This follows from the coassociativity ∆ H by the reasoning at the end of the Remark 3.4 using that the nondegeneracy Theorem 4.3 holds in the deformed case. Of course, the nondegeneracy is well known in the undeformed case. For example, if k = 2, then up to the issue of completions, it boils down to the basic fact that the Weyl algebra as defined by generators and relations is faithfully represented on the space of polynomials.
In this paper, we need the corollary in the undeformed case, as we study below the twists F l , F r on the undeformed H =Â n . If F is a candidate for a twist on undeformed H then the cocycle condition for F −1 (as a twist on deformed H g ) may be derived using Theorem 4.3 in the nontrivial deformed case. This is useful as the cocycle condition for some examples of F or F −1 is studied in the literature without previous knowledge of the cocycle condition for F.
In [17], we proved another formula for a twist deforming the coproduct ∆ 0 onÂ n into the coproduct for H g (although without bialgebroid formalism) which is however less explicit: as it is a series, where each summand involves ∆Ŝ (g) which is itself a series and not very explicit. Symbolically, using the normal ordering (x-s to the left, ∂-s to the right) operation : : one can write this formula as the normally ordered exponential   [18] are, roughly speaking, with respect to the cofiltrations on U (g) * andŜ(g * ) induced by duality from the standard filtrations on U (g) and S(g). This is essentially different from using the additional formal variable h and h-adic completions as in Xu's work [25]. Naively, to fit with his work, the Lie algebra generators (or equivalently the structure constants) should be simply rescaled by the formal variable h. For many simple purposes this gives an equivalent treatment to ours. The set of series which formally converge in two variants differs however. For the main results in the present article this is important. Namely, the twists F l , F r in Section 5 exist in both completions, the formulas as products of two exponentials make sense in our formalism, but these individual exponential factors do not exist in the h-adic completion, even after rescaling. Indeed, exp(∂ α ⊗ x α ) does not involve a small parameter (the reason is that ∂ α and x α are dual and no rescaling could make their tensor product small) and is in fact related to an infinite-dimensional version of the canonical element, while (due cancellations in the expansion) the entire twist F l equals 1 plus a series of corrections involving the small parameter. Thus, unlike the exponential factors, the final result F l does exist in both formalisms and hence can be interpreted as defining a deformation quantization in the sense used in Theorem 3.9.
Remark 4.6. The smash product algebra U (g) φŜ (g * ) can be equipped with another bialgebroid structure, namely over the commutative base algebraŜ(g * ) and with the comultiplication ∆ for which ∆ (u P ) = (u (1) 1) ⊗Ŝ (g * ) (u (2) P ) for u ∈ U (g) and P ∈Ŝ(g * ), where u → u (1) ⊗ u (2) denotes the (standard) cocommutative coproduct of U (g). While our coproduct ∆ takes values in a completed product over the noncommutative base, the coproduct ∆ is algebraic, taking values in the ordinary tensor product over the commutative base. Such scalar extensions over a commutative base were known much before [21, pp. 117-118]. This bialgebroid does not fit the physical interpretation which we intended. The coproduct onŜ(g * ) in our bialgebroid structure in this paper is dual to the algebra structure on U (g) manifest in both algebroids and the coproduct on U (g) in the other bialgebroid structure is dual to the algebra structure onŜ(g * ) manifest in both algebroids. The two entire bialgebroids are mutually not dual in some sense standard for bialgebroids, although their factors U (g) andŜ(g * ) are involved in (topological) Hopf algebra dualities.

New twist
In this section, we show that the two expressions F l and F r define (the same) twist of a completed Heisenberg double and that the twisted bialgebroid is H g from the previous sections. (i) In symmetric ordering, the deformed coproduct ∆Ŝ (g * ) onŜ(g * ) ∼ = U (g) * is given by where the right-hand side, though an element ofŜ(g * )⊗Ŝ(g * ), is calculated in the bigger algebra H⊗H (the completed tensor product over the ground field). The same equation written in a symbolic form is Proof . Part (i) is proven in [17] (and used in [22]).
(ii) To extend the identity to the formal power series just notice that the conjugation is a homomorphism of completed algebrasĤ ⊗Ĥ →Ĥ(the usual non-completed tensor product of completions). More explicitly, the deformed coproduct of the n-th order monomial in ∂-s has all summands in order n or higher. If in a formal series we replace each monomial with a formal series with nondecreased minimal order then we obtain only finitely many summands of each finite order hence again a formal series.
Proof . It is known [17] that the deformed coproduct ∆Ŝ (g * ) (∂ µ ) ∈Ŝ(g * )⊗Ŝ(g * ) has the same symmetries as the Hausdorff formula H(Z, W ) = −H(−W, −Z); thus if we interchange the tensor factors in the coproduct and multiply each partial derivative with −1 then we get the same coproduct with the overall minus sign. This rule of course applies only when the coproduct is written manifestly in terms of tensor products of formal power series in partial derivatives.
Changing the sign of each partial derivative in the realizationsx α manifest in (5.1) is equivalent to the change of φ toφ (because partial derivatives are contracted to the structure constants so it is the same as change of the sign in the structure constants), that isx α toŷ α . However, the commutators like [∂ µ ,x α ] when calculated have a different number of partials in each monomial than the raw expression. Indeed, one partial derivative and one power of x (recall thatx ρ = x σ φ σ ρ where φ σ ρ involves partial derivatives) drop out when commuting; similarly in each of the further commutators one power of x and one of the partial derivatives within theφ-s drops out. As the number of commutators in each of the left tensor factors in (5.1) is by one less than the number of partial derivatives in the raw tensor product expression, together with overall minus sign it amounts to no change in sign besides the change accounted inx µ →ŷ µ . Thus The proof that the result extends to general P ∈Ŝ(g * ) is completely analogous to the proof of part (ii) in Theorem 5.1.
The right-hand sides of the formulas (5.3), (5.4) understood as elements in H⊗H defineF l ,F r ∈ H⊗H with explicit inverses Clearly, F l =F l + I 0 , F r =F r + I 0 . We define also F −1 l :=F −1 l + I U (g) , F −1 r :=F −1 r + I U (g) (see (2.1)), but at this point we can not yet claim that they are twist inverses in the sense of Remark (3.3) as the expressions F l F −1 l etc. should be well defined what is proven only below.
. . , n. More generally, for any P ∈Ŝ(g * ) (formal power series in ∂ 1 , . . . , ∂ n ), ∆Ŝ(g * )P =F −1 Proof . By the Hadamard's formula exp(A)B exp(−A) = exp(ad A)(B), (5.1) and (5.2) read In particular, in the undeformed case when C λ µν = 0 andx α , x α andŷ α coincide, we obtain the formulas for the undeformed coproduct ∆ 0 (which can also easily be checked directly) Comparing the formulas for the deformed and for the undeformed case we obtain new formulas relating ∆ 0 to ∆Ŝ (g * ) (g * ). Indeed, comparing (5.5) and (5.7) we obtain and similarly comparing (5.6) to (5.8) we obtain To extend the identities to the formal power series proceed as in the proof of part (ii) to Theorem 5.1.
We would like to have the same identities in H⊗ U (g) H, with F instead ofF, but for this the calculation should not depend on a representative, that is F −1 L · I 0 ⊂ I U (g) , which is proven below in Proposition 5.8.
Finally, one proves that the undeformed right ideal I 0 generated by x µ ⊗ 1 − 1 ⊗ x µ after twist ends in the deformed right ideal I U (g) . In fact, Proposition 5.8.
Proof . The first line follows by subtracting (5.10) from (5.9) and the second line by subtracting (5.11) from (5.12). The third line follows by multiplying the first byF −1 r and multiplying the second line byF −1 r . The fourth line is clearly just the restatement of the third in the quotient.
F −1 l I 0 ⊂ I U (g) follows from the first line by noticing that the elements x µ ⊗ 1 − 1 ⊗ x µ , µ = 1, . . . , n generate I 0 and similarlyF −1 r I 0 ⊂ I U (g) follows from the second line. To showF l I U (g) ⊂ I 0 multiply the first line withF l from the left to obtaiñ and note that this is sufficient because the elements of the form Similarly we multiply the second line withF R from the left to obtaiñ This is sufficient to concludeF R I U (g) ⊂ I 0 after observing that the elements of the formŷ andFI U (g) ⊂ I 0 together imply the equality wheneverF −1 andF are strict inverses in H⊗H.
The assertions on F l , F r are the direct corollary: the products F l F −1 l , F r F −1 r , F −1 l F l , F −1 r F r are well defined so we can compute the representatives usingF l ,F r ,F −1 l ,F −1 r .
Proposition 5.9. For every h ∈ H = H g , Proof . By Proposition 5.8 the expressions F −1 l ∆ 0 (h)F l and F −1 r ∆ 0 (h)F r are well defined (do not depend on the representative of ∆ 0 (h)). We can compute a representative of the resulting class modulo I U (g) asF −1 ∆ 0 (h)F where ∆ 0 (h) is any representative of ∆ 0 (h) and F is F l or F r . This way the statement of the proposition for the generators h = ∂ µ follows by Lemma 5.4 and for the rest of generatorsx α by Lemma 5.7. We need to extend the statement for all h ∈ H by linearity and some sort of multiplicativity. Some care is however needed to achieve this. Namely F is a homomorphism of algebras H⊗H → H⊗H and by F −1 · I 0 = I U (g) it induces a well defined map of vector spaces F −1 (−)F : H⊗ S(g) H → H⊗ U (g) H. Regarding that H⊗ U (g) H is not an algebra, we can not say that this map is a homomorphism of algebras. Suppose h 1 , h 2 ∈ H are such that F −1 ∆ 0 (h i )F = ∆ H (h i ). Regarding that the images of ∆ H and ∆ 0 are algebras, that ∆ H , ∆ 0 viewed as corestrictions to the images are multiplicative, and FF −1 = 1 ⊗ 1 + I 0 , we calculate which would be sufficient to end the proof. However, we freely used associativity and cancellations though the factors do not belong to an associative algebra. Associativity holds for the representatives in H⊗H, hence it is enough that all the products involved are well defined up to an appropriate right ideal. In our case we inspect this sequentially from the right to the left for the products involved, using FI U (g) = I 0 , F −1 I 0 = I U (g) , ∆ 0 (h)I 0 ⊂ I 0 , ∆ H (h)I U (g) ⊂ I 0 and that I 0 , I U (g) are right ideals. Lemma 5.10 ([17]). In symmetric ordering, exp α t α x α g 1 = exp α t αxα for any formal variables t α which commute with x β ,x β . Conversely, exp α t αxα 1 = exp α t α x α . In particular, (5.21) Theorem 5.11. F l and F r are counital Drinfeld twists for S(g)-bialgebroid on completed Weyl algebraÂ n and by twisting they yield the Heisenberg double H g of the corresponding universal enveloping algebra U (g) with its canonical U (g)-bialgebroid structure.
Proof . Both F l and F r are invertible. Proposition 5.9 and Corollary 4.4 together imply that the Drinfeld cocycle condition (3.1) holds. We need to show the counitality. One has to be careful when checking this, regarding that : H g → U (g) is not a homomorphism. However, for the symmetric ordering, checking this is still not a difficult. Recall from the axioms of the bialgebroid that the undeformed counit is given by (h) = h 1. Thus because all the higher order terms have positive power of at least some ∂ α -s thus yielding zero when acting upon 1. The second counitality condition is a bit more involved; using the fact that extends the regular action of U (g) on itself and (5.21) we compute In the third line we used Lemma 5.10. Similarly, one shows that F r is counital.
The new base algebra is S(g) with the F l -twisted (or F r -twisted) star product. We need to show that it is canonically isomorphic to U (g) in the sense that where ξ : S(g) ∼ = → U (g) is the symmetrization map. Since we know that F l , F r are Drinfeld cocycles, we know that in both cases the corresponding star product is associative. Regarding that ξ transports the product in U (g) to the star product g f := ξ −1 (ξ(g) · U (g) ξ(f )) = ξ(g) f every element in S(g) is a star polynomial in the generators in g. Therefore the associativity and the star products of the form x µ f with general f (or alternatively, all star products of the form g x µ ) are sufficient to determine the star product g f for general g, f . Thus for F l it is sufficient to check that µF l ( x µ ⊗ f ) =x µ f =x φ µ (f ) for all f and µF r ( g ⊗ x µ ) =ŷ µ g =ŷ φ µ (g) for all g. When acting by we take the realization in the Weyl algebra and apply the corresponding differential operator. Thus all the higher derivatives drop out when acting on x µ and we obtain After applying µ on this equality, the first two summands cancel and we obtainx φ µ (f ). Similarly, F r ( g ⊗ x µ ) =ŷ µ (g) ⊗ 1 − g ⊗ x µ + x µ g ⊗ 1 and after applying µ the second summand cancels with the third and we obtain µF r ( g ⊗ x µ ) = y µ g = g x µ as required. Finally, we need to compute the deformed source and target map from the twist and compare them with the known source and target map in the deformed case. Regarding that we already know that F l and F r are Drinfeld twists, Xu's theorem tells us that α F l , β F l , α Fr , β Fr are algebra maps by the construction, it is sufficient to check the agreement with known deformed source and target maps on the algebra generators x µ where α 0 = β 0 is the source map in the undeformed case, which is then equal to the undeformed target map. When f = x µ , after expanding the exponentials only three summands survive, with at most one partial derivative before applying . Thus we obtain α F L (x µ ) = α(x µ ) − x µ +x µ = x µ = α(ξ(x µ )) as required. For β we first interchange the tensor factors in F l , Regarding that the second tensor factor commutes, we can compute exp(−x ρ ⊗ ∂ ρ ) exp(x σ ⊗ ∂ σ )( x µ ⊗ 1) as if ∂-s are independent formal variables. Now exp(k σx σ ) x µ = exp(k σ x σ ) x µ =ŷ µ exp(k σ x σ ) = x τφ τ µ (k) exp(k σ x σ ). Then, the exponential factors cancel and we get β F l (x µ ) = x τ φ τ µ (∂) =ŷ µ = β(ξ(x µ )) as required. α For f = x µ only the three summands with partial derivatives up to the first order survive after applying x µ . Thus as required.
Proof . In the proof of Theorem 5.11 we have shown µF l ( ⊗ )(f ⊗g) = f g = µF r ( ⊗ )(f ⊗g) for all f, g ∈Ŝ(g). ThusF l −F r ∈ I U (g) from the undeformed case of the Theorem 4.3 on nondegeneracy. Therefore for the cosets we conclude F l =F l + I U (g) =F r + I U (g) = F r .

Twisting the antipode
In this section, we discuss the twisting of the antipode. We first recall the definition of Hopf algebroids as bialgebroids with an antipode and then we recall from [18] the antipode for H g . Several nonequivalent versions of the axioms for the antipode are used in the literature (see, e.g., [4,5,13,23]). In [18] we checked for H g the axioms of the symmetric Hopf algebroid, which involve both a left and a right bialgebroid. Thus a reasonable formalism for the twisting of symmetric Hopf algebroids is expected to provide a twist for the left and another twist for the right bialgebroid, these twists satisfying some compatibilities. Instead of taking this not so obvious path, we here work with a version of the axioms for the antipode involving only the left bialgebroid. We can do this because the antipode for H g is invertible both in the undeformed and deformed case. Namely, if the antipode is invertible, as proven by Böhm, the structure of a symmetric Hopf A-algebroid H is equivalent to a left A-bialgebroid together with an antipode S which is an algebra antihomomorphism H → H having an inverse S −1 satisfying for all h ∈ H the relations If a bialgebroid twist F = F (1) ⊗ A F (2) on a Hopf algebroid H has an inverse cocycle  (2) is not the inverse of V , but worse, is not a well defined expression because µ(id ⊗ S)I A = 0. We do not know if the inverse of V F exists in general.
Proposition 6.1. Suppose that V F = SF (1) F (2) has an inverse V −1 F in H. Then define The formula h → S F h then defines an antihomomorphism of algebras S F : H → H and where the right-hand side is well defined and in particular µ(S F ⊗ id)I A = 0.
Proof . S is an antihomomorphism hence it is clear that S F given by (6.1) is an antihomomorphism as well. It follows that SF (1) SF (1) F (2) F (2) = 1. Here the primed Sweedler indices (1) , (2) refer to another copy of F Regarding that the two-sided inverse in an associative algebra is unique, we conclude (6.2) with the right-hand side in (6.2) well defined.
One would like to conclude that S F is an antipode for the twisted bialgebroid. The standard proofs for the Hopf algebras do not seem to generalize in straightforward manner.
However, in our case, for H g , we know the deformed antipode S, and one can say a bit more. For H g ∼ =Â n , the antipode S 0 for the undeformed coproduct is a continuous antihomomorphism S 0 : H g → H g given on the generators x µ , ∂ ν of the dense subalgebra A n by S 0 (x µ ) = x µ , S 0 (∂ ν ) = −∂ ν . Similarly, the antipode S : H g → H g for the deformed coproduct is determined by the formulas where we denoted ∂ ρ = ∂ ∂(∂ ρ ) . We seek for V such that S(h) = V −1 S 0 (h)V , in parallel to equation (6.1) defining S F in terms of V F . Regarding that S(∂ µ ) = S 0 (∂ µ ) = −∂ µ , this forces that [∂ µ , V ] = 0, hence V = V (∂ 1 , . . . , ∂ n ) ∈Ŝ(g * ). Moreover, Setting F µ = φ −1γ µ ∂ ρ φ ρ γ , we rewrite (6.3) as the system of formal differential equations for an unknown formal series R. In our case, F µ are analytic functions, hence the solution exists if the integrability condition ∂ ν F µ = ∂ µ F ν holds, which boils down to Any solution for V = exp(ln |V |) = exp(R) is clearly invertible with an inverse V −1 = exp(−R). The constant term V (0) of V which is viewed as a formal power series is nonzero, hence V (0) is also invertible. Therefore we can write . It is clear that V 1 is also a solution and the identity V 1 (0) = 1 holds. We conclude that without loss of generality (by passing from V to V 1 ) we may assume that V (0) = 1 and such a solution for V is unique. The condition V (0) = 1 is equivalent to the boundary condition R(0) = 0 for the "potential" R when solving the exact first order differential equation (6.4). This boundary condition guarantees the uniqueness of the solution for R. If, instead of the abstract algebra elements ∂ ν , we introduce the real variables ξ ν , we can write the solution formally R ζ 1 , . . . , ζ n = Γ F µ ξ 1 , . . . , ξ n dξ µ , where the line integral is along any path Γ from (0, . . . , 0) to (ζ 1 , . . . , ζ n ). Due to the integrability (6.5), the line integral does not depend on the path chosen.
We have some evidence that S F = S or equivalently that V F satisfies the equations for V . First of all, Proposition 6.2 below says that the element V F ∈Â n is in fact the formal power series in ∂ 1 , . . . , ∂ n (that is x 1 , . . . , x n do not appear) making sense of the equations for V . V F clearly satisfies the required boundary condition V F (0) = 1. It follows that V F is invertible, as required in Proposition 6.1, particularly in equation (6.1) defining S F . Using the Baker-Hausdorff formula for computing R F = ln |V F | directly from the definition V F = µ(S 0 ⊗ id)F, we have perturbatively checked up to the third order in ∂-s that R F satisfies the system (6.4) for R. Proposition 6.2. For F being F c = F l = F r , the element V F ∈ H g is a formal power series in ∂ 1 , . . . , ∂ n . As a corollary, S F (∂ ν ) = S 0 ∂ ν = S∂ ν .
The corollary part of the proposition follows by the defining formula (6.1) for S F recalling that ∂ 1 , . . . , ∂ n mutually commute. The first statement in Proposition 6.2 is the content of parts (v) and (vi) of the following lemma restated. (i) (S 0 ⊗ id)F l ∈ H g ⊗ H g can be written as exp(W ) where W is a formal sum of summands which are up to a rational factor equal to ∂ α 1 · · · ∂ αs ⊗ L(w α 1 , . . . , w αs ), where s ≥ 1 and w α i is either x i orx i (the choice depending on i and on the summand) and L is a Lie monomial (L differing from a summand to summand), more precisely an iterated Lie bracket involving each of the variables precisely once.
(ii) Easily follows by induction, computing inÂ n .
For the induction step one checks that if (iv) holds for ζ and ζ then it holds also for ζ .
Moving x-s to the left by commuting we bring this expression to the normally ordered form. By (iv), a nonzero contribution can possibly happen only from those terms in which each bracket of partials on the left has lost at least one power by yielding a Kronecker delta when commuted with some x. There is only one x per bracket on the right-hand side; because the ∂-groups on the left and the xζ-groups on the right match, that means that all x-s disappear in the normally ordered form.
(vi) While this is trivially equivalent to the statement in (v), there is an independent but similar proof for F c , using the defining formula (4.8) and the form of the tensors appearing in developingF c , as described in [17]. In particular, one needs to observe that for every s ≥ 1 the expression ∂ µ 1 · · · ∂ µs · [· · · [∂ λ ,x µ 1 ], . . . ,x µs ], which appears as a summand in ∆Ŝ (g * ) ∂ λ , vanishes. To this end, one uses the description in [17] of the tensors which appear in developing [· · · [∂ λ ,x µ 1 ], . . . ,x µs ]. his remarks on the paper and M. Stojić for remarks on Sections 1 and 2. We thank the referees for bringing to our attention numerous constructive suggestions, which helped extending and improving the article significantly.