{'fraction_non_alphanumeric': 0.05776415767600017, 'fraction_numerical': 0.010788799798497124, 'mean_word_length': 3.832048681541582, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, '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}}
Quantization of Hamiltonian coactions via twist

In this paper we introduce a notion of quantum Hamiltonian (co)action of Hopf algebras endowed with Drinfel'd twist structure (resp., 2-cocycles). First, we define a classical Hamiltonian action in the setting of Poisson Lie groups compatible with the 2-cocycle stucture and we discuss a concrete example. This allows us to construct, out of the classical momentum map, a quantum momentum map in the setting of Hopf coactions and to quantize it by using Drinfel'd approach.


Introduction
Deformation quantization has been introduced by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer in [3] and since then many developments occurred. A (formal) star product on a Poisson manifold M is defined as a formal associative deformation of the algebra of smooth functions C ∞ (M ) on M . Existence and classification of star products on Poisson manifolds has been proved via formality theory in [17]. In the same spirit, Drinfel'd introduced the notion of quantum groups as deformations of Hopf algebras, whose semiclassical limit are the so-called Poisson Lie groups which are Lie groups with multiplicative Poisson structures (see e.g. the textbooks [8,22] for a detailed discussion).
In this paper we focus on particular classes of star products which are induced by a (formal) Drinfel'd twist by means of universal deformation formulas (UDF) as discussed e.g. in [9,10]. Roughly speaking, a Drinfel'd twist is an element F ∈ U(g) ⊗ U(g) compatible with the Hopf algebra structure on U(g). Given a Hopf algebra action of U(g) on an associative algebra one can deform the U(g)module algebra and the deformed product turns out to be a star product. It is important to stress that the U(g)-module algebra is automatically endowed with a Poisson bracket defined as the semiclassical limit of such star product. In recent works the UDF has been further studied, e.g. [5,7,11,13,15]. Also, a twist defines a 2-cocycle on the Hopf algebra C ∞ (G) and it can be seen that the star products induced via UDF coincide with star products induced by the 2-cocycle on C ∞ (G)-comodule algebras. Finally, a non-formal version of Drinfel'd twist and its corresponding UDF has been discussed in [6].
Given a Lie algebra action ϕ : g → Γ ∞ (T M ) on a smooth manifold M , we can always obtain a Hopf algebra action U(g) × C ∞ (M ) → C ∞ (M ). Thus, Drinfel'd approach can be interpreted by saying that symmetries encoded by Lie algebra actions induce quantization. Also, this approach provides a notion of quantized action. In this paper we prove that this approach is compatible with Hamiltonian actions. In other words, given a classical Hamiltonian action our goal is to quantize it by using Drinfel'd approach and get a notion of quantum momentum map. The problem of quantizing the momentum map has been the main topic of many works, e.g. [14] and [21]. This is motivated by the fact that conserved quantities described via the momentum map lead to phase space reduction which constructs from the high-dimensional original phase space one of a smaller dimension. Thus, it is highly desirable to find an analogue in the quantum setting.
The paper is organized as follows. In Section 1 we recall the notions of Drinfel'd twist and its corresponding 2-cocycle and the construction of the universal deformation formula. Section 2 contains a definition of Hamiltonian actions in the setting of Poisson Lie groups which generalizes the one contained in [19,20]. More precisely, we need to introduce a notion of classical Hamiltonian action which is compatible with twist. It is known that the semiclassical limit of a twist gives rise to an element r ∈ g ∧ g, called r-matrix, satisfying the condition r, r = 0. It can be proved that r-matrices always induces a Lie bialgebra structure on g. Thus, the corresponding Lie group G automatically becomes a Poisson Lie group, since the Poisson tensor obtained by integrating the Lie bialgebra structure on g is multiplicative. The concept of momentum map for Poisson Lie groups acting on Poisson manifolds has been first introduced by Lu in [19,20], in the case in which the Poisson structures of G, its dual G * and M are fixed. In contrast to the ordinary momentum map it takes values in G * and the equivariance is defined in relation to the so-called dressing action of G on G * . Here we introduce a slight generalization and then focus on the case in which, in the same spirit as Drinfel'd, the Poisson structure on G * is induced by r via the dressing action and on M via the action ϕ. A concrete example has been computed, showing the difference between Lu's momentum map and our definition. Finally, in Section 3 we construct a momentum map in the setting of Hopf algebra actions and coactions and study its quantization. More precisely, given a classical Hamiltonian action ϕ : g → Γ ∞ (T M ) with momentum map J : M → G * we construct a corresponding Hopf algebra action and we prove that J * defines a momentum map for this action. This allows us to define the notion of Hamiltonian Hopf algebra action. Motivated by the significance of coactions in the theory of quantum groups in the C * -algebraic framework, we give a dual version of the above result and prove that given ϕ the corresponding Hopf algebra coaction δ Φ : is also Hamiltonian. Finally, using the UDF we obtain the quantized algebras and we prove that the quantum group coaction δ Φ :

Preliminaries
Let g be a (finite-dimensional) Lie algebra and consider the algebra U(g)[[ ]] of formal power series with coefficients in the universal enveloping algebra U(g). It can be endowed with a (topologically free) Hopf algebra structure, denoted by (U(g)[[ ]], ∆, , S).
We sometimes use the notation F = F α ⊗ F α . The semiclassical limit of a twist gives rise to a well-known structure on the Lie algebra g called r-matrix, as proved in [10] or [16,Thm. 1.14]. In facts, we have the following claim.
, the antisymmetric part of its first order is a classical r-matrix r ∈ g ∧ g.
Given a twist we can obtain a deformed Hopf algebra structure on U(g) is again a Hopf algebra denoted by U F (g).
As a consequence, the twist automatically defines a Lie bialgebra structure. Given a twist on the universal enveloping algebra, we can always define a star product on any U(g)-module algebra. In particular, let us consider the algebra C ∞ (M ) of smooth functions on a manifold M with pointwise multiplication m M and a Hopf algebra action This action can be immediately extended to formal power series, allowing the following result.

Lemma 1.4 (Universal deformation formula)
The product defined by is an associative star product quantizing the Poisson structure induced by the semiclassical limit r of F via the action.
We denote the deformed algebra by C ∞ F (M ). It is easy to see that ϕ is automatically a Poisson action. Moreover, it is important to remark that the deformed algebra C ∞ F (M ) is now a left U F (g)-module algebra We can give a dual version of the above discussion by using the notions of 2-cocycles and coactions. A more detailed discussion about 2-cocycles and their duality with twist can be found in [2]. Consider the Hopf algebra C ∞ (G), where G is the Lie group corresponding to the finite-dimensional Lie algebra g. It is known that C ∞ (G) and U(g) are dually paired Hopf algebras algebras with pairing denoted by · , · . Thus, given a twist F there corresponds an element γ : for all f, g ∈ C ∞ (G). Roughly, from the condition ii.) mentioned in Definition 1.1 it is easy to see that γ satisfies the 2-cocycle condition for any f, g, h ∈ C ∞ (G). Here we used the Sweedler notation. Thus, dualizing the deformed Hopf algebra U F (g) obtained in Proposition 1.3 we immediately obtain a twisted Hopf algebra denoted by As it will be used in the following we resume this structure in the following defining.

Definition 1.5 (Quantum group)
The quantum group corresponding to G is defined to be the Hopf where the deformed product is given by (1.6) and coproduct and counit are undeformed.
The deformed Hopf algebras C ∞ γ (G) and U F (g) are again dually paired via the same pairing.
can be easily obtained by dualizing Φ, see [22,Prop. 1.6.11]). In the same spirit of Lemma 1.4, the algebra structure of C ∞ (M ) can be deformed by considering a 2-cocycle on C ∞ (G) and pushing its deformation on C ∞ (M ) via the coaction δ. On the other hand, as discussed above the product m M can be deformed by the twist using (1.3) and the two deformations coincide.

Hamiltonian actions
In this section we introduce the notion of Hamiltonian action in the setting of Poisson Lie groups. This notion has been first defined in [19,20] in the case of a Poisson Lie group acting on a Poisson manifold with both Poisson structures fixed. In our work we are mainly interested to the case in which the Poisson structure on the manifold is the one induced by the action. This requires a slight generalization of the notion of Hamiltonian action.

Dressing generators
In the same spirit of [19,20], the notion of Hamiltonian action relies on the definition of momentum map, which provides us of a comparison tool between the dressing orbits and the orbit of the considered action. For this reason, we first focus on the dressing action and in particular on the possible descriptions of the corresponding fundamental vector fields.
Let us consider a Poisson Lie group G and denote by G * and D the corresponding dual and double, respectively. Consider g ∈ G, u ∈ G * and let ug ∈ D be their product. Since d = g ⊕ g * , elements in D close to the unit can be decomposed in a unique way as a product of an element in G and an element in G * . Then, there exist elements u g ∈ G and u g ∈ G * such that ug = u gu g . (2.1) Hence, the action of g ∈ G on u ∈ G * is given by where (ug) * G denotes the G * -factor of ug ∈ D. This defines a left action of G on G * , called dressing action. This action plays an important role in the context of Poisson actions since its orbits coincides with the symplectic leaves of G * and its linearization is the coadjoint action. Let us denote by X the corresponding fundamental vector field for X ∈ g. In the following we introduce the notion of dressing generators, which are one-forms that give us the fundamental vector fields X if contracted with the Poisson bitensor. As it will be seen in the next sections these forms are in general not globally defined, so we use the notation Ω 1 loc (G * ) to denote local forms on G * .
Definition 2.1 (Dressing generator) The map α : g → Ω 1 loc (G * ) : X → α X is said dressing generator with respect to the Poisson structure π on G * if the fundamental vector field X of the dressing action can be written as and satisfies Remark 2.2 The first example of dressing generators with respect to the standard dual Poisson structure π * is given by the left-invariant one-forms corresponding to the element X, as proved in [18, Appendix 2, pag. 66]. As already mentioned, the dressing generators with respect to a generic Poisson structure on G * are in general not globally defined (a concrete example is computed in the next section). However, the contraction with the Poisson tensor still gives rise to a smooth vector field.
Here we are interested to the case in which g is endowed with an r-matrix and we consider the Poisson structure π induced by the infinitesimal dressing action : This is a natural candidate since the contraction of π with one-forms satisfying (2.4)-(2.5) gives automatically rise to an infinitesimal Poisson action, as proved in the following Lemma.

Dressing generators on ax + b
Let s be the Lie algebra with basis H, E and commutation relation and consider the triangular r-matrix r = H ∧ E. This induces the Lie bialgebra structure on g * : As a consequence, the dual basis H * , E * satisfies the following commutation relation:  .11), is diffeomorphic to the product manifold: ii.) Within this model, the group law is given by iii.) Realizing the Lie algebra g as

15)
the exponential mapping is given by Proof: The connected simply connected Lie group H 1 corresponding to h 1 can be modelled on V × RZ with group law given by Within this setting, we observe that the symplectic group Sp(V, Ω) (which in our two-dimensional case just coincides with the group SL 2 (R)) acts by centre-fixing group-automorphisms on H 1 under: Every sub-group A of Sp(V, Ω) therefore determines the semi-direct product group modelled on the Cartesian product G = A × H 1 with group law defined by (a, a ∈ A): (a, v, z) · (a , v , z ) := (a · a , (v, z) · R a (v , z )) = (a · a , v + a(v ), z + z + 1 2 Ω(v, a(v ))). the semi-direct product is therefore the Lie group with group law given by (2.13). One then readily verifies that the given expression in (2.16) satisfies the condition exp t(a 0 , v 0 , z 0 ) · exp s(a 0 , v 0 , z 0 ) = exp(t + s)(a 0 , v 0 , z 0 ) for all s, t ∈ R. The fact that B 2 = I then implies d dt t=0 exp t(a 0 , v 0 , z 0 ) = (a 0 , v 0 , z 0 ). The computation of the Lie algebra of G is then performed using the expression of the above exponential mapping (2.16). It identifies with the one of g.
We now pass to realize s and s * in the double G. For this we start from expressing the generators at the Lie algebra level: The coordinates on s * are given by (ν, κ) * := exp νE * exp κH * where Using the group law (2.13) we get The dressing action S × S → S therefore amounts to solve the equation (a, n)(ν, κ) = (ν, κ)(a, n) for (ν, κ) as a function of a, n, κ, ν. From an easy computation it follows that the solution is given by where η is the diffeomorphism defined by η : R →] − 1, ∞[ : x → η(x) := e −2x − 1. Considering the coordinate system S → R 2 : ξ := (ν, κ) → (x, y) := (κ, η(ν)), the local right dressing action then reads: (x, y) · (a, n) := (x − ny, e −2a y). (2.32) is an open embedding. Hence locally one may set: s · ξ = ξ s · s ξ with s ξ ∈ S and ξ s ∈ S . (2.34) One then notes that for all s 1 , s 2 ∈ S and ξ ∈ S : Hence the map S × S → S : (ξ, s) → ξ s which given elements s ∈ S and ξ ∈ S expresses the S -component (local) of the product s · ξ in terms of the decomposition (2.33) is a right action of S on S . The latter globalizes under the usual matrix left-action of the affine group on the plane as Now we express the group multiplication in S within the above coordinate system: (x, y).(x , y ) := Φ Φ −1 (x, y).Φ −1 (x , y ) = (y + 1)x + x , (y + 1)y + y . (2.38) The unit consists in the vector origin (0, 0) and the inverse (which is only local at the level of the entire ambient space R 2 ) is given by (x, y) −1 = 1 y+1 (−x, −y). It is useful to rewrite the dressing action using musical notation; in this case we consider the coordinate system This implies that the dressing action is infinitesimally generated by the following fields: The next step consists in computing explicitely the dressing generators. Note that there are 3 Poisson structures involved here on the image U of S → R 2 , the dual Poisson Lie group structure the Poisson structure π induced by the action π = 2y 2 ∂ x ∧ ∂ y (2.43) and the linear one π s . It is easy to see that Finally, imposing the condition (2.3) we obtain that the dressing generators wrt to π * and π we get the expressions (2.27) and (2.28), resp.

Hamiltonian actions
Using the notion of dressing generator we give a new definition of Hamiltonian action in this context. Definition 2.6 (Momentum map) Let Φ : G × M → M be an action of (G, π G ) on (M, π) and α X the dressing generator wrt a Poisson structure π G * on G * .
i.) A momentum map for Φ is a map J : M → G * such that where ϕ(X) is the fundamental vector field of Φ. In other words, J is defined by the commutativity of the following diagram: ii.) A map J : M → G * is said to be -equivariant if it intertwines the fundamental vector field ϕ(X) and the dressing action X for any X.

Lemma 2.7 The momentum map J defined above is -equivariant if and only if is Poisson.
Proof: Let us consider generic Poisson structures π on M and π G * on G * . Thus, J is a Poisson map if and only if J * (π (J * (α))) = π G * (α).

Now the notion of Hamiltonian follows naturally:
Definition 2.8 (Hamiltonian action) An action Φ of (G, π G ) on (M, π M ) is said to be Hamiltonian if it is Poisson and is generated by a -equivariant momentum map J : M → G * .
Since in the following we mainly use the infinitesimal action ϕ, we say that it is Hamiltonian whenever the corresponding Φ is Hamiltonian.
Remark 2.9 i.) If we choose the standard dual Poisson structure on G * , the dressing generators are the left-invariant one-forms and the above definition boils down to the definition of momentum map and Hamiltonian action given by Lu in [19,20].
ii.) Let g be a triangular Lie algebra with r-matrix r, acting on a manifold M by ϕ : g → Γ ∞ (T M ). We denote by π r the Poisson structure induced by r via In this case the action ϕ and its global corresponding are automatically Poisson. (The proof is the same as the one given in Lemma 2.3).
Example 2.10 (Dressing action) The easiest example is given by the dressing action. Here the momentum map is just the identity.
Example 2.11 (Coadjoint action) Let us consider the Poisson structure π r induced by the coadjoint action. Notice that π r does not coincide with the linear one. As proved in [1, Section 3.3] one can define a map j : g * → d by j(ξ) = ξ − r(ξ, · ). Thus, the modified exponential is given by Exp : g * → G * : Exp(ξ) := pr G * (exp(j(ξ))) Alekseev showed that in contrast to the usual exponential map it intertwines the coadjoint action with the dressing action, hence it takes symplectic leaves to symplectic leaves. In other words, we have X = Exp * ϕ(X). If G is compact with the Lu-Weinstein Poisson structure, Exp is a global diffeomorphism (see [1,Remarks 3.5]). An easy computation shows that Exp is a momentum map for the coadjoint action.
Remark 2.12 From the above example we can construct other Hamiltonian actions. Given a standard momentum map µ : M → g * which is ad * -equivariant we can always construct a momentum map J : M → G * by composing µ and Exp. For instance, observing that r : g * → g intertwines adjoint and coadjoint actions we can conclude that the adjoint action is Hamiltonian with momentum map given by the composition of r with Exp.

Remark 2.13
The reduction can been obtained with various techniques (see e.g. [12]). We here remark that the preimage C = J −1 ({0}) of a -invariant momentum map is a coisotropic submanifold and I C the corresponding vanishing ideal. Thus the reduced algebra can be easily obtained by the quotient

Hamiltonian Hopf algebra (co)actions
In this section we aim to give a definition of Hamiltonian (co)action in the setting of Hopf algebra (co)actions and a possible quantization procedure. In the same spirit of Definition 2.8, given an Hopf algebra action Φ, a momentum map has to be an intertwiner between dressing action and Φ.
Lemma 3.1 Consider the infinitesimal dressing action : g → Γ ∞ (T G * ). This is equivalent to a Hopf algebra action Λ : Equivalently, it defines a Hopf algebra coaction δ Λ : Proof: The following proof holds for any Lie algebra actions on a generic manifold. The Lie algebra elements act as derivations of C ∞ (G * ), thus Λ defines a Lie algebra action : g → Γ ∞ (T G * ). Since the elements of g generate U(g), the action Λ is given by differential operators with order determined by the natural filtration of the universal enveloping algebra. Conversely, every Lie algebra action of g on G * determines via the fundamental vector fields X ∈ Γ ∞ (T G * ) a representation of g on C ∞ (G * ) by derivations which therefore extends to a Hopf algebra action Λ as above. The action Λ and the coaction δ Λ are always equivalent.
As a next step we lift the notion of dressing generator to the setting of Hopf algebra actions. We observe that, given the Lie algebra representation α : g → Ω 1 loc (G * ), we can define another Hopf algebra action by using the Lie derivative in the direction of a one-form L α which has been defined by Bhaskara and Viswanath [4]. In particular, for f ∈ C ∞ (G * ) More precisely, we have: 2 Given a dressing generator α : g → Ω 1 loc (G * ), the corresponding map given by U(g) × C ∞ (G * ) → C ∞ (G * ) : (X, f ) → L α X f is a Hopf algebra action. Furthemore we have Proof: First, as in Lemma 3.1 the map U(g) × C ∞ (G * ) → C ∞ (G * ) : (X, f ) → L α X immediately satisfies the condition to be a Hopf algebra action. Also, from the definition of dressing generator we have X = π (α X ). Thus Notice that its pullback of functions J * : is an algebra morphism. With an abuse of notation, we also refer to J * as the pullback of forms. Since the latter is always defined, we can extend J to a map J * acting on L α by Lemma 3.3 Let us consider a Hamiltonian action ϕ and its corresponding Hopf algebra action Φ(X) = L ϕ(X) . Then, J * : C ∞ (G * ) → C ∞ (M ) defines a U(g)-module algebra morphism.
Proof: We already observed that J * : is an algebra morphism. Thus, we only need to prove that it is a module morphism, i.e. the commutativity of the following diagram: In other words, we need to prove Φ(X, J * f ) = J * (Λ(X, f )). (3.6) Using (3.4) we can easily compute: Here we used the fact that, from Definition 2.8, we have ϕ(X) = π (J * (α X )).
Given the Hopf algebra action Φ we can always find the corresponding Hopf algebra coaction δ Φ , as discussed in Section 1. Thus we can immediately state the dual version of the above lemma. Proof: Dualizing the commutative diagram (3.5) we immediately get the following commutative diagram which gives the comodule morphism condition δ Φ •J * = (J * ⊗id)•δ Λ . Since J * is an algebra morphism the claim is proved.
The above discussion motivates the following definition. where Λ(X, f ) is given by (3.1). In other words, Φ is Hamiltonian is it allows an U(g)-module algebra morphism which intertwines it with the Hopf algebra action corresponding to the dressing action.
ii.) An Hopf algebra coaction δ Φ : is said Hamiltonian if there exist C ∞ (G)-module algebra morphism J, called momentum map, which intertwines it with the Hopf algebra coaction δ Λ corresponding to the dressing action.
Summing up the above lemmas we obtain the following claim.

Quantum Hamiltonian coactions via 2-cocycles
In this section we prove that, using Drinfeld approach, we obtain a quantization of the Hamiltonian coactions as in Definition 3.5. Since actions and coactions are completely equivalent we here prefer to focus only on the coaction case. Let us consider a twist F on U(g) with corresponding 2-cocycle γ on C ∞ (G). As seen in Definition 1.5, the 2-cocycle γ induces a deformed product γ and we denote by C ∞ (G) the corresponding quantum group. Furthermore, we obtain a deformed product on the comodule algebras C ∞ (M ) and C ∞ (G * ). More precisely, the action ϕ : g → Γ ∞ (T M ) induces a star product ϕ on M whose semiclassical limit is the Poisson structure π r induced by r via ϕ. Similarly, the dressing action : g → Γ ∞ (T G * ) induces a star product on G * . Let us denote by C ∞ (G * ) the deformed algebra given by the pair  Proof: Since in the Drinfeld approach the coactions do not change but they only intertwine different algebraic structures, the classical momentum map is still a comodule morphism as in Lemma 3.4. More explicitely, the diagram commutes. Thus, we only need to prove that J * : C ∞ (G * ) → C ∞ (M ) is a morphism of algebras. This can be immediately checked by using the UDF (1.3) and Lemma 3.3. We can extend the action (3.1) by Λ(F, f ⊗ g) = Λ(F α , f ) ⊗ Λ(F α , g). (3.11) As a consequence, we have: J * (f g) = J * (m(Λ(F, f ⊗ g))) = J * (m(Λ(F α , f ), Λ(F α , g))) = (J * Λ(F α , f ))(J * Λ(F α , g)) = J * f ϕ J * g.