$\kappa$-Deformed Phase Space, Hopf Algebroid and Twisting

Hopf algebroid structures on the Weyl algebra (phase space) are presented. We define the coproduct for the Weyl generators from Leibniz rule. The codomain of the coproduct is modified in order to obtain an algebra structure. We use the dual base to construct the target map and antipode. The notion of twist is analyzed for $\kappa$-deformed phase space in Hopf algebroid setting. It is outlined how the twist in the Hopf algebroid setting reproduces the full Hopf algebra structure of $\kappa$-Poincar\'e algebra. Several examples of realizations are worked out in details.


Introduction
Motivation for studying noncommutative (NC) spaces is related to the fact that general theory of relativity together with Heisenberg uncertainty principle leads to the uncertainty of position coordinates itself x µ x ν > l 2 Planck [22,23]. This uncertainty in the position can be realized via NC coordinates. There are also arguments based on quantum gravity [22,23,36], and string theory models [20,61], which suggest that the spacetime at the Planck length is quantum, i.e. noncommutative.
The symmetries of κ-Minkowski spacetime are described via Hopf algebra setting and they are encoded in the κ-Poincaré-Hopf algebra (in the same sense as are the symmetries of Minkowski spacetime encoded in the Poincaré-Hopf algebra). A Hopf algebra is a bialgebra equipped with an antipode map satisfying the Hopf axiom. The bialgebra is an (unital, associative) algebra which is also a (conunital, coassociative) coalgebra such that certain compatibility conditions are satisfied. The antipode is an antihomomorphism of the algebra structure (an antialgebra homomorphism). Hopf algebras are used in various areas of mathematics and physics for fifty years. See [8,52] for some examples.
It turns out that the notion of the Hopf algebra is too restrictive and it has to be generalized. For example, it is shown that the Weyl algebra (quantum phase space) can not have a structure of a Hopf algebra. Namely, the whole phase space (Weyl algebra) generated by p µ and x µ (orx µ ) can not be equipped with the Hopf algebra structure, since one can not include x µ in a satisfactory way, i.e. the notion of Hopf algebra is too restrictive for the whole phase space (Weyl algebra). Several types of generalizations are possible: quasi-Hopf algebras, multiplier Hopf algebras and weak Hopf algebras. Our construction is very similar to the structure of the Hopf algebroid defined by Lu in [48].
Lu was inspired by the notion of the Poisson algebroid from the Poisson geometry. Namely, some Hopf algebras are quantization of the Poisson groups. Now, Hopf algebroids can be considered as the quantization of the Poisson groupoids. Lu introduces two algebras: the base algebra A and the total algebra H. One can consider the total algebra H as the algebra over the base algebra A. The left and right multiplications are given by the source and the target maps. Hence, the coproduct is defined on the total algebra H and the image lies in H⊗ A H which is an (A, A)-bimodule but not an algebra. Namely, H ⊗ A H is the quotient of H ⊗H by the right ideal. G. Böhm and K. Szlachányi in [9] considered the same structure as Lu did, but they changed the definition of the antipode. For more comprehensive approach, see [8]. Let us mention that some ideas existed before the definition of Lu in which the base algebra or both the base algebra and the total algebra had to be commutative (see [8,48] and references therein). Bialgebroid is equivalent to the notion of × A -bialgebra introduced much earlier by Takeuchi in [62].
One can analyze the structure of the Hopf algebra by twists. See [6,7] for more details. P. Xu in [63] applies the twist to the bialgebroid (which he calls Hopf algebroid although he does not have the antipode). It is important to mention that Xu uses the definition of the bialgebroid which is equivalent to the definition from [48].
In [40] κ-Minkowski spacetime and Lorentz algebra are unified in a unique Lie algebra. Realizations and star products are defined and analyzed in general and specially, their relation to coproduct of the momenta is pointed out.
The deformation of Heisenberg algebra and the corresponding coalgebra by twist is performed in [57]. Here, the so called tensor exchange identities are introduced and coalgebras for the generalized Poincaré algebras are constructed. The exact universal R-matrix for the deformed Heisenberg (co)algebra is found.
The quantum phase space (Weyl algebra) and its Hopf algebroid structure is analyzed in [33]. Unification of κ-Poincaré algebra and κ-Minkowski spacetime is done via embedding into quantum phase space. The construction of κ-Poincaré-Hopf algebra and κ-Minkowski spacetime using Abelian twist in the Hopf algebroid approach has been elaborated.
Twists, realizations and Hopf algebroid structure of κ-deformed phase space are discussed in [34]. It is shown that starting from a given deformed coalgebra of commuting coordinates and momenta one can construct the corresponding twist operator.
In the present paper, the total algebra is the Weyl algebraĤ and the base algebra is the subalgebraÂ generated by noncommutative coordinatesx µ . The construction of the target map is obtained via dual realizations. The codomain of the coproduct is changed. We take a quotient of the image of the coproduct instead of quotient ofĤ ⊗Ĥ. As a consequence, the right ideal by which Lu [48] has taken the quotient is now two-sided and the codomain of the coproduct has the algebra structure. The notion of the counit is related to realizations. Furthermore, we manage to incorporate the twist in our construction, obtaining the Hopf algebroid structure from the twist. This paper is structured as follows. In Section 2 we introduce the κ-Minkowski spacetime and κ-deformed phase space, and we establish the connection between Leibniz rule and coproduct for the Weyl generators. Also, the dual basis is introduced and elaborated. The Hopf algebroid structure of κ-deformed phase spaceĤ and undeformed phase space H is presented in Section 3. In Section 4 we first discuss the realizations and then we provide the twist operator in the Hopf algebroid approach. It is shown that the twisted Hopf algebroid structure of phase space H is isomorphic to the Hopf algebroid structure ofĤ. Finally, in Section 5 we consider the κ-Poincaré-Hopf algebra in the natural realization (classical basis). It is outlined how the twist in Hopf algebroid setting reproduces the full Hopf algebra structure of κ-Poincaré algebra. Also, we discuss the existence and properties of twist in all types of deformations (space-, time-and light-like).

Phase space
The momentum space T = C[[p µ ]] is the commutative space generated by p µ such that is satisfied for some set of real functions ϕ µν (see [33,34,40] for details). Let us recall that lim a→0 ϕ µν = η µν 1 and det ϕ = 0. We also require that generatorsx µ and p µ satisfy Jacobi identities. This gives the set of restrictions on functions ϕ µν (see equation (11) in [40] or equation (4) in [34]). The existence of such space T is analyzed in several papers [40,54]. One particularly interesting solution is the set {p L µ } which is related to the so called left covariant realization [40,54] where ϕ µν = η µν Z −1 , i.e. (2.2) leads to Here Z denotes the shift operator defined by and for the left covariant realization is given by where we used ap L ≡ a α p L α . The phase spaceĤ is generated as an algebra byÂ and T such that (2.1) and (2.2) are satisfied.
Let be the unique action ofĤ onÂ, such thatÂ acts on itself by left multiplication and t f = [t,f ] 1 for all t ∈ T andf ∈Â.Â can be considered as anĤ-module.

Leibniz rule
We have already mentioned thatĤ does not have the structure of the Hopf algebra, but it is possible to construct the structure of the Hopf algebroid. In this subsection we do the preparation for the coproduct which will be completely defined in Section 3. The formula for the coproduct can be built from the action and the Leibniz rule (see [40,Section 2.3] and [34]). In κ-Poincaré-Hopf algebra U κ (P) (where P is generated by momenta p µ and Lorentz generators M µν ) the coproducts of momenta and Lorentz generators are unique and | Uκ(P) : U κ (P) → U κ (P) ⊗ U κ (P). However in the Hopf algebroid structure the coproduct of generators p µ andx µ are not unique, modulo the right idealK in (2.10).
It is important to emphasize that such derived coproduct is an algebra homomorphism for anyĥ 1 ,ĥ 2 ∈Ĥ which enables us to define the formula for the coproduct for all elements ofĤ.

Dual basis
In [40] we have introduced the notion of the dual basis. Let us recall some basic facts since it will be used for the definition of the target map. We define elementŝ (it would be more precise to write (O −1 ) µα ). They have some interesting properties. Sincê

13)
y µ and p µ form a basis ofĤ (it would be more correct to say that power series inŷ µ and p µ form a basis ofĤ). Elementsŷ µ satisfy commutation relations similar to (2.1): We call this basis the dual basis. It is easy to check thatx µ andŷ ν commute, i.e.
[x µ ,ŷ ν ] = 0. (2.14) Also, the straightforward calculation shows that O µν and O λρ commute. It remains to consider where C µλα = a µ η λα − a λ η µα stands for structure constants. One can easily obtain and One can easily check thatŷ µ 1 =x µ . Using (2.14) and (2.16), it is easy to obtain that y µ x ν =x νxµ and Heref op stands for the opposite polynomial ((x µxν ) op =x νxµ ). Hence, the action off (ŷ) can be understood as a multiplication from the right withf op (x). One can show that (ŷ µ ) = 1⊗ŷ µ . Note that the same construction as for κ-Minkowski space (2.1) could be generalized to arbitrary Lie algebra defined by structure constants C µνλ .

Hopf algebroid 3.1 Hopf algebroid structure ofĤ
We define the source map, target map, coproduct, counit and antipode such thatĤ has the structure of the Hopf algebroid.
In Hopf algebroid, the unit map is replaced by the source and target maps. In our caseĤ is the total algebra andÂ is the base algebra. The source mapα :Â →Ĥ is defined bŷ The target mapβ :Â →Ĥ is defined bŷ Let us recall that the source map is the homomorphism while the target map is the antihomomorphism. Relation (2.14) shows that In order to define the coproduct onĤ, we consider the subspaceB ofĤ ⊗Ĥ: where U(R µ ) denotes the universal enveloping algebra generated byR µ (see (3.1)). Here, T denotes the subalgebra ofĤ ⊗Ĥ generated by 1 ⊗ 1 and elements (p µ ). For example, we can consider p L µ and then T is generated by 1 ⊗ 1 and

4)
B is a subalgebra ofĤ ⊗Ĥ. It is obvious that (3.3) is a consequence of (3.1) and (3.2) but we write it for completeness. Now, let us consider the subspaceÎ ofB defined bŷ where U + (R µ ) is the universal enveloping algebra generated byR µ but without the unit element. Using (3.1)-(3.4) one can check thatÎ =K ∩B andÎ is the twosided ideal inB.
This map is not a homomorphism. It is easy to check that m(αˆ ⊗ 1) = 1 and m(1 ⊗βˆ ) = 1.
In order to check the first identity, we write elements ofĤ in the formf (x)g(p) and for the second identity in the formf (ŷ)g(p). The antipode S :Ĥ →Ĥ is defined by The antipode S(x µ ) can be calculated from (2.13). One obtains that It follows that S 2 (ŷ µ ) =ŷ µ + ia µ (1 − n) (and S 2 (x µ ) =x µ + ia µ (1 − n)) and S 2 (p µ ) = p µ . Previous two formulas can be written also as S 2 (ĥ) = Z 1−nĥ Z n−1 . It is enough to check it for the elementsx µ and p µ since S 2 is a homomorphism. The expression of S 2 (x µ ) can be written in terms of structure constants: A nice way to check the consistency of the antipode is to start with (2.13) and apply the antipode S (note that S(O µα ) = O −1 µα ): It produces It remains to apply expressions for (O −1 ) αβ and O µα (see (2.12) and (2.8)), use the abbreviation A L = −a α p L α = −(ap L ) and recall the identity Z = (1 − A L ) −1 (see [40]). Let P ⊂Ĥ be the enveloping algebra of the Poincaré algebra p. It is possible to define the Hopf algebra structure on the subalgebra P [40]. It is interesting to note that the coproduct and the antipode map defined above onĤ and restricted to P coincides with the coproduct and the antipode map on the Hopf algebra P [33]. For more details see Section 5.
It is easy to check that The first identity is obvious, the second one can be easily checked for the base elements and the third identity can be easily checked using the dual basis.

Hopf algebroid structure of H
Now, let us consider the case when the deformation vector a µ is equal to 0. Then (2.1) transforms to [x µ ,x ν ] = 0, the algebraĤ becomes the Weyl algebra which we denote by H and write x µ instead ofx µ . We have already mentioned that it is not possible to construct the Hopf algebra structure on H. Let us repeat the Hopf algebroid structure on H and set the terminology. Now, ϕ µν = O µν = η µν , Z = 1 andŷ µ = x µ . Let A (the base algebra) be the subalgebra of H generated by 1 and x µ . We define the action £ of H on A in the same way as we did it in Section 2.
Then A can be considered as an H-module. It is clear that the action transforms to the action £ when the vector a is equal to 0.
The source and the target map are now equal α 0 = β 0 and α 0 ; β 0 : A → H reduces to the natural inclusion.
The counit 0 : H → A is defined by In order to define the coproduct, let us define relations (R 0 ) µ by Let U[(R 0 ) µ ] be the universal enveloping algebra generated by 1 ⊗ 1 and (R 0 ) µ , U + [(R 0 ) µ ] be the universal enveloping algebra generated by (R 0 ) µ but without the unit element, and 0 T be the algebra generated by 1 ⊗ 1 and p µ ⊗ 1 + 1 ⊗ p µ . Note that T is isomorphic to 0 T . Now, we define B 0 , the subalgebra of H ⊗ H of the form and twosided ideal I 0 of B 0 by The coproduct 0 : H → B 0 /I 0 = 0 H is a homomorphism defined by One checks that the coproduct 0 and the counit 0 satisfy m(α 0 0 ⊗ 1) 0 = 1 and m(1 ⊗ β 0 0 ) 0 = 1. The antipode S 0 : H → H transforms to It is easy to check that Similarly as in the deformed case, the expression m(1⊗S 0 ) 0 is not well defined in [48], because m(1 ⊗ S 0 )K 0 = 0 and this is why the section γ is needed. In our approach, since 4 Twisting Hopf algebroid structure

Realizations
The phase space satisfying (2.1) and (2.2) can be analyzed by realizations (see [40,42,56]). In Section 3.2, we have analyzed the Weyl algebra H generated by p µ and commutative coordinates x µ satisfying Then, the noncommutative coordinatesx µ are expressed in the form such that (2.1) and (2.2) are satisfied. It is important to observe that the space H is isomorphic toĤ as an algebra. Hence, we setĤ = H and treat sets {x µ , p ν } and {x µ , p ν } as different bases of the same algebra. However, we will use both symbols,Ĥ and H in order to emphasize the basis. The action £, defined in Section 3.2 corresponds to H. However, H andĤ, considered as Hopf algebroids are different. The restriction of the counit 0 |Â, introduced in Section 3.2, defines the bijection of vector spacesÂ and A. By the abuse of notation, we denote it by 0 or £. Let us mention that the inverse map is simplyˆ | A . Then, the star product on A is defined by The algebra A equipped with the star product instead of pointwise multiplication will be denoted by A and the map 0 :Â → A is an isomorphism of algebras.
It is possible to construct the dual realizationφ µν and the dual star product φ such that is satisfied (see [40,Section 5]). Now, elementsŷ µ are given bŷ It is easy to check the following properties:

Similarity transformations
The relation between realizations is given by the similarity transformations [34]. Let us consider two realizations. The first one is denoted by x µ and p µ and given by the set of functions {ϕ µν } (and (2.2) or (4.1)). The second realization is denoted by X µ , P µ and Φ µν (x µ = X α Φ αµ (P )). The similarity transformation E is given by E = exp{x α Σ α (p)} such that lim a→0 Σ α = 0. Now, the relation between realizations is given by It is easy to see that P µ = P µ (p). Since [P µ ,x ν ] = −iΦ µν (P ), It follows that the set of functions ϕ µν can be obtained from the set of functions Φ µν and the expressions of P in terms of p. Since O µν = O µν (P (p)), it is easy to express O µν in the realization determined by x µ and p µ .

Examples
Let us consider three examples of realizations. The noncovariant λ-family of realizations is given bŷ andŷ where Z = e A (λ) and λ ∈ R. For this family we assume that a = (a 0 , 0, . . . , 0). Here, (λ) denotes the label. Generic realizations are denoted without the label. It is easy to obtain p L 0 = 1 a 0 (1 − Z −1 ) and p L k = p The left covariant realization is defined bŷ The element p L that we have mentioned in Section 2.2 corresponds to the left covariant realization. It is easy to obtain that (2.11) for the definition ofŷ µ ).
The right covariant realization is defined bŷ where Z = 1 + A R . The relation between p L µ and p R µ is given by p R µ = p L µ Z. Now, Also, it easy to calculate O µν in terms of p R µ : One should notice the duality between the left covariant and the right covariant realizations.

Twist and Hopf algebroid
For each realization, there is the corresponding twist and vice versa [34]. The relation between the star product and twist is given by for f, g ∈ A. It follows that F −1 ∈ H ⊗ H/K 0 . Now, we will use twists to reconstruct the Hopf algebroid structure described in Section 3.1, from the Hopf algebroid structure analyzed in Section 3.2. That is we will show that by twisting the Hopf algebroid structure of H one can obtain the Hopf algebroid structure ofĤ. Hence, we will consider twists F such that F : 0 H → H. Here I ∼ =Î and H ∼ = Ĥ . More precisely, I is the twosided ideal generated by elements R µ which are defined by Let us mention that the relation betweenR µ and R µ is given bŷ Also, it is easy to rebuild the realization from the twist. For the given twist F, the corresponding realization is obtained bŷ Similarly, whereF −1 is given byF −1 = τ 0 F −1 τ 0 (τ 0 stands for the flip operator with the property The noncovariant λ-family of realizations have twists of the form These twists belong to the family of Abelian twists (see [24]). The left covariant and the right covariant realizations, respectively, have twists of the form These two twists belong to the family of Jordanian twists (see [13]). Let us reconstruct the source and the target maps from the twist. First, we define α and β, α : and Now, the source and the target maps are given bŷ The counitˆ : H →Â is given bŷ The coproduct can be calculated by the formula: For the noncovariant λ-family of realizations and It is a nice exercise to expressx µ in terms of x α (see (4.2)), use (4.5)-(4.8) and obtain (3.5). Similarly, for the left covariant realization and for the right covariant realization. It remains to consider the antipode. Let where S 0 denotes the undeformed antipode map defined by (3.9) (S 0 (x µ ) = x µ and S 0 (p µ ) = −p µ ). For the similar approach regarding Hopf algebras, see [7,6]. For the noncovariant λ-family of realizations, χ has the form α (see (4.2)), use (4.10)-(4.13) and obtain (3.6). The antipode is given by and Let us recall that for the noncovariant λ-family of realizations we set a µ = (a 0 , 0, ..., 0). Now, one can compare (4.14) and (4.15) with (3.6). The formula for the antipode ofx µ can be also obtained from the formula S(ŷ µ ) =x µ , formulas for the realization ofx µ andŷ µ , (4.2) and (4.3) and formulas for S(p µ ). For all examples, it is easy to check that S(ŷ µ ) = χ(S 0 (ŷ µ ))χ −1 =x µ . For the left covariant realization For the right covariant realization There is a natural question if the antipode map on the Hopf algebroidĤ defined by (4.9) and the antipode map defined on the Hopf algebra U(igl(n)) coincide (see [38] for the formulas of the antipode). They coincide for h ∈Ĥ for which α (h) = β S 0 (h). For elements h for which α (h) = β S 0 (h), the antipode maps do not coincide. For example, S 0 (x j p j ) = −x j p j + i in the Hopf algebroid, while S 0 (x j p j ) = −x j p j in the Hopf algebra (here no summation is assumed). See also [33].
There is a question whether P µ and M µν could be obtained from twist F expressed in terms of Poincaré generators only.
1. For a µ light-like, a 2 = 0, such cocycle twist within Hopf algebra approach exists [35]  The cocycle condition for twist F (5.4) can be checked using the results by Kulish et al. [45] in the Hopf algebra setting (see also [16]).
2. For a µ time-and space-like such twist does not exist within Hopf algebra. Namely, starting from P µ = F 0 P µ F −1 and M µν = F 0 M µν F −1 one can construct an operator F = e f , where f = f 1 + f 2 + · · · is expanded in a µ and expressed in terms of Poincaré generators and dilatation only. In the first order we found that the result is not unique, namely we have a one parameter family of solutions where u ∈ R is a free parameter. However there is one solution (u = 0) that can be expressed in terms of Poincaré generators only. Also up to first order in a µ cocycle condition is satisfied and one obtains the correct classical r-matrix (see equation (65) in [25]). In the second order for f 2 we found a two parameter family of solutions. Here there is no solution without including dilatation, that is the operator F can not be expressed in terms of Poincaré generators only. We have checked that the corresponding quantum R-matrix obtained using f 1 and f 2 is correct up to the second order. The cocycle condition is no longer satisfied in the Hopf algebra approach, that is F is not a twist in the Drinfeld sense. However, after using tensor exchange identities [33,34,57] the cocycle condition is satisfied and F is a twist in Hopf algebroid approach. It also reproduces the κ-Poincaré-Hopf An axiomatic treatment of the Hopf algebroid structure on general Lie algebra type noncommutative phase spaces, involving completed tensor products, has recently been proposed in [59].
The construction of QFT suitable for κ-Minkowski spacetime is still under active research [19,39,55]. We plan to apply κ-deformed phase space, Hopf algebroid approach and twisting to NCQFT and NC (quantum) gravity.