Interpolations between Jordanian twists induced by coboundary twists

We propose a new generalization of the Jordanian twist (building on the previous idea from [J.Phys.A50,26(2017),arXiv:1612.07984]). Obtained this way, the family of the Jordanian twists allows for interpolation between two simple Jordanian twists. This new version of the twist provides an example of a new type of star product and the realization for noncommutative coordinates. Exponential formulae, used to obtain coproducts and star products, are presented with details.


Introduction
Let H = (H, ∆, S, ǫ) be a Hopf algebra and F ∈ H ⊗ H be a two-cocycle twist. Then new (twisted) Hopf algebra structure on the algebra H with deformed coproduct and antipode is denoted by H F = H, ∆ F , S F , ǫ , where ∆ F (·) = F ∆(·)F −1 .
For any invertible element ω ∈ H one can define new gauge equivalent two-cocycle twist F ω = (ω −1 ⊗ω −1 )F ∆(ω) which determines the third Hopf algebra H Fω = H, ∆ Fω , S Fω , ǫ . Notice that all three Hopf algebras share the same algebraic structure (multiplication). Its internal automorphism, defined by the similarity transformation: α (Z) = ωZω −1 , Z ∈ H establishes, at the same time, the isomorphism between two twisted Hopf algebras H F ∼ = H Fω as illustrated on the following diagram i.e.
Consider now a (left) Hopf module algebra A = (A, ⊲, ⋆) over the Hopf algebra H together with a (left) Hopf action ⊲ : H ⊗ A → A, where ⋆ denotes the multiplication in A. Changing multiplication for a, b ∈ A, one gets new module algebra A F = (A, ⊲, ⋆ F ) over H F with the same action, i.e. the module structure remains the same. It is easy to see that any invertible element ω ∈ H provides an algebra isomorphism β : (A, ⋆ Fω ) → (A, ⋆ F ), where β(a) ≡ ω⊲a, since ω⊲(a⋆ Fω b) = (ω⊲a)⋆ F ω(ω⊲b). It turns out that the invertible map β intertwines between two modules in the following sense: α(Z) ⊲ β(a) = β(Z ⊲ a), i.e. the diagram commutes. In particular, the coboundary twist T ω = (ω −1 ⊗ ω −1 )∆(ω) provides the Hopf algebra isomorphism H ∼ = H Tω as well as module algebra isomorphism A ∼ = A Tω . For group-like ω, it becomes an automorphism. It means that replacing a twist by the gauge equivalent one leads to mathematically equivalent objects. Let us explain this in the case of Jordanian deformations of Lie algebras. Let's consider a Lie algebra g. Drinfeld's quantum groups are quantized universal enveloping algebras U g [1] obtained by the methods of deformation quantization of Poisson Lie groups. More exactly, quantized objects are Hopf algebras corresponding to a given Lie bialgebra structure (g, r) [2], which in turn can be determined by the so-called classical r-matrix r ∈ g ⊗ g satisfying classical Yang-Baxter equation [[r, r]] = 0 with [[, ]] being the so-called Schouten brackets. In the triangular case r ∈ g ∧ g the quantization is provided by an invertible, two-cocycle element F r ∈ U g ⊗ U g [[γ]] called a Drinfeld twist. Here γ is a formal parameter and U g [[γ]] means topological completion in the topology of formal power series in γ (see e.g. [3,4,5] for details). Thus the classical r-matrix can be recovered from the quantum R-matrix as follows For example, in two dimensions there are only two (non-isomorphic) Lie algebra structures: Abelian ab(2) = {x, y : [x, y] = 0} and non-Abelian an(2) = {h, e : [h, e] = e} 1 . The corresponding Lie bialgebra structures are given by the Abelian : r Ab = x ∧ y or Jordanian r J = h ∧ e classical r-matrices. Embedding one of these algebras in some higher dimensional Lie algebra g as Lie subalgebra provides the twist quantization of U g : Abelian or Jordanian. In the latter, deformation can be realized by the so-called Jordanian twist of the form: This form of the twist first appeared in [6] and then a symmetrised form (i.e. where r J is the first order in expansion (5)) was proposed in [7], [8] 2 : It was obtained by applying the coboundary twist to (6). There are many possible r-symmetrizations for twists. This is due to the fact that twist deformation is defined up to the so-called gauge transformation (in Drinfeld's terminology), i.e. many twists can provide isomorphically equivalent Hopf algebraic deformations, if they differ by the so-called coboundary twist (see, e.g. [3]). The question we would like to address in the present paper is if these mathematical equivalences, in some physically relevant situations, give rise, to some extent, to physically equivalent descriptions. For example, when one considers the star product quantization.
To this aim we embed our two dimensional Lie algebra an(2) into some bigger Lie algebra which has some potential application in physics. In the present paper we focus our attention on the Lie algebra g = {P µ , D}, generated by momenta P µ (spacetime translations in n-dimensions where µ = 0, 1, ...., n − 1) and dilatation generators D with the following commutation relations: This algebra can be considered as a subalgebra of some bigger Lie algebra, e.g. Poincare-Weyl, de Sitter, etc. This embedding is realized by choosing two elements {h, e} as h = −D and e = P (P can be taken as any of P µ and the formulae in the next Section 2 will hold). However for convenience we choose the following notation: P = v α P α where v µ is the vector on Minkowski spacetime M 1,n−1 in n-dimensions such that v 2 = v α v α ∈ {−1, 0, 1}. For the correspondence with the κ-Minkowski spacetime [13], [14] we choose the deformation parameter as γ = − 1 κ . Our main aim in this work is to analyse quantum deformations corresponding to gauge equivalence of Jordanian twists (6) extending [15] and applying [16].
This paper is a sequel to [15], where we introduced a generalised form of r-symmetrised twist interpolating between Jordanian twists. The main formulae are recalled here as part of the next section 2.1. In subsection 2.2 we propose another form of r-symmetrised Jordanian twist (F R,u ) providing the interpolation between Jordanian twists as well. We present the corresponding Hopf algebra deformation, the star product form and the realization of the noncommutative coordinates. The Sec. 3 presents a relation between the two generalizations of r-symmetric twists F L,u and F R,u , including the relation between the corresponding quantum R-matrices.
We finish with brief conclusions which are followed by a series of appendices complementing the results presented in the main part of the paper. They are devoted to the explanation of the exponential formulae which are obtained from twist realization of deformed coordinate functions. Some applications for calculations of wave packets star products as well as coproducts of momenta are also considered.

Two families of twists interpolating between Jordanian twists
In our previous work [15] we have proposed a simple generalization of the locally r-symmetric Jordanian twist (7), resulting in the one-parameter family interpolating between Jordanian twists. All the proposed twists differed by the coboundary twists and produced the same Jordanian deformation of the corresponding Lie algebra. We have proposed a way, by introducing an additional parameter u, of interpolating between the two Jordanian twists: with the logarithm on the left side of the tensor product (cf. with (6)) and with the logarithm on the right side and also with the changed sign of the deformation parameter κ, τ here denotes the flip map. One can symmetrize these simple Jordanian twists, into a so-called r-symmetric form, such that at the first order in the expansion of the quantum R-matrix one gets the classical r-matrix corresponding to the given deformation. For the Jordanian deformations it is always r J . In this paper we want to present another type of such interpolation. First, in the below subsection, we recall few main formulae from [15] and then, in subsection 2.2, we shall propose another interpolation.

F L,u family of twists with dilatation on the left
The r-symmetric version of the Jordanian twist (8) was introduced in [7] and it was obtained from the coboundary twist T ω by choosing ω 0 = exp − 1 2κ DP . The formula for F T follows directly from: . In [15] we have introduced its generalization in the form of one parameter family interpolating between Jordanian twists ∀u ∈ R: This generalization simply corresponds to a modification of ω 0 to ω L = exp − u κ DP and then still differs from F 0 only by the coboundary twist T ωL . For this reason the cocycle condition (see, e.g. [3]) for F L,u is automatically satisfied. For u = 1 2 (7) is recovered. Note that now F L,u contains two parameters: one real parameter u and the other κ -the formal deformation parameter.
The reduction of F L,u , for certain values of the parameter u, to F 0 (for u = 0) and to F 1 (for u = 1) were discussed in [15].

Hopf algebra
The deformation of the Hopf algebra U g (µ, ∆, ǫ, S) of the universal enveloping algebra of g given by the twist element is provided by the deformation of the coproduct and antipode maps as follows: The coproducts, star products and realizations depend explicitly on the parameter u as well as on the parameter of deformation κ.
We recall the coalgebra sector of the Hopf algebra U F0 g for the deformation with the twisting element F 0 from [17]: The change of the twist by the coboundary twist T ω provides a new presentation for the Hopf algebra, and can be transformed by Alternatively, the deformed coproducts and antipodes can be calculated directly from the definition ∆Z = F ∆ 0 (Z) F −1 . The coalgebra sector of the Hopf algebra U FL,u g for the deformation with F L,u , recalled from [15], is as follows: The coproduct is coassociative. The antipodes are given by

Coordinate realizations and star product
As a Hopf module algebra for U g we choose the algebra of smooth (complex valued) functions on a space time (i.e. A = C ∞ (R n ) ⊗ C with an obvious algebraic structures determined by pointwise multiplication and addition). This algebra includes spacetime coordinates x µ (where x µ are considered as generators of n-dimensional Abelian Lie algebra). A natural action of g on A (i.e. action of the momenta and the dilatation operators on coordinates) is defined by This algebra becomes noncommutative due to the twist deformation once the usual multiplication is replaced by the star product multiplication (star product quantization) (3) for f, g ∈ C ∞ (R n ). The star product is associative (if the twist F satisfies cocycle condition).
The inverse of the twist F L,u provides the star product between the functions, as indicated in equation (3). This star product is associative (due to the fact that the twist F L,u satisfies the cocycle condition).
When we choose our functions to be exponential functions e ik·x and e iq·x , then we can define new function D µ (u; k, q) [18]- [22] : where k, q ∈ M 1,n−1 (in n-dimensional Minkowski spacetime). One can calculate explicitly, see appendices C.1 and C.3, that in the case of twist F L,u the function D µ (u; k, q) is given by Directly from the twist we can also calculate the coordinate realizations [23]- [28] and for the F L,u twist they have the following form: as before. These realizations are also discussed in [29] to [34].

F R,u family of twists with dilatation on the right
In this paper we want to introduce another version of the generalized Jordanian twist: where u is a real parameter u ∈ R. We point out that the sub-index R refers to the position of the dilatation generator, it is on the right with respect to momenta generators P . Due to the position of the dilatation operator with respect to momenta, this introduces a different formula than the one considered in [15] and recalled in the previous subsection 2.1, i.e. F L,u (10). This form of the family of twists F R,u can also be easily obtained from the simple Jordanian twist F 0 by the transformation with the coboundary twist T ωR however this time with the element ω R = exp − u κ P D . The twist F R,u , ∀u, satisfies the normalization and cocycle conditions. For u = 0, twist F R,u simplifies to F 0 , easily seen by just plugging in u = 0 in the (21), and for u = 1, it simplifies to the twist F 1 .
Hence F R,u provides another way of interpolating between F 0 and F 1 .

Hopf algebra
The coalgebra sector of the Hopf algebra U FR,u g for the deformation with F R,u can also be calculated and has the form: And antipodes

Coordinate realizations and star product
The inverse of the family of twists F −1 R,u provides another (new) star product between the functions (3). If we choose our functions to be exponential functions e ik·x and e iq·x , then we can define new functions D µ (u; k, q) and G(u; k, q) in the following way [16], [22]: = e iDµ(u;k,q)x µ 1 where k, q ∈ M 1,n−1 . Note the difference in the terms on the right hand side between the formula above and the one for F L,u in (18). One can calculate, see appendices C.2 and C.3, that in the case of the twist F −1 R,u the function D µ (u; k, q) is given by same as D µ (u; k, q) in equation (19). Note that the function D µ (u; k, q) can be seen as rewriting the coproduct ∆P µ without using the tensor product notation (denoting left and right leg by k and q respectively). Therefore the relation between the coproduct ∆P µ and the function D µ (u; k, q) is given by hence ∆P µ uniquely determines D µ (u; k, q), as in the case of F L,u twist. The additional function on the right hand side of (26) has the following explicit form: We refer the reader to the appendices C.2, C.3 and reference [16] for further the details of these calculations.
Note that in the case of the generalization of the Tolstoy twist (10) the function G(u; k, q) = 0 (see also [15]).
Realizations of noncommutative coordinatesx µ can be generally expressed in terms of Weyl-Heisenberg algebra generated by x µ and P µ (commutative variables).
The realization obtained from the F R,u twist has the new general form: Note that the part χ (P ) was not present in the case of F L,u twist, see subsection 2.1.2, and also [15]. Noncommutative coordinatesx µ , corresponding to the twist F R,u , are given bŷ From the last line in the above formula one can read off explicitly the form of the functions φ µ α (P ) and χ (P ) .
The noncommutative coordinatesx µ satisfy In the case u = 0,x µ = x µ + i 1 κ v µ D, while in the case u = 1,x µ = x µ (1 + 1 κ P ). The above kappa deformed Weyl-Heisenberg algebra (33) is obtained by using the realization (31). It turns out to be the same as in [15] where it was obtained from (20). Note the realizations (31) and (20) have different form.
3 Relations between two families F L,u and F R,u The two twists F L,u (10) and F R,u (21) are obviously related by the coboundary twists T ωL and T ωR in the following sense: where ω L = exp − u κ DP and ω R = exp − u κ P D . Hence and where we have used the homomorphism property of the coproduct and its deformed form. Now using the equalities: and where we chose the order with P generators on the left. Now taking the difference of these two expressions we obtain: Therefore we find, after multiplying both sides of (39) by e −uD P κ from the right: Inserting (40) into (36) we get: 41) which leads to: We point out that using star product in (18) and star product in (26) and methods introduced in [16] one can also obtain the above relation (42). Also one can write an explicit formula for the relation between R R,u and R L,u quantum Rmatrices. It has the following form:

Hopf algebras
The two twists F L,u and F R,u describe two presentations of Hopf algebra (with the same corresponding classical r-matrix). The coproducts and antipodes for momenta are the same: However this is not the case for dilatations:

Coordinate realizations and star products
Comparing the realizations we also see the difference. Equation (31) has an extra term only dependent on momenta whereas (20) does not.
Similarly in the formulae for the star products, the one coming from F R,u has an addition in the form of the function G(u; k, q) which does not appear in the case of F L,u . Nevertheless these two twists are only a change by coboundary twists from F 0 and provide equivalent Hopf algebra deformations (but with different representations).
Jordanian deformations have been of interest in some recent literature [35], [36], [37], [38], [39], [40]. In our previous paper [15] we have studied the simple generalization of the locally r-symmetric Jordanian twist. Following that idea now we have found another possible way of interpolating between the Jordanian twists F 0 and F 1 . This new version F R,u has provided a new type of star product (with the addition of the terms depending only on momenta in (26) as well as the new type of the realization of noncommuting coordinates (31)).
Even though both of the proposed twists, the previous one F L,u from [15] and the one introduced here F R,u provide the κ-Minkowski spacetime and have the support in the Poincaré-Weyl or conformal algebras as deformed symmetries of this noncommutative spacetime, the realizations and star products they induce differ. This may lead to differences in the physical phenomena resulting from star product based noncommutativity, see e.g. [41] to see how different realizations can influence modified dispersion relations between energy and momenta as well as time delay parameter.

A Exponential formula, normal ordering and Weyl-Heisenberg algebra
We recall that in the simplest case the Weyl-Heisenberg algebra W 1 (over the field C of complex numbers) can be defined abstractly as a universal associative and unital algebra over C with two generators x, p satisfying the relation where i ∈ C stands for the imaginary unit. Usually, the generator p can be identified with the derivative −i d/dx. Such realization makes it easy to remember the canonical action of W 1 onto the space of polynomial functions in one variable C[x]. Therefore, any element a ∈ W 1 admits a canonical presentation either in the form of differential operator a = ii) Moreover iii) J(k, p) turns out to be a unique (formal) solution of the (formal) partial differential equation: with the boundary condition: J(0, p) = p.
It follows that: Finally, differentiation of (49) gives This provides the differential equation (50) together with the boundary condition: J(0, p) = p. End of proof.
The last relation (67) can be shown in the following way.
Applying k α ∂ ∂kα on both sides to (66) we have Using the form of the realization forx µ = x α φ µ α (p), we obtain: In order to simplify this equation we introduce the change of variables: k α → λk α and use the identity: which leads to (67). Further generalization admits more general class of realizations. If 5 Proof of proposition 2 and it's generalizations will be given elsewhere.
C Star products C.1 Star products for F L,u -general formulas then, for action ⊲, defined by it holds: where functions K µ (k) and J µ (k, q) can be calculated from the following differential equations: with boundary conditions K µ (0) = 0 and J µ (k, 0) = K µ (k), J µ (0, q) = q µ . The star product is given by with the inverse function of K µ (k) defined as K µ K −1 (k) = K −1 µ (K (k)) = k µ .
The boundary condition is Q (k, 0) = g (k) , g (0) = Q (0, q) = 0. This gives [16], [22]: The star product is Now taking: Therefore, it follows that: e ik·x ⋆ e iq·x = e iD(k,q)·x+iG(k,q) C.3 F L,u and F R,u -explicit calculations Realizations of noncommutative coordinates for F L,u is (20): and for F R,u (31): From these realizations the form of the function φ µ α (P ) can be read as: where P = v α P α is used as a shortcut. Note that it is the same for both realizations (20), (31). Therefore, both of these realizations have the same form of the functions K(k), K −1 (k), J(k, q) and D(k, q), see below for the explicit calculations.