Interpolations between Jordanian Twists Induced by Coboundary Twists

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 054, 22 pages Interpolations between Jordanian Twists Induced by Coboundary Twists 1 2 3 Andrzej BOROWIEC † , Daniel MELJANAC † , Stjepan MELJANAC † and Anna PACHOL † †1 Institute of Theoretical Physics, University of Wroclaw, pl. M. Borna 9, 50-204 Wroclaw, Poland E-mail: andrzej.borowiec@ift.uni.wroc.pl †2 Division of Materials Physics, Ruder Bošković Institute, Bijenička c.54, HR-10002 Zagreb, Croatia E-mail: Daniel.Meljanac@irb.hr †3 Division of Theoretical Physic, Ruder Bošković Institute, Bijenička c.54, HR-10002 Zagreb, Croatia E-mail: meljanac@irb.hr †4 Queen Mary, University of London, Mile End Rd., London E1 4NS, UK E-mail: a.pachol@qmul.ac.uk 4 Received February 15, 2019, in final form July 11, 2019; Published online July 21, 2019 https://doi.org/10.3842/SIGMA.2019.054 Abstract. We propose a new generalisation of the Jordanian twist (building on the previous idea from [Meljanac S., Meljanac D., Pachol A., Pikutić D., J. Phys. A: Math. Theor. 50 (2017), 265201, 11 pages]). 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. Real forms of new Jordanian deformations are also discussed. Exponential formulae, used to obtain coproducts and star products, are presented with details. Key words: twist deformation; Hopf algebras; coboundary twists; star-products; real forms 2010 Mathematics Subject Classification: 81T75; 16T05; 17B37; 81R60 1 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 HF = H, ∆F , S F , ǫ , where ∆F (·) = F ∆(·)F −1 . For any invertible two-cocycle twist
element ω ∈ H one can define new gauge equivalent
−1 −1 F ω Fω = ω ⊗ ω F ∆(ω) which determines the third Hopf algebra H = 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 HF ∼ = H Fω as illustrated on the following diagram H ∆F −−−−−−→ α↓ H H ⊗H ↓α⊗α ∆Fω −−−→ H ⊗ H, 2 A. Borowiec, D. Meljanac, S. Meljanac and A. Pachol i.e., ∆F ◦ α = (α ⊗ α) ◦ ∆Fω . 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 m⋆ ≡ ⋆ to m⋆F ≡ ⋆F :   a ⋆F b = m⋆F (a ⊗ b) = m⋆ F −1 (⊲ ⊗ ⊲)(a ⊗ b) (1.1) for a, b ∈ A, one gets new module algebra AF = (A, ⊲, ⋆F ) over HF 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 ⊲ 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 H ⊗A ⊲ −−−−→ α⊗β ↓ H ⊗A A ↓β ⊲ −→ A
commutes. In particular, the coboundary twist Tω = ω −1 ⊗ω −1 ∆(ω) provides the Hopf algebra isomorphism H ∼ = HTω as well as module algebra isomorphism A ∼ = ATω . 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 Ug [14] 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) [15, 16], which in turn can be determined by the 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 Fr ∈ Ug ⊗ Ug [[γ]] called a Drinfeld twist. Here γ is a formal parameter and Ug [[γ]] means topological completion in the topology of formal power series in γ (see, e.g., [11, 12, 32] for details). Thus the classical r-matrix can be recovered from the quantum R-matrix as follows
Rr = Frτ Fr−1 = 1 ⊗ 1 + γr + o γ 2 , where τ denotes the flip map: F τ = F21 . 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: rAb = x ∧ y or Jordanian rJ = 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 Ug : Abelian or Jordanian. In the latter case, deformation can be realized by the Jordanian twist of the form FJ = exp(ln(1 + γe) ⊗ h). 1 It corresponds to the Lie group “ax + b” of affine transformations of the real (complex) line. (1.2) Interpolations between Jordanian Twists Induced by Coboundary Twists 3 This form of the twist first appeared in [49] and then a symmetrised form (i.e., where rJ appears in the first order in expansion of the twist) was proposed in [52, 54]2 γ   γ  FT = exp (he ⊗ 1 + 1 ⊗ he) exp (ln(1 + γe) ⊗ h) exp − ∆(he) . (1.3) 2 2 It was obtained by applying the coboundary twist to (1.2). There are many possible r-symmetrisations 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 coboundary twist (see, e.g., [32]). Jordanian deformations have been of interest for quite some time. For example, Jordanian deformations of the conformal algebra were considered in [45], as well as in [2, 3, 19]. In [2, 3, 19] the deformations of Anti de Sitter and de Sitter algebras were also investigated. Jordanian deformations have been considered in applications in AdS/CFT correspondence [25, 26, 34, 35], as integrable deformations of sigma models in relation to deformation of AdS5 and supergravity [20, 55]. Jordanian twists have been applied in deformation of spacetime metrics [5], Maxwell equations and dispersion relations [1], as well as classical and quantum mechanics [36]. Gauge theories under Jordanian deformation were also investigated in [13]. The question we would like to address in the present paper is if the mathematical equivalences of coboundary twists, in some physically relevant situations, give rise, to some extent, to physically inequivalent descriptions. For example, when one considers the star product quantization, realizations for noncommutative coordinates, or the form of Heinsenberg algebra. 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 = {Pµ , D}, generated by momenta Pµ (spacetime translations in n-dimensions where µ = 0, 1, . . . , n − 1) and dilatation generators D with the following commutation relations [Pµ , D] = Pµ , [Pµ , Pν ] = 0. This algebra can be considered as a subalgebra of some bigger Lie algebra, e.g., Poincaré–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 Pµ and the formulae in the next Section 2 will hold). However, for convenience, we choose the following notation: P = v α Pα where v µ is the vector on Minkowski spacetime M1,n−1 in n-dimensions such that v 2 = v α vα ∈ {−1, 0, 1}. For the correspondence with the κ-Minkowski spacetime [30, 31] we choose the deformation parameter as γ = − κ1 . Our main aim in this work is to analyse quantum deformations corresponding to gauge equivalence of Jordanian twists (1.2), extending [41] and applying [40]. This paper is a sequel to [41], 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 Section 2.2 we propose another generalised form of r-symmetrised Jordanian twist (FR,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. Section 3 presents a relation between the two generalisations of r-symmetric twists FL,u and FR,u , including the relation between the corresponding quantum R-matrices. In Section 4 the ∗-structure and unitarity of the twists is analysed. 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. 2 These techniques have been used before in the supersymmetric case in order to unitarize super Jordanian twist, see [6, 51, 53]. 4 A. Borowiec, D. Meljanac, S. Meljanac and A. Pachol 2 Two families of twists interpolating between Jordanian twists In our previous work [41] we have proposed a simple generalisation of the locally r-symmetric Jordanian twist (1.3), 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     1 F0 = exp − ln 1 − P ⊗ D (2.1) κ with the logarithm on the left side of the tensor product (cf. with (1.2)) and    1 τ F1 = (F0 ) |− 1 = exp −D ⊗ ln 1 + P κ κ with the logarithm on the right tensor factor and also with the changed sign of the deformation parameter κ, recall τ denotes the flip map: F τ = F21 . One can symmetrise these simple Jordanian twists, into a so-called r-symmetric form, such that at the first order in the expansion of the twist one gets the classical r-matrix corresponding to the given deformation. For the Jordanian deformations it is always rJ . In this paper we want to present another type of such interpolation. First, in the below section, we recall few main formulae from [41] and then, in Section 2.2, we shall propose another interpolation. 2.1 FL,u family of twists with dilatation on the left The r-symmetric version of the Jordanian twist (2.1) was introduced in [54] and it was obtained
1 from the coboundary twist
Tω by choosing ω0 = exp − 2κ DP . The formula for FT follows directly from: ω0−1 ⊗ ω0−1 F0 ∆(ω0 ). In [41] we have introduced its generalisation in the form of one parameter family interpolating between Jordanian twists ∀ u ∈ R      u 1 (DP ⊗ 1 + 1 ⊗ DP ) exp − ln 1 − P ⊗ D FL,u = exp κ κ    u u ∈ R. (2.2) × exp ∆ − DP , κ
This generalisation
simply corresponds to a modification of ω0 to ωL = exp − uκ DP and then FL,u = ωL−1 ⊗ ωL−1 F0 ∆(ωL ) still differs from F0 only by the coboundary twist TωL . For this reason the cocycle condition (see, e.g., [32]) for FL,u is automatically satisfied. For u = 21 (1.3) is recovered. Note that now FL,u contains two parameters: one real parameter u and the other κ -the formal deformation parameter. The reduction of FL,u , for certain values of the parameter u, to F0 (for u = 0) and to F1 (for u = 1) were discussed in [41]. 2.1.1 Hopf algebra The deformation of the Hopf algebra Ug (µ, ∆, ǫ, S) of the universal
enveloping algebra of g given by the twist element F ∈ Ug ⊗ Ug [[ κ1 ]] into UgF µ, ∆F , ǫ, S F is provided by the deformation of the coproduct and
 antipode maps as follows: ∆F (Z) = F ∆(Z)F −1 , S F (Z) =  µ((1 ⊗ S)F )]S(Z)[µ (S ⊗ 1)F −1 , where Z ∈ g. The coproducts, star products and realizations depend explicitly on the parameter u as well as on the parameter of deformation κ. Interpolations between Jordanian Twists Induced by Coboundary Twists 5 We recall the coalgebra sector of the Hopf algebra UgF0 for the deformation with the twisting element F0 from [8]  1 ∆ (Pµ ) = Pµ ⊗ 1 + 1 − P κ F0 −Pµ S (Pµ ) = , 1 − κ1 P F0  ⊗ Pµ , ∆F0 (D) = D ⊗ 1 +  1 S (D) = − 1 − P κ F0  1 ⊗ D, 1 − κ1 P D. The change of the twist by the coboundary twist
Tω provides a new presentation for the Hopf algebra, and can be transformed by α−1 ⊗ α−1 ◦ ∆F0 ◦ α = ∆Fω where α(Z) = ωZω −1 , Z ∈ H and α−1 (Z) = ω −1 Zω for chosen ω. For FL,u one needs to take ωL . Alternatively, the deformed coproducts and antipodes can be calculated directly from the F definition ∆Z = F ∆(Z)F −1 . The coalgebra sector of the Hopf algebra Ug L,u for the deformation with FL,u , recalled from [41], is as follows ∆ FL,u ∆ FL,u (Pµ ) = (D) =
Pµ ⊗ 1 + uκ P + 1 − 1 ⊗ 1 + u(1 − u) (1−u)
κ P ⊗
2 1 P ⊗P κ Pµ , 1 1 D⊗ + ⊗D 1 + u κ1 P 1 − (1 − u) κ1 P ! !  2 1 P ⊗P . 1 ⊗ 1 + u(1 − u) κ The coproduct is coassociative. The antipodes are given by Pµ , 1 − (1 − 2u) κ1 P !  1 − (1 − 2u) Pκ u  FL,u (D) = − S D 1+ P . u 1 + κP κ S FL,u (Pµ ) = − 2.1.2 Coordinate realizations and star product As a Hopf module algebra for Ug we choose the algebra of smooth (complex valued) functions on a space time (i.e., A = C ∞ (Rn ) ⊗ C with an obvious algebraic structures determined by pointwise multiplication and addition). This algebra includes spacetime coordinates xµ (where xµ 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, in terms of Weyl–Heisenberg algebra generators [∂µ , xν ] = δµν ,3 by Pµ ⊲ f (x) = −i ∂f (x) ∂ ⊲ f (x) = −i µ ∂x ∂xµ and D ⊲ f (x) = xµ ∂f (x) . ∂xµ (2.3) The algebra of coordinates becomes noncommutative due to the twist deformation once the usual multiplication is replaced by the star product multiplication (star product quantization) (1.1) for any f, g ∈ C ∞ (Rn ). This star product is associative (due to the fact that the twist FL,u satisfies the cocycle condition). When we choose the functions to be exponential functions eik·x and eiq·x , then we can define new function Dµ (u; k, q) [29, 42, 43, 44, 48]: 3 
 µ eik·x ⋆ eiq·x = m F −1 (⊲ ⊗ ⊲) eik·x ⊗ eiq·x = eiDµ (u;k,q)x , (2.4) This algebra has structure of a smash product and can be arranged in a Hopf algebroid structure [9, 10, 23, 24]. 6 A. Borowiec, D. Meljanac, S. Meljanac and A. Pachol where k, q ∈ M1,n−1 (in n-dimensional Minkowski spacetime). One can calculate explicitly, see Appendices C.1 and C.3, that in the case of twist FL,u the function Dµ (u; k, q) is given by Dµ (u; k, q) =
kµ 1 + uκ (v · q) + 1 − 1+ u(1−u) (v κ2 (1−u) κ (v · k)(v · q)
· k) qµ . (2.5) Directly from the twist we can also calculate the coordinate realizations [9, 18, 21, 22, 23, 24, 37] and for the FL,u twist they have the following form4    −1  u  i u  x̂µ = m FL,u (⊲ ⊗ 1)(xµ ⊗ 1) = xµ 1 + P + v µ (1 − u)D 1 + P κ κ κ    u i 1 + P = xα φµα (P ), = xµ + v µ (1 − u)D κ κ (2.6) where v µ is the vector on Minkowski spacetime M1,n−1 in n-dimensions such that v 2 ∈ {−1, 0, 1} as before. These realizations are also discussed in [27, 28, 38, 39, 46, 47]. 2.2 FR,u family of twists with dilatation on the right In this paper we want to introduce another version of the generalised Jordanian twist        u  u 1 FR,u = exp (P D ⊗ 1 + 1 ⊗ P D) exp − ln 1 − P ⊗ D exp ∆ − P D , (2.7) κ κ κ 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 [41] and which was recalled in the previous Section 2.1, i.e., FL,u (2.2). For the parameter u = 21 the twist FR, 1 is r-symmetric, but is not equal to FL, 1 . These two 2 2 twists differ at subleading order, see (3.4) in the following section. The form of the family of twists FR,u can also be easily obtained from the simple Jordanian twist F0 by the transformation
with the coboundary twist TωR however this time with the element ωR = exp − uκ P D . The twist FR,u , ∀ u, satisfies the normalization and cocycle conditions. For u = 0, twist FR,u simplifies to F0 , easily seen by just plugging in u = 0 in the equation (2.7), and for u = 1, it simplifies to the twist F1 . Hence FR,u provides another way of interpolating between F0 and F1 . 2.2.1 Hopf algebra F The coalgebra sector of the Hopf algebra Ug R,u for the deformation with FR,u can also be calculated and has the form

Pµ ⊗ 1 + u κ1 P + 1 − (1 − u) κ1 P ⊗ Pµ FR,u , (2.8) (Pµ ) = ∆
2 1 ⊗ 1 + u(1 − u) κ1 P ⊗ P !  2 1 ∆FR,u (D) = 1 ⊗ 1 + u(1 − u)P ⊗ P κ ! 1 1 + ⊗D . (2.9) × D⊗ 1 + u κ1 P 1 − (1 − u) κ1 P 4 For simplicity the matrix notation φα µ (P ) is written as φµ α (P ) throughout the paper. Interpolations between Jordanian Twists Induced by Coboundary Twists 7 Antipodes are −Pµ , 1 − (1 − 2u) κ1 P   P D S FR,u (D) = − 1 − (1 − u) κ S FR,u (Pµ ) = 2.2.2 (2.10) 1 − (1 − 2u) Pκ 1 − (1 − u) Pκ ! . (2.11) Coordinate realizations and star product −1 The inverse of the family of twists FR,u provides another (new) star product between the functions (1.1). If we choose our functions to be exponential functions eik·x and eiq·x , then we can define new functions Dµ (u; k, q) and G(u; k, q) in the following way [29, 40]  −1
 µ eik·x ⋆ eiq·x = m FR,u (⊲ ⊗ ⊲) eik·x ⊗ eiq·x = eiDµ (u;k,q)x +iG(u;k,q) 1 µ , (2.12) = eiDµ (u;k,q)x u(1−u) 1 + κ2 (v · k)(v · q) where k, q ∈ M1,n−1 . Note the difference in the terms on the right hand side between the formula above and the one for FL,u in (2.4). One can calculate, see Appendices C.2 and C.3, −1 that in the case of the twist FR,u the function Dµ (u; k, q) is the same as in equation (2.5). Note that the function Dµ (u; k, q) can be seen as rewriting the coproduct ∆(Pµ ) without using the tensor product notation (denoting left and right leg by k and q respectively). Therefore the relation between the coproduct ∆(Pµ ) and the function Dµ (u; k, q) is given by ∆(Pµ ) = Dµ (u; P ⊗ 1, 1 ⊗ P ), hence ∆(Pµ ) uniquely determines Dµ (u; k, q), as in the case of FL,u twist. The additional function on the right hand side of (2.12) has the following explicit form   u(1 − u) G(u; k, q) = i ln 1 + (v · k)(v · q) . (2.13) κ2 We refer the reader to the Appendices C.2, C.3 and [40] for further the details of these calculations. Note that in the case of the generalisation of the twist FT (1.3) into FL,u (2.2), the function G(u; k, q) = 0 (see also [41]). Realizations of noncommutative coordinates x̂µ can be generally expressed in terms of Weyl– Heisenberg algebra generated by xµ and Pµ (commutative variables). The realization obtained from the FR,u twist has the new general form x̂µ = xα φµα (P ) + χµ (P ). Note that the part χµ (P ) was not present in the case of FL,u twist, see equation (2.6) in Section 2.1.2, and also [41]. Noncommutative coordinates x̂µ , corresponding to the twist FR,u , are given by    −1  u  u  i x̂µ = m FR,u (⊲ ⊗ 1)(xµ ⊗ 1) = xµ 1 + P + v µ (1 − u) 1 + P D κ κ κ    i i u = xµ + v µ (1 − u)D 1 + P + u(1 − u) 2 v µ P. κ κ κ (2.14) From the last line in the above formula one can read off explicitly the form of the functions 1 µ µ µ and χµ (P ). In the
case when u = 0, we have x̂ = x + i κ v D, while in the case when 1 µ µ u = 1, x̂ = x 1 + κ P . φµα (P ) 8 A. Borowiec, D. Meljanac, S. Meljanac and A. Pachol The noncommutative coordinates x̂µ satisfy
 i µ ν v x̂ − v ν x̂µ , x̂µ , x̂ν = κ      1 ν 1 ν ν Pµ , x̂ = −iδµ + i v (1 − u)Pµ 1+u P . κ κ  (2.15) The above kappa deformed Weyl–Heisenberg algebra (2.15) is obtained by using the realization (2.14). It turns out to be the same as in [41] where it was obtained from (2.6). Summarising, the realizations (2.14) and (2.6) differ by the term χµ (P ) which does not change the form of (2.15). 3 Relations between two families FL,u and FR,u The two twists FL,u (2.2) and FR,u (2.7) are obviously related by the coboundary twists TωL and TωR in the following sense

−1 −1 FR,u = ωR ωL ⊗ ωR ωL FL,u ∆ ωL−1 ωR ,

where ωL = exp − uκ DP and ωR = exp − uκ P D . Hence and  u   u  u  −1 −1 FR,u = exp ∆ exp P D exp −∆ DP FL,u (DP ⊗ 1 + 1 ⊗ DP ) κ  κ  κu × exp − (P D ⊗ 1 + 1 ⊗ P D) κ −1 FL,u FR,u    u  P P exp (DP ⊗ 1 + 1 ⊗ DP ) exp u D exp −uD =∆ κ κ κ  u  × exp − (P D ⊗ 1 + 1 ⊗ P D) , κ FL,u   (3.1) where we have used the homomorphism property of the coproduct and its deformed form. We can use the equalities5
n    ∞ ∞  X X u Pκ D P n D uP D κ = e = u n! κ n n=0 n=0 and e uD P κ ∞ X uD Pκ = n! n=0
n    ∞  X P n D−1 = u , κ n n=0 where we chose the order with P generators on the left. Now taking the difference of these two expressions we obtain e uP D κ −e uD P κ         X ∞  ∞  X D−1 P n D−1 P n D − = = u u κ κ n n−1 n n=1 n=0 P P = u euD κ . κ 5 Here D n
denotes the generalised binomial coefficient: (3.2) D n
= Dn n! = D(D−1)(D−2)...(D−(n−1)) . n(n−1)(n−2)···1 Interpolations between Jordanian Twists Induced by Coboundary Twists 9 P Therefore, we find, after multiplying both sides of (3.2) by e−uD κ from the right P P eu κ D e−uD κ = 1 + u P . κ (3.3) Inserting (3.3) into (3.1) we get −1 FL,u FR,u = ∆FL,u =  P 1+u κ  1 1 ⊗ 1 + u Pκ ⊗ 1 1 1 ⊗ 1 + u(1 − u) Pκ ⊗ P κ ! 1 1⊗1+u·1⊗ P κ ! , which leads to −1 −1 FR,u = FL,u 1 1⊗1+ u(1−u) P κ2 ⊗P . (3.4) Note that the twists FL,u and FR,u agree in the leading order of the deformation parameter, but are different at higher orders. We point out that using star product in (2.4) and star product in (2.12) and methods introduced in [40] one can also obtain the above relation (3.4). Also one can write an explicit formula for the relation between RR,u and RL,u quantum R-matrices. It has the following form   1 u(1 − u) RL,u = P ⊗ P . 1 ⊗ 1 + R R,u κ2 1 ⊗ 1 + u(1−u) P ⊗ P 2 κ Hopf algebras The two twists FL,u and FR,u describe two presentations of Hopf algebra (with the same corresponding classical r-matrix). The coproducts and antipodes for momenta are the same ∆FL,u (Pµ ) = ∆FR,u (Pµ ), S FL,u (Pµ ) = S FR,u (Pµ ). However this is not the case for dilatations ∆FL,u (D) 6= ∆FR,u (D), S FL,u (D) 6= S FR,u (D). Coordinate realizations and star products Comparing the realizations we also see the difference. Equation (2.14) has an extra term only dependent on momenta whereas (2.6) does not. Similarly in the formulae for the star products, the one coming from FR,u has an addition in the form of the function G(u; k, q) which does not appear in the star product coming from FL,u . Nevertheless, these two twists are only a change by coboundary twists from F0 and provide equivalent Hopf algebra deformations (but with different representations). 4 Discussion on the real forms of the Jordanian deformations For physical applications it is important to address the question if the symmetry Hopf algebras and the deformed Weyl–Heisenberg algebra can be endowed with (compatible) ∗-structures. In general, ∗-structure (real Lie algebra structure) can be introduced by an antilinear involutive anti-automorphism ∗ : g → g acting on the complex Lie algebra g. Thus the real coboundary Lie 10 A. Borowiec, D. Meljanac, S. Meljanac and A. Pachol bialgebra can be considered as a triple (g, ∗, r), where the skew-symmetric element r is assumed to be anti-Hermitian, i.e.,6 r∗⊗∗ = −r = rτ . The ∗-operation extends, by the property (XY )∗ = Y ∗ X ∗ (i.e., as an antilinear antiautomorphism), to the enveloping algebra Ug , as well as to quantized enveloping algebra, making both of them associative ∗-algebras. Therefore, any quantized enveloping algebra admits a natural ∗-structure, inherited from the corresponding Lie bialgebra structure, which can be preserved under twist deformation provided that the twisting element is ∗-unitary, i.e., F ∗⊗∗ = F −1 . (4.1) More exactly, we recall that a complex Hopf algebra H = (H, ∆, S, ǫ) endowed with an antilinear involutive anti-automorphism ∗ : H → H is called a real Hopf algebra or Hopf ∗-algebra if the following compatibility conditions for coproducts, antipodes and counits are satisfied ∆(X ∗ ) = (∆(X))∗⊗∗ , S((S(X ∗ ))∗ ) = X, ǫ(X ∗ ) = ǫ(X) ∀ X ∈ H, where the standard (’unflipped’) way of ∗-operation acting on a tensor product (i.e., coproduct) is assumed (X ⊗ Y )∗ = X ∗ ⊗ Y ∗ . In general, the twist deformation of quasitriangular Hopf algebra (H, R) give rise to quasitriangular Hopf algebra with new universal R-matrix RF = F τ RF −1 . If the real form of quasitriangular Hopf algebra is twisted by unitary twist then the deformed Hopf algebra is also real quasitriangular Hopf algebra. More precisely, if (H, ∆, S, ǫ, R, ∗) is a quasitriangular Hopf ∗-algebra with R being real universal R-matrix (i.e., satisfying R∗⊗∗ = Rτ ) (resp. antireal if satisfying R∗⊗∗ = R−1 ), then for any unitary, normalized 2-cocycle twist F = (F −1 )⋆⊗⋆ ∈ H ⊗H the quantized algebra (H, ∆F , S F , ǫ, RF , ∗) is a quasitriangular Hopf ∗-algebra such that RF = F τ RF −1 is real (resp. antireal). In our case only two Jordanian twists F0 and F1 are unitary, i.e., satisfy the condition (4.1) provided that we assume the following ∗-conjugations Pµ∗ = Pµ , and D∗ = −D, (4.2) which are compatible with the commutation relations [Pµ , D] = Pµ , [Pµ , Pν ] = 0.7 Therefore, the corresponding twisted deformations are the Hopf ∗-algebras with the same ∗-structure. Instead, for generic value of the parameter u ∈ R two families of twists are related to each other by ∗  F ∗⊗∗  F −1 ∗⊗∗ S L,u (X) = S FR,1−u (X ∗ )|−κ . = ∆FR,u (X ∗ ), , ∆ L,u (X) = FR,u FL,u Therefore, the well-known purely Jordanian twists F0 and F1 remain only unitary with respect to the conjugation (4.2). This situation can change, if we use the method proposed by S. Majid [32, Proposition 2.3.7, p. 59] and admit deformation of the original ∗-structure. Before, one needs to check if the twist satisfies the following condition
(S ⊗ S) F ∗⊗∗ = F τ , (4.3) 6 More detailed exposition of the background material for the present section, as well as more complete list of references, can be found in [7]. 7 It can be observed that these commutation relations determine the generator D up an additive constant while each generator Pµ can be multiplied by a constant. For this reason the conjugation D∗ = −D + c, Pµ∗ = eib Pµ (c, b are real constants) would be admissible too. In fact, our Weyl–Heisenberg ∗-algebra realisation (2.3) is compatible with the assumption that D∗ = −D − n, Pµ∗ = Pµ , where n denotes the spacetime dimension. Interpolations between Jordanian Twists Induced by Coboundary Twists 11 where the antipode map S and the conjugations ∗ are taken before deformation. If it does, then one can define new quantized †-structure: ( )† = S −1 (U )(
)∗ S −1 U −1 , which makes the twisted Hopf algebra real. Here U is related with the twist by the formula P P (1) −(1)
−(2) (2)
fi with the short cut notation for the twist and U −1 = S fi fi S fi U = i i P (1) P −(1) (2) −(2) introduced as follows: F = fi ⊗ fi and its inverse as: F −1 = fi ⊗ fi . i i New quantized †-structure for FL,u= 1 twist 4.1 2 One can explicitly check that only for one value of the parameter u, namely u = tion (4.3) for the Majid’s method is satisfied. By direct calculation one gets

∗⊗∗ = FL, 1 l.h.s. = (S ⊗ S) FL, 1 2 2 and 18 2 the condi- −κ      1 1 τ r.h.s. = (FL, 1 ) = exp (DP ⊗ 1 + 1 ⊗ DP ) (F0 ) exp ∆ − DP 2 2κ 2κ      1 1 (DP ⊗ 1 + 1 ⊗ DP ) (F1 )|−κ exp ∆ − DP = exp 2κ 2κ       1 1 = exp − (DP ⊗ 1 + 1 ⊗ DP ) F1 exp ∆ |−κ DP 2κ 2κ      
1 1 (DP ⊗ 1 + 1 ⊗ DP ) F0 exp −∆ DP |−κ = FL, 1 |−κ . = exp 2 2κ 2κ τ Hence

∗⊗∗ = FL, 1 (S ⊗ S) FL, 1 2 2 −κ = FL, 1 2 Therefore X † = −S FL, 1 2 (X ∗ ),
τ . ∀ X ∈ H. Analogously, one can show that FR, 1 also satisfies the condition (4.3) 2

∗⊗∗ (S ⊗ S) FR, = FR, 1 1 2 2 −κ = FR, 1 and the new star structure X † = −S FR, 1 2 (X ∗ ), 2
τ ∀ X ∈ H. It shows that for u = 21 we can introduce an exotic Hopf †-algebra structure which, however, in the classical limit reduces to the standard ∗ one. 8 Note that only for this value of the parameter u the twists FR,u and FL,u are r-symmetric. 12 4.2 A. Borowiec, D. Meljanac, S. Meljanac and A. Pachol Coboundary Hopf ∗-algebra Actually, the best way to obtain coboundary unitary twist preserving a given ∗-Hopf algebra structure is by using a unitary coboundary element ω ∗ = ω −1 . As a particular example let us consider a one-parameter family of such elements   u ωLR = exp − (DP + P D) . 2κ Then the new generalised family of the Jordanian twists
−1 −1 F0 ∆(ωLR ) FLR,u = ωLR ⊗ ωLR is unitary for any real parameter u and satisfies ∗⊗∗ −1 FLR,u = FLR,u with Pµ∗ = Pµ , D∗ = −D. Note that FLR,u is r-symmetric again only for u = 12 and differs from both FL, 1 and FR, 1 . 2 2 FLR,u provides, via the usual twist deformation, a new Hopf ∗-algebra r 1 u(1 − u) P ⊗ P ∆FL,u (Pµ ) q ∆FLR,u (Pµ ) = 1 ⊗ 1 + 2 κ P ⊗P 1 ⊗ 1 + u(1−u) κ2

Pµ ⊗ 1 + uκ P + 1 − (1−u) κ P ⊗ Pµ = , u(1−u) 1 ⊗ 1 + κ2 P ⊗ P r u(1 − u) 1 FLR,u (D) = 1 ⊗ 1 + ∆ P ⊗ P ∆FL,u (D) q κ2 u(1−u) 1 ⊗ 1 + κ2 P ⊗ P ! r 1 u(1 − u) 1 + = 1⊗1+ P ⊗P D⊗ ⊗D κ2 1 + uκ P 1 − (1−u) κ P r u(1 − u) P ⊗P × 1⊗1+ κ2 and Pµ , 1 − (1 − 2u) κ1 P s

  s  1 − (1 − u) Pκ 1 + uκ P P P FLR,u
1 − (1 − 2u) D 1 − (1 − 2u) . (D) = − S κ κ 1 + uκ P 1 − (1 − u) Pκ S FLR,u (Pµ ) = − These formulae can be calculated by using the following relation between the twists FLR,u , FL,u and FR,u : r 1 u(1 − u) −1 −1 −1 FLR,u = FL,u q P ⊗ P. = FR,u 1 ⊗ 1 + κ2 P ⊗ P 1 ⊗ 1 + u(1−u) κ2 We can also provide the relation between quantum R-matrices r 1 u(1 − u) RLR,u = 1 ⊗ 1 + . P ⊗ P RL,u q 2 κ 1 ⊗ 1 + u(1−u) P ⊗ P 2 κ Interpolations between Jordanian Twists Induced by Coboundary Twists 5 13 Conclusions Jordanian deformations have been of interest in some recent literature [1, 2, 3, 5, 13, 19, 20, 25, 26, 34, 35, 36, 45, 50, 55]. In our previous paper [41] we have studied the simple generalisation of the locally r-symmetric Jordanian twist. Following that idea now we have found another possible way of interpolating between two Jordanian twists F0 and F1 . In both cases, we have introduced one real valued parameter u and obtained the family of Jordanian twists which provides interpolation between the original unitary Jordanian twist (for u = 0) and its flipped version (for u = 1, up to minus sign in the deformation parameter). If D∗ = −D and Pµ∗ = Pµ both twists F0 and F1 are ∗-unitary while those for u 6= 0, 1 are not. Only in the case of u = 12 the interpolating twists provide the real Hopf algebra structures with the deformed ∗-structures (obtained by deformation techniques from [32]). Also only for this value of the parameter u the twists FR,u and FL,u and FLR,u are r-symmetric. However, using unitary coboundary elements ω ∗ = ω −1 one can interpolate preserving the original real ∗-structure. The new family of Jordanian twists FR,u (2.7), as all Jordanian twists mentioned in this work, provides the socalled κ-Minkowski noncommutative spacetime and has the support in the Poincaré–Weyl or conformal algebras as deformed symmetries of this noncommutative spacetime. In this paper, starting from the twist (2.7), we have found deformed Hopf algebra symmetry (2.8)–(2.11), star products (2.12) and corresponding realizations for noncommutative coordinates x̂µ (2.14). Noncommutative coordinates induce the deformation of Weyl–Heisenberg algebra (2.15) as well. Even though both of the proposed twists, the previous one FL,u from [41] and the one introduced here FR,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. The new type of star product (2.12) obtained here contains additional term depending only on momenta G(u; k, q). The additional term χµ (P ) also appears in the realization of noncommuting coordinates in (2.14). However, the Weyl–Heisenberg algebra form is the same in both FL,u and FR,u deformed cases (2.15). Therefore, mathematically equivalent deformations, may lead to differences in the physical phenomena. Many authors [1, 13, 36] have used star products to discuss physical consequences of deformations. In those cases the two families of Jordanian twists FL,u , FR,u would lead to different outcomes. It would be interesting, for example, to investigate the deformation of differential and integral calculus in the context of the new versions of Jordanian twists proposed here and their application to second quantization [17]. Also the differences in realizations of noncommutative coordinates could lead to different physical predictions, see, e.g., [4] where the influence of different realizations on modified dispersion relations between energy and momenta as well as time delay parameter was investigated. A Exponential formula, normal ordering and Weyl–Heisenberg algebra We recall that in the simplest case the Weyl–Heisenberg algebra W1 (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 xp − px = i, 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 W1 onto the space of polynomial functions in one variable C[x]. Therefore, any element a ∈ W1 Np P admits a canonical presentation either in the form of differential operator a = as (x)ps , s=0 14 A. Borowiec, D. Meljanac, S. Meljanac and A. Pachol where as (x) ∈ C[x] or in the normal ordered form a = Nx P xr ar (p), where ar (p) ∈ C[p], see, r=0 e.g., [33]. This algebra can be also defined as a smash product of two polynomial algebras, i.e., W1 = C[x] ⋊ C[p], where C[x] plays a role of module algebra over the Hopf algebra C[p].9 Using smash product definition one can naturally extend W1 to C[[x]] ⋊ C[p], i.e., replacing the polynomial algebra C[x] by its x-adic extension C[[x]] of formal power series (see, e.g., [12]), which c1 bears the structure of (left) C[[x]] module. Its topological completion provides the algebra W ∞ P r with elements having a form x ar (p), ar (p) ∈ C[p]. r=0 For our purposes, however, one needs further extension by introducing a new commuting c1 [[k]] with elements of the form formal variable k and taking W ∞ X xr ar (k, p) = r=0 ∞ X xr k s ar,s (p), r,s=0 where ar,s (p) ∈ C[p] as before and ar (k, p) = ∞ P ar,s (p)k s ∈ C[p][[k]]. Now we are in position to s=0 formulate the following Proposition A.1. In this framework: i) For any element φ ≡ φ(p) ∈ C[p] there exists a unique element Φ ≡ Φ(k, p) ∈ C[p][[k]] such that ∞ X (ix)r exp(ikxφ(p)) = [Φ(k, p)]r r! r=0 c1 [[k]]. One should notice that the last expression can be denoted as holds true in W : exp(ixΦ(k, p)):, where : . . . : denotes normal ordering of the generators x, p (i.e., x’s left from p’s). ii) Moreover Φ(k, p) = J(k, p) − p, where J(k, p) = e−ikxφ(p) peikxφ(p) . (A.1) iii) J(k, p) turns out to be a unique (formal) solution of the (formal) partial differential equation ∂ J(k, p) = φ(J(k, p)) (A.2) ∂k with the boundary condition: J(0, p) = p. c1 [[k]], one can employ the adjoint action adp a = [p, a] to calcuProof . Since exp(ikxφ(p)) ∈ W late ∞ X (ix)n Φn (k, p), exp(ikxφ(p)) = n! n=0 (adp )m exp(ikxφ(p)) = ∞ X (ix)n−m Φn (k, p), (n − m)! n=m (adp )m exp(ikxφ(p))|x=0 = Φm (k, p). 9 For the definition see, e.g., [10] and references therein. Interpolations between Jordanian Twists Induced by Coboundary Twists 15 Hence exp(ikxφ(p)) = ∞ X (ix)n n=0 (adp )n (exp(ikxφ(p)) n!
x=0 . (A.3) On the other hand, introducing J(k, p) by formula (A.1), one gets (cf. (A.3))
 
adp eikxφ(p) = p, eikxφ(p) = eikxφ(p) e−ikxφ(p) peikxφ(p) − p = eikxφ(p) (J(k, p) − p) = eikxφ(p) Φ(k, p), i.e., Φ(k, p) = J(k, p) − p. Similarly, by induction
(adp )n eikxφ(p) = eikxφ(p) (Φ(k, p))n . It follows that Φn (k, p) = (adp )n eikxφ(p) hence exp(ikxφ(p)) = ∞ X (ix)n n=0 n!
x=0 = (Φ(k, p))n = (J(k, p) − p)n , (Φ(k, p))n = : exp(ikxΦ(k, p)):. Finally, differentiation of (A.1) gives ∂ J(k, p) = e−ikxφ(p) [−ix, p]φ(p)eikxφ(p) = e−ikxφ(p) φ(p)eikxφ(p) = φ(J(k, p)). ∂k This provides the equation (A.2) together with the boundary condition: J(0, p) = p. B  The multidimensional case Replacing W1 by the Weyl–Heisenberg algebra WN with 2N generators xα , pβ : xα xβ − xβ xα = pα pβ − pβ pα = 0, xα pβ − pβ xα = iδβα , where α, β = 0, 1, . . . , N − 1, allows for the following generalisation [48] Proposition B.1.10 For arbitrary realization φ(p) for the noncommutative coordinates x̂µ = xα φµα (p) it holds: i)

exp ikα x̂α = : exp ixα Φα (k, p) :, where : . . . : denotes normal ordering of the generators x, p (i.e., x’s left from p’s). ii) α α Φµ (k, p) = e−ikα x̂ pµ eikα x̂ − pµ = Jµ (k, p) − pµ , i.e., α α Jµ (k, p) = e−ikα x̂ pµ eikα x̂ . 10 Proof of Proposition B.1 and it’s generalisations will be given elsewhere. (B.1) 16 A. Borowiec, D. Meljanac, S. Meljanac and A. Pachol iii) Jµ (k, p) satisfies d Jµ (λk, p) = kα φαµ (J(λk, p)) dλ (B.2) with the boundary condition: Jµ (0, p) = pµ . The last relation (B.2) can be shown in the following way. Applying kα ∂k∂α on both sides to (B.1) we have e−ikx̂ [−ikx̂, pµ ]eikx̂ = kα ∂ (Jµ (k, p)). ∂kα Using the form of the realization for x̂µ = xα φµα (p), we obtain kα ∂ (Jµ (k, p)) = kα φαµ (J(k, p)). ∂kα In order to simplify this equation we introduce the change of variables: kα → λkα and use the identity λ d ∂ ∂ = λkα = kα , dλ ∂(λkα ) ∂kα which leads to (B.2). Further generalisation admits more general class of realizations. If x̂µ = xα φµα (p) + χµ (p) for φµα (p) and χµ (p) as in (2.14) then we have

exp ikα x̂α = : exp ixα (Jα (k, p) − pα ) : exp(iQ(k, p)). Therefore we get e−ikα x β φα (p) β pµ eikx̂ = Jµ (k, p)eiQ(k,p) . Applying kα ∂k∂α on both sides we finally obtain kα ∂ Q(k, p) = kα χα (J(k, p)), ∂kα i.e., after changing variables kα → λkα we find d Q(λk, p) = kα χα (J(λk, p)). dλ Note that if χα (p) 6= 0 then α β α +iQ(k,q) eikα x̂ ⊲ eiqβ x = eiJα (k,q)x , where the action is defined in (C.1) below. For kα = 0 it holds: Jµ (0, q) = qµ and Q(0, q) = 0. If qµ = 0 then Jµ (k, 0) = Kµ (k) and Q(k, 0) = g(k). Interpolations between Jordanian Twists Induced by Coboundary Twists C 17 Star products C.1 Star products for FL,u : general formulas If we start with realization x̂µ = xα φµα (P ), then, for action ⊲, defined by xµ ⊲ f (x) = xµ f (x), Pµ ⊲ f (x) = −i ∂f (x) , ∂xµ (C.1) it holds eik·x̂ ⊲ 1 = eikµ x α φµ (P ) α eik·x̂ ⊲ eiq·x = eikµ x ⊲ 1 = eiK(k)·x , α φµ (P ) α ⊲ eiq·x = eiJ(k,q)·x , where functions Kµ (k) and Jµ (k, q) can be calculated from the following differential equations dK µ (λk) = φµα (K(λk))k α , dλ dJ µ (λk, q) = φµα (J(λk, q))k α , dλ (C.2) (C.3) with boundary conditions Kµ (0) = 0 and Jµ (k, 0) = Kµ (k), Jµ (0, q) = q µ . The star product is given by eik·x ⋆ eiq·x = eiK −1 (k)·x̂ ⊲ eiq·x = eiJ(K −1 (k),q)·x = eiD(k,q)·x , where D(k, q) = J K −1 (k), q

with the inverse function of Kµ (k) defined as Kµ K −1 (k) = Kµ−1 (K(k)) = kµ . C.2 Star products for FR,u : general formulas For a more general realization given by x̂µ = xα φµα (P ) + χµ (P ), it holds eik·x̂ ⊲ 1 = eiK(k)·x+ig(k) , eik·x̂ ⊲ eiq·x = eiJ(k,q)·x+iQ(k,q) . K(k), J(k, q) satisfy the same differential equation as in Appendix C.1. Similarly we can determine g(k) and Q(k, q) by differentiating with respect to λ [29, 40] dg(λk) = k · χ(K(λk)), dλ dQ(λk, q) = k · χ(J(λk, q)). dλ (C.4) (C.5) 18 A. Borowiec, D. Meljanac, S. Meljanac and A. Pachol The boundary condition is Q(k, 0) = g(k), g(0) = Q(0, q) = 0. This gives [29, 40] Z 1 g(k) = Q(k, 0) = dλ[k · χ(K(λk))], 0 Z 1 dλ[k · χ(J(λk, q))]. Q(k, q) = 0 The star product is eik·x ⋆ eiq·x = eiK −1 (k)·x̂−g(K −1 (k)) ⊲ eiq·x = eiJ(K −1 (k),q)·x+iQ(K −1 (k),q)−iQ(K −1 (k),0) . Now we take
D(k, q) = J K −1 (k), q ,

G(k, q) = Q K −1 (k), q − Q K −1 (k), 0 . (C.6) Therefore, it follows that eik·x ⋆ eiq·x = eiD(k,q)·x+iG(k,q) . C.3 FL,u and FR,u : explicit calculations Realizations of noncommutative coordinates for FL,u is (2.6)   u  i(1 − u) µ v D 1+ P , u ∈ [0, 1], x̂µ = xµ + κ κ and for FR,u (2.14)   i(1 − u) µ u  iu(1 − u) µ x̂µ = xµ + v D v P, 1+ P + κ κ κ2 u ∈ [0, 1]. From these realizations the form of the function φµα (P ) can be read as   (1 − u) µ u  φµα (P ) = δαµ − 1+ P , v Pα κ κ where P = v α Pα is used as a shortcut. Note that it is the same for both realizations (2.6), (2.14). 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. To obtain K µ (k) and J µ (λk, q), the following differential equations (C.2), (C.3) need to be solved    (1 − u) µ dK µ (λk) u = φµα (K(λk))k α = δαµ − v Kα (λk) (C.7) 1 + K(λk) k α , dλ κ κ    (1 − u) µ u dJ µ (λk, q) α µ µ = φα (J(λk, q))k = δα − v Jα (λk, q) (C.8) 1 + J(λk, q) k α , dλ κ κ after using the explicit form of φ. Here the shortcut notation P = v α Pα has been extended to K(λk) = v α Kα (λk) and J(λk, q) = v α Jα (λk, q). The solution of the first equation (C.7) is µ K (k) = k µe 1 v·k κ 1 κv −1 1 . 1 · k (1 − u)e κ v·k + u (C.9) Interpolations between Jordanian Twists Induced by Coboundary Twists The inverse function K −1 K
−1 µ
µ 1 ln (k) = k 1 κv · k µ 19 (k) is 1 + uκ (v · k) 1− (1−u) κ (v · k) ! . The function J µ (k, q) is calculated similarly. The solution of the second equation (C.8) is

Kµ (k) 1 + uκ (v · q) + 1 − (1−u) κ (v · K(k)) qµ (C.10) Jµ (k, q) = 1 + u(1−u) (v · K(k))(v · q) 2 κ and K µ (k) = Jµ (k, 0). v µ P , so we can determine g(k) and Q(k, q) by differFor FR,u case, we have χµ (P ) = iu(1−u) κ2 entiating with respect to λ, i.e., using (C.4), (C.5), dg(λk) iu(1 − u) = k · χ(K(λk)) = (k · v)(v · K(λk)), dλ κ2 iu(1 − u) dQ(λk, q) = k · χ(J(λk, q)) = (k · v)(v · J(λk, q)). dλ κ2 To find the solution to the first equation we use (C.9)
1 κ e κ v·k − 1 , v · K(k) = 1 (1 − u)e κ v·k + u and for the second equation we take (C.10)

(v · K(k)) 1 + uκ (v · q) + 1 − (1−u) κ (v · K(k)) (v · q) v · J(k, q) = . 1 + u(1−u) (v · K(k))(v · q) 2 κ So g(k) and Q(k, q) are calculated as Z 1 iu(1 − u) dλ[v · K(λk)], (k · v) g(k) = Q(k, 0) = κ2 0 Z 1 iu(1 − u) Q(k, q) = dλ[v · J(λk, q)]. (k · v) κ2 0 The solutions are   (1−u) u g(k) = Q(k, 0) = i ln ue− κ v·k + (1 − u)e κ v·k ,       (1−u) u 1 1 − v·k v·k + (1 − u) 1 + u v · q e κ Q(k, q) = i ln u 1 − (1 − u) v · q e κ . κ κ Using equations (C.6), it follows that

kµ 1 + uκ (v · q) + 1 − (1−u) κ (v · k) qµ Dµ (u; k, q) = 1 + u(1−u) (v · k)(v · q) κ2 and  u(1 − u) (v · k)(v · q) , G(u; k, q) = i ln 1 + κ2  i.e., eiG(u;k,q) = 1 1+ u(1−u) (v κ2 · k)(v · q) . Compare the above with (2.5) and (2.13). 20 A. Borowiec, D. Meljanac, S. Meljanac and A. Pachol Acknowledgments This work has been supported by COST (European Cooperation in Science and Technology) Action MP1405 QSPACE. AB is supported by Polish National Science Center (NCN), project UMO-2017/27/B/ST2/01902. We are grateful to Zoran Škoda for his comments. We would like to thank the referees for their constructive input. References [1] Aschieri P., Borowiec A., Pachol A., Observables and dispersion relations in κ-Minkowski spacetime, J. High Energy Phys. 2017 (2017), no. 10, 152, 27 pages, arXiv:1703.08726. [2] Ballesteros A., Bruno N.R., Herranz F.J., A non-commutative Minkowskian spacetime from a quantum AdS algebra, Phys. Lett. B 574 (2003), 276–282, arXiv:hep-th/0306089. 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