Realizations of $\kappa$-Minkowski space, Drinfeld twists and related symmetry algebras

Realizations of $\kappa$-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of $\mathfrak{gl}(n)$ generators. There are three one-parameter families of linear realizations for time-like and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between deformed Heisenberg algebra, star product, coproduct of momenta and twist operator is presented. It is proved that for each linear realization there exists Drinfeld twist satisfying normalization and cocycle conditions. $\kappa$-deformed $\mathfrak{igl}(n)$-Hopf algebras are presented for all cases. The $\kappa$-Poincar\'e-Weyl and $\kappa$-Poincar\'e-Hopf algebras are discussed. Left-right dual $\kappa$-Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All known Drinfeld twists related to $\kappa$-Minkowski space are obtained from our construction. Finally, some physical applications are discussed.

One of the biggest problems in fundamental theoretical physics is a great difficulty to reconcile quantum mechanics and general theory of relativity in order to formulate consistent theory of quantum gravity. It is argued that at very high energies the gravitational effects can no longer be neglected and that the spacetime is no longer a smooth manifold, but rather a fuzzy, or better to say a noncommutative space [1]. Physical theories on such noncommutative manifolds require a new framework. This new framework is provided by noncommutative geometry [2]. In this framework, the search for generalized (quantum) symmetries that leave the physical action invariant leads to deformation of Poincaré symmetry, with κ-Poincaré symmetry being among the most extensively studied [3][4][5][6].
κ-deformed Poincaré symmetry is algebraically described by the κ-Poincaré-Hopf algebra and is an example of deformed relativistic symmetry that can possibly describe the physical reality at the Planck scale. κ is the deformation parameter usually interpreted as the Planck mass or some quantum gravity scale. It was shown that quantum field theory with κ-Poincaré symmetry emerges in a certain limit of quantum gravity coupled to matter fields after integrating out the gravitational/topological degrees of freedom [7]. This amounts to an effective theory in the form of a noncommutative field theory on the κ-deformed Minkowski space.
It is known [8] that the deformations of the symmetry group can be realized through the application of the Drinfeld twist on that symmetry group [9][10][11][12]. The main virtue of the twist formulation is that the deformed (twisted) symmetry algebra is the same as the original undeformed one and that there is only a change in the coalgebra structure which then leads to the same free field structure as the corresponding commutative field theory.
In [13] it was shown that the coproduct of D = 2 and D = 4 quantum κ-Poincaré algebra in the classical basis can not be obtained by the cochain twist depending only on the Poincaré generators (even if the coassociativity condition is relaxed). However the deformation used in [13] is the so called time-like type of deformation, and it is known [14] that for light-like deformation such a twist indeed exists [15][16][17][18].
In this work, we go other way round. Starting from κ-Minkowski space, we obtain its linear realizations, then coproducts of momenta from realizations and finally, we present a method for obtaining corresponding twists from those coproducts. We show that, for linear realizations, those twists are Drinfeld twists, satisfying normalization and cocycle conditions. The method for obtaining Drinfeld twists corresponding to each linear realization is elaborated and it is shown how these twists generate new Hopf algebras. The resulting symmetry algebras are κ-deformed igl(n) Hopf algebras. In special cases we obtain κ-Poincaré-Weyl-Hopf algebra and κ-Poincaré-Hopf algebra, but the former is obtained only for the case of light-like deformation.
The paper is organized as follows. In the second section, κ-Minkowski spacetime with deformation vector a µ in various directions (timelike, spacelike and lightlike) is introduced. In section III, notion of linear realizations is introduced and all linear realizations in n dimensions for n > 2 are found. Those realizations are then expressed in terms of generators of gl(n) algebra. In section IV, deformed Heisenberg algebra is presented, along with star product and coproducts of momenta.
At the end of this section, the twist operator is introduced and the relation between star product, twist operator and coproduct of momenta is given. In section V it is shown that the twist operator from the previous section is a Drinfeld twist, satisfying normalization and cocycle conditions. It is shown that initial linear realizations follow from these twists, which confirms the consistency of our approach. At the end of the section V, R-matrix is presented. In the section VI, κ-deformed igl(n) Hopf algebra is presented, in general and for four special cases. In section VII, left-right dual κ-Minkowski algebra is constructed from the transposed twists. Alternatively, κ-Minkowski algebra is obtained from transposed twists with a µ → −a µ . The corresponding realizations are non-linear.
In section VIII, nonlinear realizations of κ-Minkowski space and related Drinfeld twists, known in the literature so far, are presented. Finally, in section IX, outlook and discussion are given.
Note that Lie algebra (2) is independent of metric. However, we point out that our physical requirement is that in the limit a µ → 0, we get ordinary Minkowski spacetime. Hence, it is natural to assume and treat a µ as a vector in undeformed Minkowski space. There are two possibilities.
One is to fix real parameters a µ and the other is when a µ are not fixed (transforming together with noncommutative coordinatesx µ ) [21]. In this paper we chose the first possibility.
a · b = a µ b µ = η µν a µ b ν and a 2 = a · a = a µ a µ = η µν a µ a ν for any vectors a µ and b µ .
The deformation vector can be timelike (a 2 < 0), lightlike (a 2 = 0) and spacelike (a 2 > 0), so it can be written like: where κ −1 is expansion parameter and u 2 ∈ {−1, 0, 1}, which corresponds to the previously mentioned three cases. Light-like deformation a 2 = 0 was first treated in context of null-plane quantum Poincaré algebra [22]. Depending on the sign of a 2 , κ-Minkowski Lie algebra is invariant under the following little groups: • If a µ is timelike (a 2 < 0), the little group is SO(n − 1) • If a µ is lightlike (a 2 = 0), the little group is E(n − 2) • If a µ is spacelike (a 2 > 0), the little group is SO(n − 2, 1) It is useful to introduce enveloping algebraÂ, generated by the elementsx µ of κ-Minkowski algebra.

III. LINEAR REALIZATIONS
Commutative coordinates x µ and momenta p µ generate an undeformed Heisenberg algebra H given by: Analogously toÂ in the previous section, commutative coordinates x µ generate enveloping algebra A, which is subalgebra of undeformed Heisenberg algebra, i.e. A ⊂ H. Momenta p µ generate algebra T , which is also subalgebra of undeformed Heisenberg algebra, i.e. T ⊂ H. Undeformed Heisenberg algebra is, symbolically, H = AT .
In general, realization of NC space is given by: where ϕ α µ (p) is a function of p µ which should reduce to δ α µ in the limit when deformation goes to zero [23][24][25].
We are looking for linear realizations of κ-Minkowski space, that is the realizations where the function ϕ α µ (p) is linear in p µ . They can be written in the form where l µ is linear in momentum p µ . It is given by: where K βµ α ∈ R. Inserting it in (2) gives that K µν α has to satisfy: It also follows that l µ satisfies the same commutation relations asx µ :

A. Classification of linear realizations
Since we assume that equations (8) and (9) transform under Lorentz algebra, the most general covariant ansatz for K µν α in terms of deformation vector a µ for arbitrary number of dimensions 4 n > 2 is: 4 In 2 dimensions there are additional terms constructed with two dimensional Levi-Civita tensor ǫµν . For example, there is a solution K µνλ = aµaν a 2 (c1aα + c2ǫ αβ a β ), where c1, c2 ∈ R are parameters and a 2 = 0.
leading to the following l µ : From equation (8) it follows that Using (9) in combination with (13) yields the following equations: Those equations have four solutions: where c ∈ R is a free parameter. We will denote these four types of realizations by C 1 , C 2 , C 3 and C 4 respectively 5 . Explicitly for the tensor K µνα we have if a 2 = 0, 5 Where C stands for covariant.
Inserting (17) into (6) and (7) gives: Linear realizations for κ-deformed Euclidean space were studied in [25]. However, in κ-Minkowski spacetime, we have found four new linear realizations corresponding to light-like deformations (a 2 = 0). Only one of them, C 4 , corresponds to κ-Poincaré Hopf algebra [17]. C 1 and C 2 are equivalent for c = 1 while C 1 and C 3 are equivalent for c = −1. It is important to note that the first three solutions C 1 , C 2 and C 3 are valid for all a 2 ∈ R, and the fourth solution C 4 is only valid in the case of light-like deformation.

B. Symmetry algebra igl(n)
For fixed solution K µνλ we define undeformed igl(n) algebra generated by p µ and L µν : In addition to commutation relations (21), also holds.
Linear realizations can be written in terms of L µν : Particularly, for (18): where M µν = L µν − L νµ generate Lorentz algebra. Note that C 4 is the only solution that can be written in terms of Lorentz generators.
Commutation relations between generators of igl(n) algebra withx µ are Algebra generated by L µν , p µ andx µ , satisfies all the Jacobi relations. Only for solution C 4 this is also true for algebra generated by M µν , p µ andx µ .
At the end of this section let us introduce anti-involution operator † by λ † =λ, for λ ∈ C and bar denoting the ordinary complex conjugation ( (2), (4), (25) and (27) remain unchanged (i.e. they are invariant) under the action of †. Note that realizations C 1 , C 2 and C 3 are generally not hermitian. In order to get the hermitian realizations, one has to make following substitutions: throughout the whole paper [28].
IV. DEFORMED HEISENBERG ALGEBRA, STAR PRODUCT AND TWIST OPERA-TOR Non-commutative κ-Minkowski coordinatesx µ and momenta p µ generate a deformed Heisenberg algebraĤ given by [29]: From previous section, it follows thatĤ is isomorphic to H. AlgebraÂ is a subalgebra ofĤ, i.e.
A. Actions ◮ and ⊲ Action ◮ is a map ◮:Ĥ ⊗Â →Â satisfying the following properties: It follows thatĤ In a complete analogy, the action ⊲ is a map ⊲ : H ⊗ A → A satisfying the following properties: Also, it follows that ◮ and ⊲ are actions, so they satisfy:
Using this identification, star product ⋆ : A ⊗ A → A is defined by: For κ-Minkowski space, the star product is associative: Star product defines algebra A ⋆ , which is defined like A, but with non-commutative star product instead of ordinary multiplication. Algebras A ⋆ andÂ are isomorphic as algebras, not only as vector spaces. It follows that: It can be shown that: It can also be shown that there is a function P µ (k 1 , k 2 ), such that The star product between such exponentials is then given by: Note that K µ (k) = P µ (k, 0) and D µ (k 1 , k 2 ) = P µ (K −1 (k 1 ), k 2 ). D µ (k 1 , k 2 ) describes deformed addition of momenta (k 1 ) µ ⊕(k 2 ) µ = D µ (k 1 , k 2 ) (for more details see [30]). Calculation of P µ (k 1 , k 2 ), D µ (k 1 , k 2 ) and K µ (k) for linear realizations (described in previous section) is given in Appendix A.
For elements f, g ∈ A which can be Fourier transformed we find corresponding elementsf ,ĝ ∈Â Then the star product f ⋆ g can be written in the following way: The undeformed coproduct ∆ 0 : T → T ⊗ T for momentum p µ is: Deformed coproduct for momenta ∆ : T → T ⊗ T is [31][32][33]: Using results from Appendix A, we have: and We also have: In order to specify ∆p µ , we have to express K −1 µ (p) ≡ p W µ in terms of momenta p µ (see Appendix A). Momentum p µ acts on e ik·x and e ik·x with ⊲ and ◮ respectively in the following way: and momentum p W µ acts as: It is useful to introduce the shift operator Z, with properties: Explicitly, for C 1 , C 2 , C 3 and C 4 , coproducts of momenta are: • Case C 1 : • Case C 4 : D. Relation between star product, twist operator and coproduct The star product is related to the twist operator F −1 in the following way: wheref ∈Â is expressed in terms of x, p ∈ H.
Using the above expression for star product eq. (52) and (89), the twist operator can be written as [30,34,35] where t ∈ R and is generally defined up to the right ideal I 0 ⊂ H ⊗ H defined by
where p W µ is given in the subsection IV C after equation (64) and in Appendix A, and l µ = −iK βµα L αβ , where K βµα satisfies (8) and L µν generate gl(n) algebra, see equation (21).
The classical r-matrix r cl , related to twist (93) is: For C 1 , C 2 , C 3 and C 4 , twists are: Note that only for the case C 4 (a 2 = 0), the corresponding twist operator can be expressed in terms of Poincaré generators only [17,36].

B. Normalization condition
Now we show that these twists satisfy normalization condition and cocycle condition, i.e. that they are Drinfeld twists.

C. Cocycle condition
Cocycle condition is We shall prove it using factorization properties of twist F where The first factorization property (110) can be proven to hold in a following way: This holds because l µ generates the same algebra asx µ and e ik 1 ·x e ik 2 ·x = e iD W (k 1 ,k 2 )·x .
Up to the second order we have: where Commutator [p W µ ,x ν ] is given in equation (A21) and inverse matrices ϕ −1 µν are given in (19) and relation between p µ and p W µ is given in equation (A20). Generally, classical matrix r cl = ln R up to the first order in 1 κ and the classical r cl matrices can be written in terms of igl(n) generators as where K βµα are given in (17). Using (125) we find the classical r cl -matrices for twists (96), (97), (98) and (99): is traceless symmetric part of L µν , D = L α α and M µν = L µν − L νµ . Note that for the case C 4 , r (C 4 ) cl in (129) coincides with the r cl for light-cone case discussed in [16]. Also r (C 1 ) cl = r for c = −1, which is consistent with discussion in section III.
We point out that generators l µ (see equation (7)) close κ-Minkowski algebra (see equation (10)) and [l µ , p ν ] = iK αµν p α . Note that twists can be expressed in terms of l µ and p µ . From this it follows The counit is unchanged: and the antipode S, obtained from coproduct and counit via m is given by The deformed Hopf algebra acting onx µ ⊗ 1, i.e. using gf = m ∆g(◮ ⊗1)(f ⊗ 1) , ∀g ∈ igl(n) andf ∈Â leads to which also leads to (2).
Let us consider special cases. For the case S 1 , the twist operator is: and coproducts and antipodes of p µ , D ≡ L α α and M µν , obtained from the twist (143), are: The coproduct and antipode of l µ are: The symmetry of this case is Poincaré-Weyl symmetry. The case S 1 corresponds to the right covari- , see equation (20), and is related to [38], but with interchanged left and right side in tensor product and with a µ → −a µ .
For the case S 2 , the twist operator is: and coproducts and antipodes of p µ and L µν are: The coproduct and antipode of l µ are: The case S 2 corresponds to the left covariant realizationx µ = x µ [1 + (a · p)], see equation (20).
For the case S 3 , the twist operator is: and and coproduct and antipode of p µ are: Similarly one finds ∆L µν and S(L µν ) using equations (133) and (138) respectively. The coproduct and antipode of l µ are: The case S 3 corresponds tox µ = x µ − a µ (x · p) − (a · x)p µ , see equation (20).
For the case C 4 , i.e. for the light-like κ deformation of Poincaré Hopf algebra, the twist operator is: and coproducts and antipodes of p µ and M µν , obtained from the twist (154), are: The coproduct and antipode of l µ are: The case C 4 corresponds to the natural realizationx µ = x µ [1 + (a · p)] − (a · x)p µ , see equation (18). It is the only solution compatible with κ-Poincaré Hopf algebra [16,17,29,39].
left and right side in coproducts ∆h are interchanged and a µ is replaced by −a µ .
We point out that solution C 4 , and its transposed case, are of special interest because they lead to light-like κ-Poincaré Hopf algebra. They are related to the result by Borowiec and Pachoł [16] (comparison is given is section VIII B).

DRINFELD TWISTS
We shall also present a few families of non-linear realizations and corresponding Drinfeld twist operators known in the literature so far.
i) The first case is ψ(A) = 1, with arbitrary ϕ(A) and γ(A) = ϕ ′ (A) ϕ(A) + 1, see equation (170). The coproducts of momenta are: The twist operator is Abelian [35] F ϕ = exp (N ⊗ 1) ln where N = ix i p i and [N, A] = 0. Since this twist is Abelian, it automatically satisfies cocycle condition, and therefore it is a Drinfeld twist. Special case is presented in [40,41] ii) In the second case, leading to Jordanian twists, given by Borowiec and Pachoł in [42], ψ(A) is a linear function, i.e. ψ(A) = 1 + rA, where r ∈ R, and γ(A) = 0, which leads to The coproducts of momenta are: The family of corresponding twist operators is: Special Jordanian twist was studied by Bu, Yee and Kim in [38] and corresponds to S 1 , but with interchanged left and right side of the tensor product, and with a 0 → −a 0 and c = r + 1, i.e.

B. Light-cone deformation
In the light-cone basis, the κ-Poincaré algebra was studied in [15] and [16], the corresponding twist is extended Jordanian twist, written in terms of two exponential factors. It is identical to transposed twist of F C 4 with a µ → −a µ , i.e.F C 4 aµ→−aµ .
Extended Jordanian twist corresponding to light-cone deformation is: where [P µ ,x µ ] = −iη µν [1 + (a · P )] + ia µ P ν , see equation (18), and If we define and A, B and α generate algebra (C1), given in Appendix C. Using result (C2) from Appendix C, it follows that F LC , written as one exponential function, is given by Using our notation, this result is which thus proves the relation The twist F LC leads to nonlinear realization (157) and the corresponding coproduct is transposed coproduct with a µ → −a µ . In this paper we were dealing mostly with linear realizations and the corresponding Drinfeld twists. However, for any realization, in general, one can construct a twist operator that does not have to satisfy the cocycle condition in the Hopf algebra sense (i.e. not a Drinfeld twist), rather it satisfies the cocycle condition in a more general sense (up to tensor exchange identities [27]), i.e.
in the framework of Hopf algebroids [27]. It is crucial to notice that the κ-Minkowski space can be embedded into a Heisenberg algebra which has a natural Hopf algebroid structure. One can show that the star product resulting from this generalized twist operator is associative and that the corresponding symmetry algebra is a certain deformation of igl(n)-Hopf algebra. This general framework is more suitable to address the questions of quantum gravity [43,44] and related new effects to Planck scale physics.
The problem of finding all possible linear realizations is closely related to classification of bicovariant differential calculi on κ-Minkowski space [36]. Namely, the requirement that the differential calculus is bicovariant leads to finding all possible algebras between NC coordinates and NC oneforms that are closed (linear) in these NC one-forms. The corresponding equations for the structure constants (from the super-Jacobi identities) are exactly the same as eqs. (8,9). The linear realizations elaborated in this paper are expressed in terms of Heisenberg algebra, but one can extend this to super-Heisenberg algebra, by introducing Grassmann coordinates and momenta. This way one can construct the extended twists [36,45] which have the same desired properties, but also give the whole differential calculi.
With linear realization it is much easier to understand and to perform practical calculation in the NC space. In [46] it is proposed that the NC metric should be a central element of the whole differential algebra (generated by NC coordinates and NC one-forms). This NC metric should encode some of the main properties of the quantum theory of gravity. We hope that using the tool of linear realizations one can perform such calculations for a large class of deformations, and for all types of bicovariant differential calculi and predict new contributions to the physics of quantum black holes and the quantum origin of the cosmological constant [47].
Recently [18], the Drinfeld twist corresponding to C 4 was analyzed and the corresponding scalar field theory was discussed. We are planing to further analyse the properties of quantum field theories [48], especially gauge theories that arise from this twist, but we are also interested in pursuing further investigations on the physical aspects of C 1,2,3 cases.
Appendix A: Derivation of coproduct ∆p µ Here we present construction of equations for K µ (k) and P µ (k 1 , k 2 ) and their solutions for linear realizations.
The solution for P µ (k 1 , k 2 ) is: Solution for K µ (k) = P µ (k, 0) is simply: It is useful to define Inserting this definition and solution (A16) into (A5) we get The momentum p W µ = K −1 µ (p), introduced in section IV C (see eq. (66)), is related to p µ via and is given in closed form in (72), (77), (82) and (87) for the solutions C 1 , C 2 , C 3 and C 4 respectively.
For p W µ , it is useful to define: Using this definition, and solution (A19) with the equation (A4), finally leads to the coproduct: This identity is a generalization of the result presented in [30] in equations (A. 16) and (A. 17). See also Section 2 in [49].
Twists can be calculated from the known coproducts of momenta using the equation (91). We would like to write the twist in the following form: and f s ∈ U [igl(n)] ⊗ U [igl(n)] is contribution to f in s-th order of 1 κ . Inserting (55) into (91), for t = 0 we get: From equation (B1) it follows Since Λ µν = e K µν , we find the twist: Since K µν = −K µαν (p W ) α and l µ = −iK βµα L αβ , the result can also be written as: Let us consider algebra generated by A, B and α: