Quantum $\kappa$-Poincare in Any Dimensions

The $\kappa$-deformation of the D-dimensional Poincar\'e algebra $(D\geq 2)$ with any signature is given. Further the quadratic Poisson brackets, determined by the classical $r$-matrix are calculated, and the quantum Poincar\'e group"with noncommuting parameters"is obtained.


Introduction
The quantum deformation of semisimple Lie algebras has been given by Drinfeld [1] and Jimbo [2]. However such a general solution does not exist for non semisimple Lie algebras. Examples of quantum inhomogeneous rotation algebras with Euclidean or Minkowski metric have been given for the dimensions two [3] three [4] and four [4,5], and shown to be Hopf algebras. The aim of this paper is to point out the common features of these examples and to generalize them to any dimension.
The method used in these examples is the contraction procedure, applied to the Drinfeld-Jimbo deformation of the semisimple rotation algebras in three, four and five dimensions, denoted by U q (so(D)), D = 3, 4, 5. In this way we obtain the Hopf algebra U q (so(D − 1) ⊃ T D−1 ), i.e. the deformation of the semidirect sum of rotations and translations in D − 1 dimensions. The presence of the "de Sitter" radius R (which is ultimately taken to infinity) explains that the deformation parameter q is replaced by a dimensionful parameter which we called κ [4,5] for D − 1 = 4. For this reason we denoted the quantum deformation of the enveloping algebra of the Poincaré algebra P 4 in four dimension by U κ (P 4 ). In the limit κ → ∞ one recovers P 4 from the Cartan-Weyl basis.
Another way to get U κ (P 4 ) is to start with standard P 4 and replace the right hand side of the commutation relations involving the boosts by functions of the momenta (with some restrictions to be specified later). The Jacobi identities restrict these functions in such a way that one of the solutions is precisely U κ (P 4 ) [6,7]. For dimensional reasons, the momenta appearing on the RHS of the commutator of two boosts must be divided by a dimensionful constant which turns out to be the deformation parameter κ.
One can show that it is not possible to write a quantum Poincaré algebra as a semidirect sum of a quantum Lorentz algebra and a quantum fourmomentum with standard (Drinfeld-Jimbo) deformation of the Lorentz sector unless one adds the dilatation generator [8,9]. An explicit example of such a quantum deformation of the Poincaré group as well as Poincaré algebra with nonstandard deformation of the Lorentz sector was given recently by Chaichian and Demichev [10]. Together with the quantum κ-Poincaré algebra [4,5], these are the only Poincaré Hopf algebras with 10 generators so far known.
From the point of view of physics it is interesting to compare the predictions of U κ (P 4 ) with those of the P 4 . The modification of g-2 [11] imposes a lower limit of 10 7 Gev for κ, while the change in the quadratic Casimir operator requires κ > 10 12 Gev [12]. For other predictions see [13,14].
An interesting result was obtained by Maslanka [15]. He performed a non linear change in the Cartan-Weyl basis of the classical Poincaré Lie algebra. The new generators obtained in this way belong to U (P 4 ) but they obey the commutation relations of the generators of U κ (P 4 ). From this one obtains the κ-Poincaré algebra with (complicated) cocommutative coproducts. Such a coproduct is actually preferred by some authors [6] because of the difficulties of the physical interpretation of a non cocommutative coproduct (for dimensions of space-time larger than two). In the present paper we shall assume however that the coproduct is noncocommutative, i.e. we shall assume a genuine quantum deformation of the Poincaré algebra.

Quantum κ-Poincaré in 4-dimensions.
In order to abstract the common features of the quantum deformations of inhomogeneous rotation algebras we start with the explicit example of U κ (P 4 ) for a real value of κ [5]. We choose the Minkowski metric (-+++), but it is easy to change the metric to the Euclidean (++++) or to (++--) [7].
The Hopf algebra structure is the following: (we use M for rotations, N for boosts, P µ for momenta).
Algebra structure: It is seen that the commutation relations involving the rotations M i and the time translation P 0 are classical, and the momenta commute. The quantum deformation appears only when the boosts are involved. For dimension D = 3 the classical so(3) is replaced by so (2).
For D = 2 there is only one boost, and (2.4) is trivial, while (2.3) is the same for D = 2, 3, 4.
For D = 3, (2.4) is linear in M 3 cosh P 0 /κ, but the term X ij linear in M and bilinear in P is absent, because there are only two space momenta and one rotation M 3 .
We now discuss the Coalgebra structure The coproducts for M i and P 0 are classical.
All the axioms of a Hopf algebra are satisfied: associativity of the product m defined on the algebra, coassociativity of the coproduct ∆ of the coalgebra; ∆ and the counit ǫ are homomorphisms of the algebra. The following relations are satisfied.
where i is the unit map of the algebra. For D = 2, the term linear in P i for the antipode S(N i ) is missing.

Quantum κ-Poincaré in any dimension.
We have seen that :  D)), the inhomogeneous rotation algebra. We require the "rotation" subalgebra o(D-1) to be classical, as well as the "time" translation P 0 . The "space" translations P a , a = 1..D − 1, while commuting with P 0 and among themselves, will have a non trivial coproduct.
The metric tensor g AB , A, B = 0..D − 1, will be kept general (diagonal). We shall call "rotations" the generators M ab = −M ba , a, b, = 1...D − 1, and "boosts" the D-1 generators M a0 = −M 0a . The sign convention is chosen in such a way that for the "Minkowski" metric g AB = diag (-++...+), we recover for D = 4 the κ-Poincaré algebra with M ij = ǫ ijk M k , i, j, k = 1, 2, 3 and M i0 = N i . With these notations we postulate the following algebra structure X ab has to satisfy the Jacobi identities. This implies To our surprise we found that X ab is linear in M ab and bilinear in P a for any dimensions! Thus the case D=4 is already the general case. The following satisfies the Jacobi identities: and reduces to (2.4) for D = 4. It is now easy to guess the Coalgebra structure The expression for Y a is obtained from the homomorphism property of the coproduct.
which is of course the same as (2.7) for g 00 = −1.

From Classical r-matrix to quantum Poincaré group
From the κ-Poincaré quantum algebra, with the coproducts given by (3.8-10), one can obtain the classical r-matrix, i.e. one gets the classical Poincaré bialgebra. Because such an r-matrix satisfies the (modified) classical YB equation, one can construct the quadratic Poisson brackets on the classical Poincaré group ISO(D)) dual to the Poincaré algebra. The quantization of these Poisson brackets provide the quantum Poincaré group with non commuting parameters satisfying quadratic relations.
Following Zakrzewski [16] who first provided the quantization scheme described above for D = 4, we define the cocommutator on the algebra i(o(D)) by the antisymmetric part of the coproducts (3.8-10) 2) It appears that the cobrackets (4.2) can be expressed as follows (XǫU (io(D))) where r ǫ U (io(D)) ⊗ U (io(D)) is the classical r−matrix for the Poincaré algebra.
The "canonical" Poisson brackets (4.14) on the D-dimensional Poincaré group are compatible with the standard comultiplication If we perform the standard quantizations of the Poisson brackets (4.14), by replacing one obtains the following set of commutation relations: with the coproduct (4.15) valid for the non-commutative parametersR AB ,v A (we recall that t jD =t jD = δ jD ). It should be stressed that the quantization procedure given in this paragraph takes into consideration only the linear terms in the deformation parameter 1 κ , and consequently the commutators (4.17) can be expressed by the standard relation R(t ⊗ t) = τ (t ⊗ t)R where R = exp ir. It has been shown in [20] that for D = 2 such a quantization is related by a nonlinear transformations of the parameters to the full quantization, obtained by dualizing the D = 2 κ-Poincaré-algebra (see also [21]). For D > 2 the relation between the quantization obtained from the Poisson bracket (4.11) and the complete one, by dualizing the full κ-Poincaré algebra, is under investigation.

Conclusions.
In the present paper, we described the quantum κ-Poincaré-algebra in any dimension and a version of quantum κ-Poincaré group obtained by the quantization of the quadratic r-matrix Poisson brackets. We would like finally to mention that : i) The universal R-matrix is known only for the case D = 3 [6] which is the only dimension (for D ≥ 2) for which the quantum de-Sitter contraction limit of the universal R-matrix for U q (o(D + 1)) has provided after suitable renormalization the universal R-matrix for U q (io(D)) .
ii) In order to apply the κ-Poincaré algebra to the description of the deformed D-dimensional space-time symmetries one should introduce the space-time coordinates x A . Two possible ways of introducing space-time coordinates can be proposed: 1) by introducing the commuting space-time coordinates via standard Fourier transform of the commuting momentum variables.
If we introduce the translated space-time coordinatesx ′ A =x A ⊕v A , the relations (5.1) are preserved provided [x A ,x B ] = 0.
It is interesting to develop the κ-Poincaré covariant differential calculus on the noncommutative coordinates (5.1).
iii) Recently the κ-deformation of the D = 4 super Poincaré algebra has been obtained by suitable contraction of the quantum super-de-Sitter algebra U q (Osp(1; 4)) [22]. It is known that (for 3 ≤ D ≤ 10) the super-de-Sitter algebra exists only for D = 3, 4, 6 and 10. It would be interesting to show using e.g. the algebraic methods [6,7] that the quantum κ-Poincar superalgebra exists for all D-also for those for which the contraction method from quantum super-de-Sitter algebra cannot be applied.