A non-commutative Minkowskian spacetime from a quantum AdS algebra

A quantum deformation of the conformal algebra of the Minkowskian spacetime in $(3+1)$ dimensions is identified with a deformation of the $(4+1)$-dimensional AdS algebra. Both Minkowskian and AdS first-order non-commutative spaces are explicitly obtained, and the former coincides with the well known $\kappa$-Minkowski space. Next, by working in the conformal basis, a new non-commutative Minkowskian spacetime is constructed through the full (all orders) dual quantum group spanned by deformed Poincar\'e and dilation symmetries. Although Lorentz invariance is lost, the resulting non-commutative spacetime is quantum group covariant, preserves space isotropy and, furthermore, can be interpreted as a generalization of the $\kappa$-Minkowski space in which a variable fundamental scale (Planck length) appears.


Introduction
One of the most relevant applications of quantum groups in physics is the construction of deformed spacetime symmetries that generalize classical Poincaré kinematics beyond Lie algebras, such as the well known κ-Poincaré [1,2,3] and the quantum null-plane (or lightcone) Poincaré [4,5] algebras. For all these cases, the deformation parameter has been interpreted as a fundamental scale which may be related with the Planck length. In fact, these results can be seen as different attempts to develop new approaches to physics at the Planck scale, an idea that was early presented in [6]. A further physical development of the κ-Poincaré algebra has led to the so called doubly special relativity (DSR) theories [7,8,9,10,11] that analyse the fundamental role assigned to the deformation parameter/Planck length as an observer-independent length scale to be considered together with the usual observer-independent velocity scale c, in such a manner that Lorentz invariance is preserved [12,13,14].
From a dual quantum group perspective, when the quantum spacetime coordinatesx µ conjugated to the κ-Poincaré momentum-space P µ (translations) are considered, the noncommutative κ-Minkowski spacetime arises [15,16,17,18]. Some field theories on such a space have been proposed (see [19] and references therein) and its role in DSR theories has been analysed [20]. More general non-commutative Minkowskian spacetimes can be expressed by means of the following Lie algebra commutation rules [21]: where a µ is a constant four-vector in the Minkowskian space.
In the context of κ-deformations, which are understood as quantum algebras with a dimensionful deformation parameter related with the Planck length, Poincaré symmetry should be taken only as a first stage that should be embedded in some way within more general structures such as deformed conformal or AdS/dS symmetries, that is, quantum so(4, 2)/so(5, 1) algebras. Thus, it is natural to think that a non-commutative Minkowskian space of the form (1) could also be either embedded or generalized. In this respect, by considering the simplest quantum deformation of the Weyl-Poincaré algebra (isometries plus dilations), U τ (WP) [22], a new DSR proposal has been presented in [23]. Such a quantum algebra arises as a Hopf subalgebra of a 'mass-like' quantum deformation U τ (so(4, 2)) of the conformal algebra of the (3 + 1)D Minkowskian space.
The aim of this letter is to analyse the first-order (in both the deformation parameter and non-commutative coordinates) quantum group dual to U τ (so(4, 2)), to construct the complete (all orders) Hopf algebra dual to U τ (WP) and, afterwards, to extract some physical implications of the associated non-commutative Minkowskian spacetime.
In the next section we identify the deformation U τ (so(4, 2)), formerly obtained in a conformal basis, with a (4 + 1)D quantum AdS algebra at both algebra and dual group levels. By using the Hopf subalgebra spanned by the deformed Poincaré and dilation generators we compute in section 3 the associated dual quantum group by making use of the quantum R-matrix. The last section is devoted to derive the physical consequences conveyed by the resulting non-commutative Minlowskian spacetime which is covariant under quantum group transformations and does preserve space isotropy, although its Lorentz invariance is lost. Moreover, this new non-commutative spacetime generalizes κ-Minkowski space since it is defined through Lie algebra commutation rules whose structure constants are the quantum group entries associated to the Lorentz sector.
Some properties of U τ (so(4, 2)) can be unveiled by studying its Lie bialgebra structure, that is, the first-order relations in the deformation parameter, generators and dual coordinates. If we write the deformed coproduct ∆ as formal power series in τ , the cocommutator δ is given by the skewsymmetric part of the first-order deformation, where σ(a ⊗ b) = b ⊗ a. In our case, from (4) we find that where ∧ denotes the skewsymmetric tensor product. Next, Lie bialgebra duality [24,25] leads to the dual commutation rules in the form where Y i is a generic generator andŷ i its associated dual quantum group coordinate ful- (7) the following non-vanishing first-order quantum group commutation rules: The non-commutative Minkowskian spacetime is then characterized by which coincides with the usual κ-Minkowski space (1) with a µ = (1, 0, 0, 0), provided that τ = −1/κ. Nevertheless, in the next section we shall compute the full (all orders) dual quantum group to U τ (WP), and the associated non-commutative spacetime will generalize the first-order relations (10).
We stress that all the above expressions can be rewritten in terms of deformed symme- be the Lorentz and translations undeformed generators obeying such that η = (η AB ) = diag (−1, 1, 1, 1, 1) is the Lorentz metric associated to so(4, 1), L 0B are the four boosts in AdS 4+1 and R is the AdS radius related with the cosmological constant by Λ = 6/R 2 . Then the following change of basis (i = 1, 2, 3): connects CM 3+1 with AdS 4+1 and can be taken in the deformed case as a way to identify (2)-(4) as the quantum deformation U τ (so(4, 2)) ≡ U τ (AdS 4+1 ). The dual Lorentzl AB and spacetimet A quantum AdS-coordinates can also be written in terms of the conformal Hence the (first-order) non-vanishing commutation rules for the non-commutative AdS 4+1 spacetime turn out to be which involve both the boostl 0,i+1 and the rotationl 1,i+1 quantum coordinates. A Lie bialgebra contraction analysis [26] shows that the contraction R → ∞ from U τ (AdS 4+1 ) and its dual to a (4 + 1)D quantum Poincaré algebra/group is well defined whenever the deformation parameter is transformed as τ → τ R 2 .
Therefore, the maps (12) and (13) can be used to express the same quantum deformation of so(4, 2) within two physically different frameworks: In fact, such a quantum group relationship might further be applied in order to analyse the role that quantum deformations of so(4, 2) could play in relation with the "AdS-CFT correspondence" that relates local QFT on AdS (d−1)+1 with a conformal QFT on the (compactified) Minkowskian spacetime CM (d−2)+1 [27,28,29] (in our case up to d = 5).
We also remark that a more general (three-parameter) quantum deformation of o(3, 2) can be found in [30], where the connection between the corresponding quantum CM 2+1 and AdS 3+1 algebras is explicitly described.

Quantum Weyl-Poincaré group
The classical r-matrix associated to U τ (so(4, 2)) reads which satisfies the classical Yang-Baxter equation [31]. Therefore, U τ (so(4, 2)) is a triangular (or twisting) quantum deformation, different to the Drinfeld-Jimbo type, which is supported by the Hopf subalgebra spanned by {D, P 0 }. The universal T -matrix for the latter can be written as [32] T = ed D ex while the R-matrix reads Since this element is also a universal R-matrix for both U τ (WP) ⊂ U τ (so(4, 2)) [22], the corresponding dual quantum groups can be deduced explicitly by applying the FRT procedure [33]. This requires a matrix representation R for (17) as well as to choose a matrix element T of the quantum group with non-commutative entries. However, the consideration of the complete U τ (so(4, 2)) structure (in both conformal and AdS bases) precludes a clear identification of the non-commutative spacetime coordinates as these would appear as arguments of functions that also depend on many other coordinates. In this respect see, for instance, [34] for the construction of a quantum AdS space from a q-SO(3, 2) of Drinfeld-Jimbo type.
Since we are mainly interested in the structure and physical consequences of the as- where e ab (a, b = 0, . . . , 5) is the matrix with entries δ ab . We construct the quantum group element T in such a representation by considering the following matrix product, which is consistent with the exponential form of the universal T -matrix (16) for the carrier subalgebra {D, P 0 }: where the non-commutative entries are the quantum Minkowskian coordinatesx µ and 1, 1, 1). (20) Note that quantum rotation and boost coordinates are jointly expressed through the formal Lorentz entriesΛ µ ν .
Since P 3 0 vanishes, R reads where 1 is the 6 × 6 unit matrix. Next in pursuing the FRT program we impose that where T 1 = T ⊗ 1 and T 2 = 1 ⊗ T . This matrix equation provides the commutation rules among all the entries in (19), which by taking into account (20) can then be reduced to where the quantum Lorentz entriesΛ µ 0 are given bŷ The coproduct for all the entries in T is just ∆(T ) = T⊗T . By using again (20) the coproduct for {d,x µ ,Λ µ ν } can consistently be found: which is a homomorphism of (23). Thus the expressions (23)-(25) together with the counit ǫ(T ) = 1 and antipode S(T ) = T −1 determine the Hopf algebra structure of the quantum Weyl-Poincaré group dual to U τ (WP), which is the restrictionĉ µ ≡ 0 of that dual to U τ (CM 3+1 ).
We remark that the commutation relations (9) (withĉ µ ≡ 0) can be recovered from (23) by only taking the first-order in all the quantum coordinates (notice that in this case, Recall that the FRT approach was also used in the construction of the null-plane quantum Poincaré group [5]. However, as κ-Poincaré has no universal R-matrix, the associated quantum group was obtained [15,16,17,18] through a direct quantization of the semiclassical Poisson-Lie algebra coming from the κ-Poincaré classical r-matrix.

A new non-commutative Minkowskian spacetime
Now we focus our attention on some structural physical consequences of the new noncommutative spacetime that comes out from (23) which can be seen as a generalization of (1) through a µ →Λ µ 0 (ξ). Firstly, we stress that sinceΛ µ 0 (24) only depend on the quantum boost parameters and the quantum rotation coordiantesθ i do not play any role in the spacetime noncommutativity, the isotropy of the space is thus preserved. Furthermore, by taking into account the commutation rules (23),Λ µ 0 can be considered to play the role of the structure constants within the quantum space (26). In fact, the quantum boost coordinatesξ i can be regarded as scalars (usual commutative parameters) within the quantum Weyl-Poincaré group. From this viewpoint, relations (26) would define a Lie algebraic non-commutative spacetime of the type (1). However, this situation changes in the full quantum conformal group since, as shown in (9),ξ i andΛ µ 0 no longer commute with the dilation parameterd. Secondly, relations (26) show that different observers in relative motion with respect to quantum group transformations have a different perception of the spacetime noncommutativity, i.e., Lorentz invariance is lost. Nevertheless, we remark that, in this context, covariance under quantum group transformations is ensured by construction. Explicitly, in the commutative case the T -matrix (19) is just a matrix representation of the transformation group of the spacetime, and the coproduct (25) represents the multiplication of two different elements of the group. A similar interpretation holds in the quantum group case, for which (25) provides the transformation law for the non-commutative spacetime coordinates that can be rewritten aŝ where the tensor product notation has been replaced by two different copies of the noncommutative coordinates (x ⊗ 1 ≡x, 1 ⊗x ≡x ′ ). The fact that the spacetime (26) is quantum group covariant is a direct consequence of the Hopf algebra structure which implies that where the new Lorentz entries are also given by (25): Therefore, if we assume that two "observers" are actually related through a quantum group transformation (27), they will be "affected" by different structure constants for the spacetime commutation rule (26), yet the latter is manifestly quantum group covariant.
To end with, we also stress that if the following new space variablesX i in the (3 + 1)D spacetime (26) are considered the transformed commutation rules for the quantum spacetime are given by which, in turn, can be interpreted as a generalization of the κ-Minkowski space (10) with a "variable" Planck length τ ′ = τΛ 0 0 (ξ) that does depend on all the quantum boost parameters (in the (1 + 1)D case, this yields τ ′ = τ coshξ). This result is a direct consequence of imposing a larger quantum group symmetry than Poincaré. Moreover, if the quantum conformal transformations and parameters are taken into account and the corresponding quantum group is constructed, thenΛ 0 0 becomes a non-central operator in such a manner that (35) (and also (26)) defines a quadratic non-commutative spacetime. This suggests that a further study of non-Lie spacetime algebras derived from conformal or AdS quantum symmetries could be meaningful.