On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions
aa r X i v : . [ h e p - t h ] A ug On quantum deformations of (anti-)de Sitter algebrasin (2+1) dimensions ´Angel Ballesteros , Francisco J. Herranz and Fabio Musso Departamento de F´ısica, Universidad de Burgos, 09001 Burgos, Spain Dipartimento di Fisica ‘Edoardo Amaldi’, Universit´a degli Studi di Roma Tre, 00146 Roma,ItalyE-mail: [email protected], [email protected], musso@fis.uniroma3.it
Abstract
Quantum deformations of (anti-)de Sitter (A)dS algebras in (2+1) dimensions are re-visited, and several features of these quantum structures are reviewed. In particular, theclassification problem of (2+1) (A)dS Lie bialgebras is presented and the associated non-commutative quantum (A)dS spaces are also analysed. Moreover, the flat limit (or vanishingcosmological constant) of all these structures leading to (2+1) quantum Poincar´e algebrasand groups is simultaneously given by considering the cosmological constant as an explicitLie algebra parameter in the (A)dS algebras. By making use of this classification, a three-parameter generalization of the κ -deformation for the (2+1) (A)dS algebras and quantumspacetimes is given. Finally, the same problem is studied in (3+1) dimensions, where atwo-parameter generalization of the κ -(A)dS deformation that preserves the space isotropyis found. Introduction
Quantum groups and algebras were introduced in the eighties as Hopf algebra deformations ofLie groups and algebras, respectively (see [1, 2, 3, 4, 5, 6, 7] and references therein). Since then,the construction of quantum deformations of the kinematical groups of spacetimes opened thepossibility of introducing consistent proposals of the mathematical descriptions of the ‘quantum’spacetime that is commonly assumed to arise in quantum gravity theories when the Planckenergy is approached (see [8]). Essentially, this idea has been worked out so far in the literatureunder two dual approaches: • Quantum kinematical algebras : they provide quantum deformations of the symmetry al-gebras of spacetimes (see, for instance, [9, 10, 11]) in which the ‘quantum’ deformationparameter would be a second invariant scale related with the Planck energy (or length) andin which q -deformed Casimir operators generate the kind of modified dispersion relationsthat are expected to arise in the Planck regime. This viewpoint gave rise to the so-called‘double special relativity’ theories [12, 13, 14, 15, 16]. • Quantum kinematical groups : they are the Hopf algebra dual of the previous quantum al-gebras [4, 5, 17] and provide a self-consistent mathematical foundation of the so-called non-commutative spacetimes, in which the noncommutativity of the spacetime coordinates isgenerated by a non-vanishing quantum deformation parameter that should describe Planckscale effects. As a consequence, uncertainty relations between these noncommuting coor-dinate operators arise naturally, and the intrinsic Planck scale ‘fuzziness’, ‘discretization’or ‘quantization’ of the spacetime itself can be mathematically described (see [18, 19, 20]).At this point it is important to stress that for a given Lie group or algebra there exist(many) different Hopf algebra deformations. In fact, quantum algebras are in correspondencewith Lie bialgebra structures (and quantum groups with Poisson–Lie structures), and the explicitclassification problem for Lie bialgebras (or, equivalently, for Poisson–Lie groups) is by no meansa simple problem from a computational viewpoint. Therefore, it seems natural that some criteriashould be taken into account in order to select the type of quantum deformation that could besuitable in the abovementioned specific physical settings.In the particular case of (2 + 1) quantum gravity, it was (heuristically) stated in [21] that theperturbations of the vacuum state of a Chern–Simons quantum gravity theory with cosmologicalconstant Λ are invariant under transformations that close a certain quantum deformation of the(A)dS algebra. In fact, the low energy regime/zero-curvature limit of this algebra was found tobe the known κ -Poincar´e quantum algebra [9, 22, 23, 24].Also, it is well known that Poisson–Lie (PL) structures on the isometry groups of (2+1)spaces with constant curvature seem to play a relevant role as phase spaces in Chern–Simonstheory. Moreover, in the Lorentzian case, these PL structures (given by certain classical r -matrices that have to be ‘compatible’ with the Chern-Simons pairings) are just the classicalcounterparts of certain quantum deformations of (A)dS and Poincar´e groups in (2+1) dimensions(see [25, 26, 27, 28] and references therein). These PL groups are in one-to-one correspondencewith certain three-dimensional Lie bialgebras and their associated Drinfel’d doubles, in such away that an explicit connection between (2+1) Chern-Simons symmetries and Drinfel’d doublescoming from three-dimensional quantum deformations has been recently proposed [29, 30]. Inthis setting the (2+1) (A)dS and Poincar´e r -matrices associated to such Drinfel’d doubles are2ot the ones defining κ -deformations and, therefore, quantum groups corresponding to these(multiparametric) r -matrices seem to be physically interesting to construct and analyse.The aim of this paper is to address the classification problem for the quantum deformationsof the (2+1) (A)dS and Poincar´e algebras by considering the three Lie algebras as particularcases of a one-parametric Lie algebra, the (2+1) AdS ω algebra, in which ω = − Λ. In this way,the ω → ω willdistinguish between the AdS ( ω >
0) and dS ( ω <
0) cases. Moreover, we will show that thetype of approach here presented makes feasible to tackle the same problem in (3+1) dimensions.The paper is organized as follows. In the next section we will briefly recall the basics ofquantum deformations and their classification in terms of their associated Lie bialgebras. Insection 3 we will review the known results concerning the κ -deformation of the (2+1) AdS ω algebra, including the full PL structure that provides the semiclassical (Poisson) counterpart ofthe noncommutative (2+1) AdS ω spacetime, where the nonvanishing cosmological constant Λgenerates nonlinear commutation relations among the quantum coordinates. In section 4 themost generic Lie bialgebra for the (2+1) AdS ω algebra is computed, and this result gives a clearidea of the plethora of different quantum deformations that there exist. The possible general-izations of the κ -deformation are identified in section 5 by imposing on the generic deformationthat the rotation J and time translation P generators have to be primitive ones, a constraintderived from dimensionality arguments. In this way we find that two more parameters canbe added to the κ -deformation, and we explicitly describe the corresponding first-order non-commutative spacetimes, which are generalizations of the κ -Minkowski spacetime. The sameapproach is used in section 6 in order to study the possible generalizations of κ -deformationsin (3+1) dimensions. In this case we find two disjoint families of two-parametric deformationsthat preserve the primitive nature of the generators P and J . Finally, we demonstrate thatone of these generalized κ -deformations preserve space isotropy, since under some appropriateautomorphism the symmetric role of the three rotation generators can be manifestly shown. Quantum algebras are Hopf algebra deformations of universal enveloping algebras. This meansthat we consider the algebra U z ( g ) of formal power series in the deformation parameter z andcoefficients in the universal enveloping algebra U ( g ) of a given Lie algebra g , and we endow itwith a Hopf algebra structure by finding an algebra homomorphism called the coproduct map∆ U : U z ( g ) −→ U z ( g ) ⊗ U z ( g ) , as well as its associated counit ǫ U and antipode γ U mappings.Any quantum universal enveloping algebra can be thought of as a Hopf algebra deformationof U ( g ) ‘in the direction’ of a certain Lie bialgebra ( g, δ ). Such a Lie bialgebra provides thefirst-order deformation (in z ) of the coproduct∆ U = ∆ + δ + o [ z ] , where ∆ ( X ) = X ⊗ ⊗ X is the primitive (nondeformed) coproduct and δ ( X ), the so-called cocommutator, will be linear in z . Therefore, a precise Lie bialgebra structure ( g, δ )3ill be associated to each possible quantum coproduct ∆ U , and the equivalence classes (underautomorphisms) of Lie bialgebra structures on a given Lie algebra g will provide the chart of allpossible quantum deformations of U ( g ).Complementarily, quantum groups ( A, ∆ A ) are just noncommutative algebras of functionsdefined as the dual Hopf algebras to the quantum algebras ( U z ( g ) , ∆ U ). More explicitly, if wedenote by m A and m U the noncommutative products in A and U z ( g ), respectively, the dualitybetween the Hopf algebras ( A, m A , ∆ A ) and ( U z ( g ) , m U , ∆ U ) is established through the existenceof a canonical pairing h , i : A × U z ( g ) → C between them such that h m A ( f ⊗ g ) , a i = h f ⊗ g, ∆ U ( a ) i , (1) h ∆ A ( f ) , a ⊗ b i = h f, m U ( a ⊗ b ) i , (2)where a, b ∈ U z ( g ); f, g ∈ A , and h f ⊗ g, a ⊗ b i = h f, a i h g, b i . It is important to stress that the duality relation (1) implies that the product in A is definedby the coproduct in U z ( g ), and conversely relation (2) implies that the coproduct in A is givenby the product in U z ( g ). Since the first-order deformation of the coproduct ∆ U is defined by theLie bialgebra map δ , the first-order noncommutativity for the quantum group A will be givenby (the dual of) δ . Therefore, the Lie bialgebra structure associated to a given quantum algebraprovides immediately the first-order information about the noncommutative algebra of quantumgroup coordinates. Moreover, the Lie bialgebra ( g, δ ) is in one-to-one correspondence with thePL structure whose quantization provides the full quantum group ( A, ∆ A ). As a consequence,the classification of all possible Lie bialgebra structures for g constitute the first and mostrelevant step for the analysis and explicit construction of its quantum algebra deformations. Let us summarize all the basic facts about Lie bialgebras that will be needed in the sequel. ALie bialgebra ( g, δ ) is Lie algebra g with structure tensor c kij [ X i , X j ] = c kij X k , (3)that is endowed with a skewsymmetric cocommutator map δ : g → g ⊗ g fulfilling the two following conditions: • i) δ is a 1-cocycle, i.e. , δ ([ X, Y ]) = [ δ ( X ) , Y ⊗ ⊗ Y ] + [ X ⊗ ⊗ X, δ ( Y )] , ∀ X, Y ∈ g. • ii) The dual map δ ∗ : g ∗ ⊗ g ∗ → g ∗ is a Lie bracket on g ∗ .Therefore any cocommutator δ will be of the form δ ( X i ) = f jki X j ∧ X k , (4)4here f jki is the structure tensor of the dual Lie algebra g ∗ defined by[ ˆ ξ j , ˆ ξ k ] = f jki ˆ ξ i , (5)where h ˆ ξ j , X k i = δ jk . Therefore, a Lie bialgebra is a pair of ‘matched’ Lie algebras of the samedimension, since the 1-cocycle condition implies the following compatibility equations f abk c kij = f aki c bkj + f kbi c akj + f akj c bik + f kbj c aik . (6)As it could be expected, for some Lie bialgebras the 1-cocycle δ is a coboundary δ ( X ) = [ X ⊗ ⊗ X, r ] , ∀ X ∈ g, (7)where r is a skewsymmetric element of g ⊗ g (the classical r -matrix) r = r ab X a ∧ X b , that has to be a solution of the modified classical Yang–Baxter equation (mCYBE)[ X ⊗ ⊗ ⊗ X ⊗ ⊗ ⊗ X, [[ r, r ]] ] = 0 , ∀ X ∈ g, (8)in which the Schouten bracket [[ r, r ]] is defined as[[ r, r ]] := [ r , r ] + [ r , r ] + [ r , r ] , (9)where r = r ab X a ⊗ X b ⊗ , r = r ab X a ⊗ ⊗ X b , r = r ab ⊗ X a ⊗ X b . Recall that [[ r, r ]] = 0is just the classical Yang–Baxter equation (CYBE).For semisimple Lie algebras all Lie bialgebra structures are coboundaries, and that is also thecase for the Poincar´e algebra in (2+1) and (3+1) dimensions. On the contrary, for solvable andnilpotent Lie algebras many of their Lie bialgebra structures are non-coboundaries (see [31, 32]and references therein). g With the previous definitions in mind, the algorithm for the explicit computation and classifi-cation of all the possible Lie bialgebra structures for a given Lie algebra g (3) in the basis X i can be sketched as follows: • The most generic (pre)-cocommutator map is given in the form (4). • The 1-cocycle condition (6) is imposed onto f jki . • Further, the (quadratic) condition that the dual map δ ∗ : g ∗ ⊗ g ∗ → g ∗ is a Lie bracket on g ∗ is imposed. Note that this constraint is nothing but the Jacobi identity for f jki . • Finally, the solutions for f jki can be classified into equivalence classes under automorphismsof the Lie algebra g .Note that since Lie bialgebra structures for (2+1) (A)dS and Poincar´e algebras are known tobe always coboundaries, this procedure is completely equivalent to the classification of constantclassical r -matrices on these algebras. 5 The κ -deformation of the AdS ω algebra In this section we exemplify the theory of quantum deformations by recalling the main resultsconcerning the so-called κ -deformation of the (2+1) (anti-)de Sitter and Poincar´e algebras. Wewill use a unified approach to the three Lie algebras by making use of the six-dimensional Liealgebra AdS ω which is given in terms of the generators { J, P , P i , K i } ( i = 1 ,
2) (rotation, timetranslation, space translations and boosts) as[
J, P i ] = ǫ ij P j , [ J, K i ] = ǫ ij K j , [ J, P ] = 0 , [ P i , K j ] = − δ ij P , [ P , K i ] = − P i , [ K , K ] = − J, [ P , P i ] = ωK i , [ P , P ] = − ωJ, where i, j = 1 , ǫ = 1 and the parameter ω is just the constant sectional curvature ofthe corresponding classical spacetime (so this is related to the cosmological constant through ω = − Λ). Therefore, these Lie brackets encompass: • The AdS Lie algebra, so (2 , ω = +1 /R > • The dS Lie algebra, so (3 , ω = − /R < • The Poincar´e Lie algebra, iso (2 , ω = 0. Note that this case corresponds tothe flat contraction obtained via the limit of the universe radius R → ∞ connecting so (2 , → iso (2 , ← so (3 , ω algebra are given by C = P − P + ω ( J − K ) , W = − J P + K P − K P , where C comes from the Killing–Cartan form and it is related to the energy of the particle, while W is the Pauli–Lubanski vector. κ -deformation As we mentioned before, all Lie bialgebra structures for the (A)dS and Poincar´e Lie algebrasin (2+1) dimensions are coboundaries. This means that the Lie bialgebra is fully specifiedby a certain skewsymmetric solution of the mCYBE (or of the CYBE). In the case of the κ -deformation such a classical r -matrix reads r = z ( K ∧ P + K ∧ P ) , (10)where the quantum deformation parameter z is related to the usual Planck energy scale κ through z = 1 /κ . Once r is fixed, the cocommutator δ providing the first-order deformation ofthe coproduct is given by the coboundary relation (7), and turns out to be δ ( P ) = 0 , δ ( J ) = 0 ,δ ( P i ) = z ( P i ∧ P − ωǫ ij K j ∧ J ) ,δ ( K i ) = z ( K i ∧ P + ǫ ij P j ∧ J ) . (11)A ‘quantum group local coordinate’ ˆ y i will be defined as dual to the generator Y i through theHopf algebra pairing h ˆ y i , Y j i = δ ij . In particular, we will denote { ˆ θ, ˆ x µ , ˆ ξ i } the noncommutative6oordinates which are dual to the generators { J, P µ , K i } ( µ = 0 , , z ) of thecoproduct given by δ (11), we get the first-order quantum group relations for the κ -deformationof the AdS ω algebra, namely:[ˆ x , ˆ x i ] = − z ˆ x i , [ˆ θ, ˆ x i ] = zǫ ij ˆ ξ j , [ˆ x , ˆ x ] = 0 , [ˆ θ, ˆ x ] = 0 , [ˆ θ, ˆ ξ i ] = − zωǫ ij ˆ x j , [ˆ x , ˆ ξ i ] = − z ˆ ξ i , [ ˆ ξ , ˆ ξ ] = 0 , [ˆ x i , ˆ ξ j ] = 0 . We realize that in these relations the ‘quantum’ time and space translation parameters closea non-Abelian subalgebra[ˆ x , ˆ x i ] = − z ˆ x i , [ˆ x , ˆ x ] = 0 , i = 1 , , (12)that in the case of the quantum Poincar´e group is known as the (2+1) κ -Minkowski noncommu-tative spacetime M z [9, 22, 23, 24], since no higher order corrections have to be incorporatedwhen the full quantum Poincar´e group is constructed. Note that commutativity is always re-covered in the limit z → i.e. , when the quantum deformation vanishes. It is also worthstressing that the first-order relations (12) do not depend on the curvature ω , so the threefirst-order (A)dS and Minkowskian noncommutative spacetimes coincide. As we shall see later,higher order corrections depending on ω will appear when the full quantum (A)dS groups areconsidered. κ -AdS ω algebra The full (all orders in z ) quantum universal enveloping algebra corresponding to the κ -defor-mation of AdS ω was constructed for the first time in [10]. Its quantum coproduct reads (hereafterwe will use the ρ parameter defined as ω = ρ )∆( P ) = P ⊗ ⊗ P , ∆( J ) = J ⊗ ⊗ J, ∆( P i ) = P i ⊗ e z P cosh( z ρJ ) + e − z P cosh( z ρJ ) ⊗ P i − ρ ǫ ij K j ⊗ e z P sinh( z ρJ ) + ρ e − z P sinh( z ρJ ) ⊗ ǫ ij K j , ∆( K i ) = K i ⊗ e z P cosh( z ρJ ) + e − z P cosh( z ρJ ) ⊗ K i + ǫ ij P j ⊗ e z P sinh( z ρJ ) ρ − e − z P sinh( z ρJ ) ρ ⊗ ǫ ij P j , the compatible set of deformed commutation rules are[ J, P i ] = ǫ ij P j , [ J, K i ] = ǫ ij K j , [ J, P ] = 0 , [ P i , K j ] = − δ ij sinh( zP ) z cosh( zρJ ) , [ P , K i ] = − P i , [ K , K ] = − cosh( zP ) sinh( zρJ ) zρ , [ P , P i ] = ωK i , [ P , P ] = − ω cosh( zP ) sinh( zρJ ) zρ , and the deformed Casimir operators have the form C z = 4 cos( zρ ) ( sinh ( z P ) z cosh (cid:0) z ρJ (cid:1) + sinh ( z ρJ ) z cosh (cid:0) z P (cid:1)) − sin( zρ ) zρ (cid:0) P + ω K (cid:1) , W z = − cos( zρ ) sinh( zρJ ) zρ sinh( zP ) z + sin( zρ ) zρ ( K P − K P ) .
7t is worth mentioning that this is exactly the quantum (A)dS algebra proposed in [21] as thesymmetry algebra of the vacuum excitations in (2+1) quantum gravity. By construction, thelimit ω → κ -Poincar´e algebra. Note also that the physical dimensions of the quantum deformation parameter z are inherited from P , since [ z ] = [ P ] − . If c = 1, this means that z can be interpreted as afundamental length parameter, which in the usual double special relativity theories is consideredto be of the order of the Planck length l p . Algebraically, this link between z and P is directlyrelated to the fact that P and J remain nondeformed (primitive) at the level of the coproduct,which allows the deformation to contain formal power series of these two generators (see [33] fora dimensional analysis of the deformation parameters from the viewpoint of contraction theory). κ -AdS ω Poisson–Lie group
The full κ -AdS ω quantum group could be obtained by computing the Hopf algebra dual to thequantum κ -AdS ω algebra that we have just introduced. This requires a lengthy and cumbersomecomputation, but many features of this quantum group can be extracted from its semiclassicallimit, i.e. , the PL structure on the classical AdS ω group that is in one-to-one correspondencewith the Lie bialgebra that characterizes this quantum deformation.The construction of the κ -AdS ω PL group was fully performed in [34], where we refer to theinterested reader for details. In particular, under the appropriate parametrization of the group,it is found that the local coordinate functions corresponding to the translation parameters closethe following PL subalgebra { x , x } = − z tanh ρx ρ cosh ρx , { x , x } = − z tanh ρx ρ , { x , x } = 0 , whose quantization (in the usual ‘quantum-mechanical’ sense) would give rise to the full non-commutative κ -AdS ω spacetime. Since { x , x } = 0, no ordering ambiguities appear in thequantization process, and the quantum (2+1) noncommutative AdS ω spacetime reads[ˆ x , ˆ x ] = − z tanh ρ ˆ x ρ cosh ρ ˆ x = − z ˆ x + zω ˆ x + zω ˆ x ˆ x + o ( ω ) , [ˆ x , ˆ x ] = − z tanh ρ ˆ x ρ = − z ˆ x + zω ˆ x + o ( ω ) , [ˆ x , ˆ x ] = 0 . Therefore, the κ -Minkowski space M z is the first-order noncommutative spacetime for allthe AdS ω quantum groups. However, when the curvature is not zero ( ω = 0) higher order con-tributions appear, and the quantum (A)dS spacetimes turn out to be nonlinear noncommutativealgebras. Note also that, in any case, (see [34] for the full expressions) the quantum spacetimecoordinates { ˆ x , ˆ x , ˆ x } close a subalgebra. ω deformation The aim of the present section is to show that the κ -deformation is a very particular one-parametric choice amongst all the possible quantum deformations of the AdS ω algebra. In thesequel we will present the chart of Lie bialgebra structures of the AdS ω algebra, that can be8btained by taking into account that the Lie bialgebra structures of so (2 , so (3 ,
1) and the(2+1) Poincar´e algebra are always coboundary structures. Therefore, they come from classical r -matrices, and the most general form for a constant classical r -matrix on AdS ω is r = a J ∧ P + a J ∧ K + a P ∧ P + a P ∧ K + a P ∧ K + a P ∧ K + b J ∧ P + b J ∧ K + b P ∧ P + b P ∧ K + b P ∧ K + b P ∧ K (13)+ c J ∧ P + c K ∧ K + c P ∧ P , where we have initially 15 possible ‘deformation parameters’, that will have to fulfil the con-straints coming form the mCYBE (8). We recall that pure non-standard/twist AdS ω deforma-tions will be obtained when the Schouten bracket [[ r, r ]] = 0 ( i.e. , if their r -matrices are solutionsof the CYBE).Explicitly, the mCYBE (8) leads to the following nonlinear constraints involving the 15 (real)Lie bialgebra parameters and the curvature ω :( a a − a a + a b − a b + a b + b b + a c − ωa c ) = 0 , ( a a − a b + b b + a b + b b + a c − b c − ωa b ) = 0 , ( − a a − a b + a b − b b + a c + ωa a − ωa b − ωa c ) = 0 , ( − a a + a a − a b − a b + a c + b c + b c + ωa c ) = 0 , ( a a + a b − a b − a c + ωa a − ωa c ) = ω ( a a + a b + b b − a b − a c − b c ) , ( a a + b b + a b − a c − c c + ωa b − ωa c + ωc c ) = 0 , ( − a a − b b + b c + a c − b c + ωa b − ωa b − ωa c ) = 0 , ( a a − a b − a b + b b + a b + b b + b c − ωb c ) = 0 , ( a a − a a + b b + a b − a b − a c + b c + ωa b ) = 0 , ( a a − a b + b b + a b − b c − ωa b + ωb b + ωb c ) = 0 , ( − a b + b b + a b + b b + a c − b c + a c − ωb c ) = 0 , ( − a b + a b − b b + b c − ωb b + ωb c )= − ω ( a a + a b + b b − a b − a c − b c ) , ( a a − a b + b b + b c − c c − ωa b + ωb c + ωc c ) = 0 , ( − a a − b b − a c + a c + b c + ωa b − ωa b − ωb c ) = 0 , ( a a + b b − a c − b c + ωa b − ωa b ) = ω ( a a + a b + b b − a b − a c − b c ) ,a + b − a b + a b − a b + a b − a c + b c − ωc = ( a − a + a b − a b − a b + a c + b c − c c + ωa ) , ( − a − b + c − ωa b + ωa b + ωa b − ωa b − ωa c + ωb c )= − ω ( a − a + a b − a b − a b + a c + b c − c c + ωa ) , ( a b − a b + b − b − a b − a c − b c − c c + ωb )= ( a − a + a b − a b − a b + a c + b c − c c + ωa ) . (14)9herefore, the most generic cocommutator is given by (7), namely: δ ( J ) = a J ∧ P + a J ∧ K + a P ∧ P + a P ∧ K + ( b − a )( K ∧ P − P ∧ K )+( a + b )( K ∧ P + P ∧ K ) + b P ∧ J + b K ∧ J + b P ∧ P + b K ∧ P ,δ ( P ) = ωa J ∧ K + a P ∧ J + ωa P ∧ K + a P ∧ P + ( a − b )( P ∧ P + ωK ∧ K )+ ωb J ∧ K + b P ∧ J + ωb P ∧ K + b P ∧ P + ( ωc − c )( K ∧ P + P ∧ K ) ,δ ( P ) = ( b + c )( P ∧ P + ωK ∧ J ) + ωb K ∧ J − ( a − ωb )( J ∧ P + P ∧ K )+ ωc J ∧ P + ωb K ∧ K + ωa P ∧ K + c K ∧ P + b K ∧ P + a P ∧ P + a P ∧ P ,δ ( P ) = ωa J ∧ K + ( a − c )( ωJ ∧ K + P ∧ P ) + ( b + ωa )( P ∧ J + P ∧ K )+ ωc J ∧ P + ωa K ∧ K + c P ∧ K + a P ∧ K + b P ∧ P + b P ∧ P + ωb P ∧ K ,δ ( K ) = c J ∧ K + ( a + b )( J ∧ P + P ∧ K ) + ( a + c )( J ∧ P + P ∧ K ) + b J ∧ P + a K ∧ K + a P ∧ K + c P ∧ P + a P ∧ K + b P ∧ P + b P ∧ K ,δ ( K ) = c J ∧ K + b K ∧ K + a K ∧ P + ( a − b )( P ∧ J + P ∧ K ) + b P ∧ K +( b − c )( P ∧ K + P ∧ J ) + a P ∧ J + c P ∧ P + b P ∧ K + a P ∧ P , (15)where all the parameters appearing in these expressions must satisfy the constraints (14). Ob-viously, some of the parameters could be shown to be inessential through the appropriate au-tomorphisms, and physical restrictions on the parameters should be also imposed in order toobtain some tractable deformations. Anyhow, these expressions give a clear idea about the sizeof the zoo of possible quantum deformations of the AdS ω algebra. Note also that the dual of δ (15) would provide the most generic first-order AdS ω noncommutative spacetime. κ -deformation in (2+1) dimensions The generalizations of the κ -deformation that are contained in the previous characterization ofall the quantum AdS ω deformations can be identified by imposing that, like in (11), δ ( J ) = 0 , δ ( P ) = 0 . These conditions imply that the deformation parameters have to fulfil the relations a i = b i = 0 ( i = 1 , , , a = b ; a = b = 0; c = ωc . Under these conditions the resultant r -matrix only depends on three parameters r = b ( P ∧ K + P ∧ K ) + c J ∧ P + c ( ωK ∧ K + P ∧ P ) , (16)where the κ -AdS ω Planck length parameter is b = − z (see (10)). These three parameters arefree ones, since (16) is already a solution of the mCYBE. Moreover, the Schouten bracket forthe r -matrix (16) reads[[ r, r ]] = − b c ( P ∧ P ∧ P + ωP ∧ K ∧ K + ωP ∧ K ∧ J + ωP ∧ K ∧ J ) − ( b + ωc )( P ∧ K ∧ P + P ∧ P ∧ J + ωK ∧ K ∧ J + P ∧ P ∧ K ) , and this implies that if ω = 0, non-standard (twist) deformations are obtained when b = c = 0( i.e. , the term J ∧ P generates a Reshetikhin twist). In the Poincar´e case ( ω = 0), the P ∧ P term provides a second twist. 10he full cocommutator that corresponds to the three-parametric r -matrix (16) is δ ( J ) = δ ( P ) = 0 ,δ ( P ) = c ( P ∧ P + ωK ∧ J ) − z ( ωK ∧ J + P ∧ P ) + ωc ( J ∧ P + K ∧ P ) ,δ ( P ) = − c ( ωJ ∧ K + P ∧ P ) − z ( ωJ ∧ K + P ∧ P ) + ωc ( J ∧ P + P ∧ K ) ,δ ( K ) = c ( J ∧ P + P ∧ K ) − z ( J ∧ P + P ∧ K ) + c ( P ∧ P + ωJ ∧ K ) ,δ ( K ) = − c ( P ∧ K + P ∧ J ) − z ( P ∧ K + P ∧ J ) + c ( P ∧ P + ωJ ∧ K ) , which leads to the first-order noncommutative spacetime given by[ˆ x , ˆ x ] = − z ˆ x − c ˆ x − c ˆ ξ , [ˆ x , ˆ x ] = − z ˆ x + c ˆ x + c ˆ ξ , [ˆ x , ˆ x ] = 0 , thus providing a two-parametric generalization of the κ -Minkowski spacetime (12).To the best of our knowledge, the full quantum AdS ω algebra generated by the r -matrix(16) is not known yet and constitutes and interesting open problem. As a first step, the Poissoncounterpart of the corresponding all-orders noncommutative spacetimes could be obtained bycomputing the PL brackets on the AdS ω group through the Sklyanin bracket given by (16).Again, the spacetime brackets { x i , x j } so obtained will be no longer linear when the curvature ω is turned on. The same Lie bialgebra approach can be applied to study the quantum deformations of the(3+1) AdS ω Lie algebras, with generators { J i , P , P i , K i } , ( i = 1 , ,
3) and whose commutationrules are [ J i , J j ] = ε ijk J k , [ J i , P j ] = ε ijk P k , [ J i , K j ] = ε ijk K k , [ P i , P j ] = − ω ε ijk J k , [ P i , K j ] = − δ ij P , [ K i , K j ] = − ε ijk J k , [ P , P i ] = ωK i , [ P , K i ] = − P i , [ P , J i ] = 0 , (17)where i, j = 1 , , ε = 1 and the ω parameter is again the constant sectional curvature ofthe associated spacetime such that ω = − Λ. Therefore, when ω > so (3 , ω < so (4 , ω = 0gives rise to the (3+1) Poincar´e Lie algebra iso (3 , r -matrix, r = r ij Y i ∧ Y j that in this case would depend on 45 deformationparameters r ij (we recall that again in the (3+1) case all AdS ω Lie bialgebras are coboundaries).In particular, if we are interested in generalizations of the κ -deformation for the AdS ω algebra,the minimal constraint that we have to impose is that one deformation parameter has to playthe role of a Planck length or, equivalently, that the coproduct of the quantum P generator hasto be primitive (∆( P ) = P ⊗ ⊗ P ). As we know, this means that we have to impose δ ( P ) = [ P ⊗ ⊗ P , r ij Y i ∧ Y j ] = 0 , and, as a result, this assumption elliminates 30 of the r ij parameters. If, by following the (2+1) κ -AdS ω case, we further impose that J also remains primitive under deformation, we get a11ve-parametric candidate for the r -matrix r = z ( K ∧ P + K ∧ P ) + z ( P ∧ P + ωK ∧ K )+ z P ∧ J + z K ∧ P + z J ∧ J , although now the mCYBE is not automatically fulfilled and leads to the following relationsamong the quantum deformation parameters: z z = 0 , ( z − z ) z − ωz z = 0 ,z z + z ( z − z ) + ωz = 0 ,z z − z ( z − z ) = 0 ,z + ω ( z z − z z ) = 0 . Finally, the solution of these equations leads to two disjoint families of two-parametric AdS ω Lie bialgebras, that are generated by the two following classical r -matrices [35]: r z ,z = z (cid:0) K ∧ P + K ∧ P + K ∧ P ± √ ω J ∧ J (cid:1) + z P ∧ J , (18) r z ,z = z (cid:0) P ∧ P + ωK ∧ K − ω J ∧ J ± √ ωP ∧ K (cid:1) + z P ∧ J , (19)whose Schouten brackets are ( l = 1 , r z l ,z , r z l ,z ]] = A l (cid:18) ω J ∧ J ∧ J − ε ijk ( ωJ i ∧ K j ∧ K k + J i ∧ P j ∧ P k ) (cid:19) + A l X i =1 K i ∧ P i ∧ P , where A = z , A = z ω. Some comments on these two r -matrices are pertinent: • The r z , matrix generates the κ -AdS ω deformation in (3+1) dimensions, but the explicitform of its all-orders quantization when ω = 0 is still an open problem. • When ω = 0, the r z ,z matrix generates the twisted κ -deformation of the (3+1) Poincar´ealgebra whose full expressions were given in [36]. • If we compare the (3+1) r -matrix (18) with the (2+1) r -matrix (16), we realize that the c and the b terms of the latter are not compatible with the κ -deformation in (3+1)dimensions, and can only be simultaneously considered in the deformation of the secondtype generated by (19). The full quantization of the latter is also unknown yet.The first-order noncommutative AdS ω spaces associated to these two families of Lie bialge-bras are deduced from the dual cocommutators, and are given by the following commutationrules involving the (3+1) quantum coordinates { ˆ θ, ˆ x µ , ˆ ξ i } dual to { J, P µ , K i } ( µ = 0 , , , r z ,z : [ˆ x , ˆ x i ] = − z ˆ x i + z ε ij ˆ x j , [ˆ x i , ˆ x j ] = 0 .r z ,z : [ˆ x , ˆ x I ] = − z ε IJ ˆ ξ J + z ε IJ ˆ x J , [ˆ x , ˆ x ] = ± z √ ω ˆ x , [ˆ x I , ˆ x ] = z ˆ θ I , [ˆ x , ˆ x ] = 0 , I, J = 1 , . Note that the first deformation gives rise to a generalization of the κ -Minkowski space, that isrecovered when z = 0. Again, no dependence on the curvature ω appears in this first-order12uantum group, although a nonlinearity ruled by the cosmological constant is expected when thefull quantum group is constructed. On the contrary, the quantum spacetimes coming from thesecond deformation have a completely different shape: the deformation linked to z introducesthe quantum rotation and boost coordinates into the spacetime commutation rules, that areno longer a subalgebra and, moreover, the cosmological constant is already manifest at thefirst-order of these quantum group relations.Finally, we would like to recall that one of the main features of the (3+1) κ -Poincar´e al-gebra is the fact that the rotation subalgebra remains undeformed. Therefore, the isotropy ofthe ‘quantum space’ is preserved, since the κ -Minkowski spacetime has the same type of com-mutation rules for the three quantum space coordinates. This seems to be a quite reasonablephysical condition and, apparently, the generalization of the κ -Poincar´e deformation providedby r z ,z would break this ‘quantum isotropy’ property. However, we would like to stress that ifwe consider the following automorphism of the AdS ω algebra e Y = − √ Y + 1 √ Y , Y = 1 √ (cid:16) e Y − e Y (cid:17) , e Y = 1 √ Y + 1 √ Y + 1 √ Y , Y = 1 √ (cid:16) − e Y + e Y + e Y (cid:17) , e Y = − √ Y + 1 √ Y + 1 √ Y , Y = 1 √ (cid:16) e Y + e Y + e Y (cid:17) , for Y i ∈ { J i , P i , K i } , e P = P , where ˜ Y i denote the transformed generators, and if we take ˜ z = z / √
3, we get r z , ˜ z = z (cid:18) ˜ K ∧ ˜ P + ˜ K ∧ ˜ P + ˜ K ∧ ˜ P ± √ ω √ (cid:16) ˜ J ∧ ˜ J + ˜ J ∧ ˜ J + ˜ J ∧ ˜ J (cid:17)(cid:19) +˜ z ˜ P ∧ (cid:16) ˜ J + ˜ J + ˜ J (cid:17) . This means that in this new AdS ω basis, the ‘broken’ isotropy of the r z ,z deformation can berestored, since the role of the three rotation generators is exactly the same. This indicates thatsome new multiparametric deformations of the (A)dS and Poincar´e algebras could be consideredas reasonable symmetries on a physical basis.Summarizing, in this work we have shown that the classification of the quantum deformationsof the kinematical symmetries of (2+1) and (3+1) relativistic spacetimes can be approached in avery efficient and computationally tractable way by analyzing the first-order deformation givenby the corresponding Lie bialgebra structures. We think that this viewpoint could be usefulin the near future in order to explore the role that multiparametric quantum groups with non-vanishing cosmological constant could play in a quantum gravity context. Acknowledgments
This work was partially supported by the Spanish MICINN under grant MTM2010-18556 (withEU-FEDER support). 13 eferences [1] Kulish P P and Reshetikhin N 1981
Proc. Steklov Math. Inst.
Quantum Groups in Proc. Int. Congress of Math. (Berkeley 1986) edA V Gleason (Providence, RI: American Mathematical Society) p 798[3] Jimbo M 1985
Lett. Math. Phys . Introduction to quantum group and integrable massive models of Quan-tum Field Theory , eds Mo-Lin Ge and Bao-Heng Zao (Singapore: World Scientific) p 69[5] Reshetikhin N Y, Takhtadzhyan L A and Faddeev L D 1990
Leningrad Math. J. A Guide to Quantum Groups (Cambridge: Cambridge Uni-versity Press)[7] Majid S 1995
Foundations of Quantum Group Theory (Cambridge: Cambridge UniversityPress)[8] Garay L J 1995
Int. J. Mod. Phys. A Phys. Lett. B
J. Phys. A: Math. Gen. J. Phys. A: Math. Gen. Int. J. Mod. Phys. D ibid Phys. Rev. Lett. Phys. Lett. B ibid
Class. QuantumGrav. Phys. Rev. D Symmetry Commun. Math. Phys.
Phys. Lett. B
Commun. Math. Phys.
Phys. Lett. B
Class. Quantum Grav. J. Phys. A: Math. Gen. L1251[23] Majid S and Ruegg H 1994
Phys. Lett. B
J. Phys. A: Math. Gen. Commun. Math. Phys. ibid
Am. Math. Soc. Transl.
J. Math. Phys. Nucl. Phys. B
Phys. Lett. B
Clas. Quantum Grav. J. Math. Phys. J. Phys. A: Math. Gen. J. Math.Phys. Czech. J. Phys. Int. J. Mod. Phys A23