The κ (A)dS quantum algebra in (3+1) dimensions

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Physics

Letters

B

www.elsevier.com/locate/physletb

The

κ

-(A)dS

quantum

algebra

in

(3

+

1)

dimensions

Ángel Ballesteros

a

,

∗

,

Francisco

J. Herranz

a

,

Fabio Musso

b

,

Pedro Naranjo

a

a_{DepartamentodeFísica,UniversidaddeBurgos,E-09001Burgos,Spain}

b_{IstitutoComprensivo“LeonardodaVinci”,ViadellaGrandeMuraglia37,I-0014Rome,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received9December2016 Accepted11January2017 Availableonline16January2017 Editor:M.Cvetiˇc

Keywords:

Anti-deSitter Cosmologicalconstant Quantumgroups Poisson–Liegroups Liebialgebras

Quantumdualityprinciple

ThequantumdualityprincipleisusedtoobtainexplicitlythePoissonanalogueofthe

κ

-(A)dSquantum algebra in (3+1) dimensions as the corresponding Poisson–Lie structure on the dual solvable Lie group.The constructionis fullyperformedinakinematicalbasisand deformed Casimirfunctionsare also explicitlyobtained. The cosmological constant is included as aPoisson–Lie group contraction parameter,andthelimit→0 leadstothewell-known

κ

-Poincaréalgebrainthebicrossproductbasis. AtwistedversionwithDrinfel’ddoublestructureofthis

κ

-(A)dSdeformationissketched.

©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Since Classical Gravity is essentially a theory describing the geometryof spacetime, it seems natural to consider that a suit-able definition of a “quantum” spacetime geometry emerging at thePlanckenergyregime could bea reasonable featureof Quan-tum Gravity. Indeed, specific mathematical frameworks for such “quantumgeometry”havetobeproposed.Inparticular,“quantum spacetime” is frequently introduced as a noncommutative alge-brawhose noncommutativityisgoverned by a parameter related tothe Planckscale, thusleading to minimumlength frameworks through generalized spacetime uncertainty relations (see, for in-stance,[1–5]andreferencestherein).

Inthiscontext,quantumgroups[6–8]provideaconsistent ap-proach to noncommutative spacetimes, since the latter are ob-tained asnoncommutative algebras that are covariant under the action of quantum kinematical groups. For instance, the well-known

κ

-Minkowskispacetime[9–12]was obtainedasa byprod-uct of the

κ

-Poincaré quantum algebra, which was introduced in [13] (see also [14–16]) by making use of quantum group contraction techniques [17,18]. One of the main features of the

κ

-Poincaréquantumalgebra(whichistheHopfalgebradualtothe quantumPoincaré group, andis definedasa deformation of the

*

Correspondingauthor.

E-mailaddresses:[email protected](Á. Ballesteros),[email protected](F.J. Herranz), [email protected](F. Musso),[email protected](P. Naranjo).

Poincaré algebra intermsof thedimensionful parameter

κ

) con-sistsinitsassociateddeformedsecond-orderCasimir,whichleads to a modified energy-momentum dispersion relation. From the phenomenologicalside,thistypeofdeformeddispersionrelations havebeenproposedaspossibleexperimentallytestablefootprints ofquantumgravity effectsinverydifferentcontexts (see[19–21] andreferencestherein).

Moreover, if the interplay between quantum spacetime and gravity atcosmologicaldistances isto be modeled,then the cur-vature of spacetime cannot be neglected and models with non-vanishing cosmological constant have to be considered [22–25]. Thus, therelevantkinematicalgroups(and spacetimes)wouldbe the (anti-)de Sitter ones, hereafter (A)dS, and the construction of quantum (A)dS groups should be faced. In (1

+

1) and (2

+

1) dimensions, the corresponding

κ

-deformations have been con-structed [26,27] (see also [28–30] for classification approaches). In fact, it is worth stressing that the

κ

-(A)dS deformation intro-ducedin[27] was proposedin[31] asthealgebra ofsymmetries for(2

+

1)quantum gravity(seealso [32]),andcompatibility con-ditions imposed by the Chern–Simons approachto (2

+

1) gravity have been recently used [33] in order to identify certain privi-leged(A)dSquantumdeformations[34](amongthem,thetwisted

κ

-(A)dSalgebra[35–38]).

Concerning (3

+

1) dimensions, we recall that in the papers [13–16] the

κ

-Poincaré algebra was obtained asa contraction of the Drinfel’d–Jimbo quantum deformation [6,39] of the

so

(

3

,

2

)

and

so

(

4

,

1

)

Liealgebras,bystartingfromthelatterwritteninthe

http://dx.doi.org/10.1016/j.physletb.2017.01.020

Cartan–Weyl or Cartan–Chevalley basis, and then obtaining suit-able real forms of the corresponding quantum complex simple Lie algebras. However, to the best of our knowledge, no explicit expression of the (3

+

1)

κ

-(A)dS algebras in a kinematical basis (rotations J, boosts K, translations P) andincluding the cosmo-logicalconstant

hasbeenpresentedsofar,thus preventingthe appropriate physicalanalysisoftheinterplaybetween

andthe quantumdeformation.Moreover,explicitexpressionsforthe(3

+

1)

κ

-(A)dSCasimiroperatorscannotbefoundintheliterature.

Theaimofthispaperistofillthisgapandtoprovidethe Pois-son version ofsuch (3

+

1)

κ

-(A)dS algebra together withits two deformedCasimirs withnon-vanishingcosmologicalconstant:the deformedsecond-order invariant–whichisrelatedtothe energy-momentum dispersion relation– as well as thedeformed fourth-orderCasimir, whichwould be relatedtothe spin/helicityofthe particles[40,41].We recallthatinthecase

=

0 suchdeformed Pauli–Lubanski4-vectorwasobtainedforthe(3

+

1)

κ

-Poincaré al-gebrain[42].

From a technical perspective, the quantum (Poisson) algebra willbefullyconstructedwithoutmakinguseneitherofrealforms nor of contractiontechniques, thus solving the kinematical basis problemsince the basis is fully fixed from theinitial data.As it waspresentedin[43],bymakinguseofthePoissonversionofthe quantumduality principle(see [6,44–46] andreferencestherein), itis possibleto constructthe fullPoisson–Hopf algebra structure fromtheLiebialgebrathatdefinesthecocomutator

δ

(i.e., the first-order of the coproduct

) of the

κ

-(A)dS deformation. In short, it can be said that a quantumPoisson algebra isjust a Poisson– Liestructure on thedual Lie group G∗, whichhas asLiealgebra

g∗ (the one obtainedby dualizing the cocommutator

δ

). In fact, thismethodwasalreadyusedin[43]inordertorecoverthe Pois-son

κ

-Poincaré algebrain (3

+

1) dimensions. Moreover, inallthe expressions ofthe

κ

-(A)dS algebrathat we willpresent, the cos-mologicalconstant

willbeintroducedasanexplicitparameter, andthe

→

0 limitwillautomaticallyprovidethe

κ

-Poincaré al-gebraintheso-calledbicrossproductbasis[8,10,47,48].

Evidently, the proper quantum

κ

-(A)dS algebra wouldbe ob-tainedbysubstitutingthePoissonbracketshereobtainedby com-mutatorsandbyreplacingthePoissonalgebrageneratorsby non-commutingoperators.Thiswouldleadto manyordering ambigu-ities that have tobe solved by the appropriate choice ofa sym-metrization prescription, which has to be also implemented on thecoproductmap (see thediscussion in[43]).Nevertheless,the Poissonapproachherepresentedisenoughtogetaquite compre-hensible description of the deep changes that the non-vanishing cosmologicalconstantgenerateswithinthe

κ

-deformation.In par-ticular,theinfluenceofthecosmologicalconstantonthedeformed Casimir operators andon the noncommutative spacetime can be explicitlyevaluated. Moreover, the interplay betweenthe cosmo-logicalconstantandthe(1

+

1)

κ

-dS Poissonalgebra hasbeen re-cently presentedin [49,50] inthe context ofcurved momentum spacesandrelative localityframeworks[51–54].Thus,the results herepresentedprovidethetoolsneededinordertogetadeeper insightabouttheremnants ofthequantumgeometrythatare en-codedatthePoissonlevelinthemorerealistic(3

+

1)case.

The paper is structured as follows. In the next section, the kinematicalbasisforthe(A)dSandPoincaréalgebrasin(3

+

1) di-mensionsisrevisitedbyconsideringtheone-parameterLiealgebra AdSωwhoseparameter

ω

isrelatedwiththecosmologicalconstant inthe form

ω

= −

.In Section 3,the Poisson

κ

-deformation of the(3

+

1)AdSωalgebraisfullyconstructedasaPoisson–Lie struc-ture on the dual Poisson–Lie group defined by the Lie bialgebra structurethatunderliesthe

κ

-deformation.Theexplicitexpression ofthetwoCasimirfunctionsisthenobtainedandthe

κ

-Poincaré limit is straightforwardly computed. This method can be further

applied to the twisted

κ

-AdSω algebra arising from a Drinfel’d doublestructurein(3

+

1)dimensions,andwhosefirst-order non-commutativespacetimehasbeenrecentlyintroducedin[55].A fi-nalsectionincludingseveralcommentsandopenproblemscloses thepaper.

2. The(3

+

1)AdSωalgebra

Let us consider the three (3

+

1) Lorentzian Lie algebras as the one-parameter family AdSω, where

ω

is a real (graded) contraction–deformation parameter [41]. In the kinematical ba-sis

{

P0

,

Pa

,

Ka

,

Ja

}

(

a

=

1

,

2

,

3

)

of generators of time translation,

spacetranslations,boosts androtations,respectively, the commu-tationrulesforAdSωread

[

Ja

,

Jb

] =

abcJc

,

[

Ja

,

Pb

] =

abcPc

,

[

Ja

,

Kb

] =

abcKc

,

[

Ka

,

P0

] =

Pa

,

[

Ka

,

Pb

] =

δ

abP0

,

[

Ka

,

Kb

] = −

abcJc

,

[

P0

,

Pa

] =

ω

Ka

,

[

Pa

,

Pb

] = −

ω

abcJc

,

[

P0

,

Ja

] =

0

,

(2.1)

wherehereaftera

,

b

,

c

=

1

,

2

,

3 andsumoverrepeatedindiceswill be assumed. Inparticular, thecontraction parameter

ω

isrelated with the cosmologicalconstant in the form

ω

= −

,thus AdSω containstheAdSLiealgebra

so

(

3

,

2

)

when

ω

>

0,thedSLie alge-bra

so

(

4

,

1

)

for

ω

<

0,andthePoincaréone

iso

(

3

,

1

)

for

ω

=

0.

The Lie algebra AdSω is endowedwith two Casimiroperators (see, e.g., [41,56]). The quadratic one comes from the Killing– Cartanformandisgivenby

C

=

P₀2

−

P2

+

ω

J2

−

K2

,

(2.2)

where P₀2

−

P2 is the square of the energy-momentum 4-vector

(

P0

,

P

)

.For

ω

=

0,(2.2)givesthesquareofthePoincaréinvariant restmass.

ThesecondAdSω Casimirisafourth-orderinvariantthatreads

W

=

W₀2

−

W2

+

ω

(

J

·

K

)

2

,

W0

=

J

·

P

,

Wa

= −

JaP0

+

abcKbPc

,

(2.3)

where

(

W0

,

W

)

are just the components of the Poincaré Pauli– Lubanski4-vector.For

ω

=

0,theinvariant W2

0

−

W2 providesthe squareofthespin/helicityoperator(see[40,41]),whichintherest frame itis proportionalto thesquare ofthe angularmomentum. Asaconsequence,theCasimir

W

(2.3)generalizesthisoperatorto thecasewithanon-vanishingcosmologicalconstant.

Notice that setting

ω

=

0 in all the above expressions corre-sponds to apply an Inönü–Wigner contraction leading to the flat

→

0 limit[18].TheCartandecompositionofAdSω isgivenby

AdSω

=

h

⊕

p

,

h

=

Span

{

K

,

J

}

so

(

3

,

1

),

p

=

Span

{

P0

,

P

}

,

where

h

is the Lorentz subalgebra. Thus, the family of the three(3

+

1)Lorentziansymmetrichomogeneousspaceswith con-stant sectional curvature

ω

is definedby the quotientAdS3_ω+1

≡

SOω

(

3

,

2

)/

SO

(

3

,

1

)

, where the Lie groups SO

(

3

,

1

)

and SOω

(

3

,

2

)

have

h

andAdSω asLiealgebras, respectively.Thespecific spaces are:

•

ω

>

0

,

<

0:AdSspacetimeAdS3+1

≡

SO

(

3

,

2

)/

SO

(

3

,

1

)

.

•

ω

<

0

,

>

0:dSspacetimedS3+1

≡

SO

(

4

,

1

)/

SO

(

3

,

1

)

.

•

ω

=

=

0:MinkowskispacetimeM3+1

≡

ISO

(

3

,

1

)/

SO

(

3

,

1

)

.

3. ThePoisson

κ

-deformationofthe(3

+

1)AdSωalgebra

The approach presented in [43] for the construction of the Poisson version of quantum algebras is fully determined by the first-order(in the deformation parameter z) of the quantum de-formation,whichisprovided bythecocommutatormap

δ

.Inour case, the classical r-matrix for the

κ

-deformation of the (3

+

1) AdSω,givenintermsofthekinematicalgenerators,is[18]

r

=

z

K1

∧

P1

+

K2

∧

P2

+

K3

∧

P3

+

Here the zerocosmological constant limit corresponds to setting

ω

=

0,and thequantum deformation parameters q,

κ

and z are relatedasq

=

ez_andz

₌

₁

_/κ

_.

Now,the quantumdualityprinciple[6,44–46]impliesthatthe PoissonversionofthequantumAdSω algebraisjustaPoisson–Lie structureonthedualgroup G∗_ω,whoseLiealgebra

g

∗_ω isobtained asthedualofthecocommutatormap(3.2).Moreover,the coprod-uctmapfor

κ

-AdSω isexactlythedualoftheproductonG∗ω,i.e., the multiplication lawfor the dual group in terms of the corre-spondinglocalcoordinates.Therefore,if

{

p0

,

p

,

k

,

j

}

isthebasisfor thedual coordinates to

{

P0

,

P

,

K

,

J

}

, then

δ

(3.2) defines the fol-lowing(solvable)10-dimensionaldualLiealgebra

g

∗_ω:

[j1

,

j2]

=

0

,

[j1

,

j3]

=

z

√

ω

j1

,

Indeed,thelimitz

→

0 givesrisetoanAbelianLiealgebrawhose group law is additive for all coordinates, thus givingrise to the undeformed(primitive)coproduct

(

X

)

=

X

⊗

1

+

1

⊗

X forthe non-deformedquantumalgebra.

Therefore,atthePoissonlevelthedeformationinduces a non-Abelian dual group G∗_ω, whose group law provides the explicit

coproduct for the Poisson–Hopf algebra AdSω. Since the adjoint representation

ρ

of

g

∗ω isfaithful,wecanuseittoobtainthe cor-responding10-dimensionalmatrixrepresentation ofageneric Lie groupelementintheform:

G∗_ω

=

exp

−

J3

ρ

(

j3

)

t

Alongbutstraightforwardcomputation(see[43,57]fordetails about thespecificprocedure)leadstothegrouplawforG∗_ω,which can be written as the following coproduct forthe Poisson AdSω algebra:

We stress that we have obtained a two-parametric deformation, which is ruled by a “quantum” deformation parameter z

=

1

/κ

together witha “classical” deformation parameter

ω

(the cosmo-logical constant). Notice also that this coproduct is written in a “bicrossproduct-type”basisthatgeneralizestheonecorresponding tothe(2

+

1)

κ

-AdSω algebra[38].

SincethisdualLiebialgebraisnotacoboundaryone,we haveno Sklyaninbracket available andthe Poisson algebra relations have tobeobtainedbymakinguseofthecomputational procedure in-troducedin[43].In particular,we will assume that such Poisson tensor is quadratic in the “elementary” functions that appear in thepreviouscoproduct,namely

andthemostgeneralantisymmetricquadraticPoissonbracket de-pendingonthesefunctionshastobeconstructed.Afterwards,we impose onthisbracketthe Poissonhomomorphism condition for thedeformed

(3.4) that we have previously obtained. This re-quirementgivesrisetoahugesystemoflinearequations,thatcan be solved byusing asymbolicmanipulation program. Finally,we requirethatthelinearizationofthistentativePoissontensorgives backtheAdSω brackets(2.1).Afterenforcingthislastrequirement wegettheuniquelydeterminedPoissonbrackets,whichfulfillthe Jacobiidentityandareexplicitlygivenby:

{

J1

,

J2

} =

ThisdeformedPoissonalgebra,togetherwiththedeformed co-product

(3.4),providesthePoissonversionofthe

κ

-AdSω alge-bra in(3

+

1) dimensions. Moreover, the deformedcounterpart of thesecond-orderCasimirinvariant(2.2)isfoundtobe

C

z

=

W

z

= −

sinh2

(

z P0

)

z2

sinh2

(

z

√

ω

J3

)

z2

_ω

+

1 2

1

+

e−2z√ωJ3

₍

_J2 1

+

J22

)

+

sinh2

(

z

√

ω

J3

)

2z2

_ω

×

2e2z P0

₍

_P2

3

+

ω

K32

)

−

(

e2z P0

−

1

)(

P2

+

ω

K2

)

−

1 4e

2z P0

sinh

(

z

√

ω

J3

)

√

ω

(

P

2

₊

_ω

_K2

₎

₋

_2R 3cosh

(

z

√

ω

J3

)

2

+

(

e2z P0

−

1

)

sinh

(

2z

√

ω

J3

)

2z2

√

_ω

R3

−

e2z P0

_R2

1

+

R22

−

2e−2z √

ωJ3_J

1J2

(

P1P2

+

ω

K1K2

)

+

e2z P0

1

−

e−2z√ωJ3

z

√

ω

+

z

√

ω

e−2z√ωJ3

₍

_J2 1

+

J22

)

×

(

J1P1

+

J2P2

)

P3

+

ω

(

J1K1

+

J2K2

)

K3

+

e2z P0

−

1 2z

1

+

e−2z√ωJ3

₍

_J

1R1

+

J2R2

)

+

e2z P0_e−2z√ωJ3

_J2

1

(

P12

+

ω

K21

)

+

J22

P2₂

+

ω

K₂2

−

1 4

(

J

2 1

+

J22

)

(

e2z P0

−

₁

₎₍

_L

−

+

e−2z√ωJ3

₍

_P2

+

_ω

_K2

₎₎

+

2e2z P0

₍

_e−2z√ωJ3

−

₁

₎₍

_P2 3

+

ω

K32

)

+

z 2e

2z P0_e−2z√ωJ3

₍

_J

1R1

+

J2R2

)

×

L₊

+

e2z√ωJ3_L

−

+

ω

(

1

−

e−2z P0

₎₍

_J2 1

+

J22

)

−

z2

ω

e2z P0_e−2z√ωJ3

₍

_J

1R1

+

J2R2

)

2

−

z2 8e

2z P0

₍

_J2

1

+

J22

)

L−

(

L−

+

e−2z √

ωJ3_L +

)

−

z2 4

ω

e

−2z√ωJ3

₍

_J2 1

+

J22

)

2

×

sinh2

(

z P0

)

z2

−

e 2z P0

₍

_P2

3

+

ω

K32

)

+

e2z P0

−

₁ 2 L−

+

z3 2

ω

e

2z P0_e−2z√ωJ3

₍

_J2

1

+

J22

)

L−

×

J1R1

+

J2R2

−

z

8

(

J 2

1

+

J22

)

L−

,

(3.7)

wherewehaveusedthenotation L_±

=

P2

₊

_ω

_K2

_±

₂

√

_ω

_R 3.Both the“classical” and“flat”limitsz

→

0 and

ω

→

0 arealways well-defined.

Indeed,whatwehaveobtainedisacommutativePoisson–Hopf algebra,whosequantizationhastobeperformedinordertoobtain the proper

κ

-AdSω quantum algebra. Once this had been com-pleted,a (presumablynonlinearandcomplicated)changeofbasis should existbetween thekinematical basis andthe Cartan–Weyl orCartan–Chevalley basis used in [13–16].Nevertheless, mostof thephysically relevantfeatures of thisquantum deformation can alreadybe extractedfromthe kinematicalPoisson structurehere obtained.

3.1.ThePoisson

κ

-Poincaréalgebra

Underthe

ω

→

0 limittheclassical r-matrix(3.1)andthe co-commutators(3.2)reduceto

r

=

z

(

K1

∧

P1

+

K2

∧

P2

+

K3

∧

P3

) ,

δ(

P0

)

=

0

,

δ(

Ja

)

=

0

,

δ(

Pa

)

=

z Pa

∧

P0

,

δ(

Ka

)

=

z

(

Ka

∧

P0

+

abcPb

⊗

Jc

) ,

whichmeansthatthedualgroupis“moreAbelian”thanthe

ω

=

0 one. Therefore the dual group law is simpler, andthe

ω

→

0 limitof(3.4)and(3.5)leadtothefollowingcoproductandPoisson brackets

(

P0

)

=

P0

⊗

1

+

1

⊗

P0

,

(

Ja

)

=

Ja

⊗

1

+

1

⊗

Ja

,

(

Pa

)

=

Pa

⊗

1

+

e−z P0

⊗

Pa

,

(

Ka

)

=

Ka

⊗

1

+

e−z P0

⊗

Ka

+

z

abcPb

⊗

Jc

,

{

Ja

,

Jb

} =

abcJc

,

{

Ja

,

Pb

} =

abcPc

,

{

Ja

,

Kb

} =

abcKc

,

{

Ka

,

P0

} =

Pa

,

{

Ka

,

Kb

} = −

abcJc

,

{

P0

,

Ja

} =

0

,

{

P0

,

Pa

} =

0

,

{

Pa

,

Pb

} =

0

,

{

Ka

,

Pb

} =

δ

ab

1 2z

1

−

e−2z P0

+

z 2P

2

−

z PaPb

.

SinceallthecoordinatefunctionsPoisson-commute,ordering am-biguities do not exist andthe quantizationof this Poisson–Hopf algebra is immediate, thus giving exactly the

κ

-Poincaré alge-brainthebicrossproductbasis[10,47,48].Thedeformedquadratic Casimir function providing the deformed mass-shell condition is obtainedthroughthe

ω

→

0 limitof(3.6);namely

C

z

=

2

z2[cosh

(

z P0

)

−

1]

−

e

z P0_P2

=

4

z2sinh 2

₍

_{z P}

0

/

2

)

−

ez P0P2

.

(3.8)

The deformed Pauli–Lubanski invariant, coming from the

ω

→

0 limitof(3.7),canbewrittenintheform

W

z

=

cosh

(

z P0

)

−

z2 4 e

z P0_P2

W2_z,0

−

W2_z

,

Wz,0

=

e

z

2P0_J

·

_P

_,

Wz,a

= −

Ja

sinh

(

z P0

)

z

+

e

z P0

abc

Kb

+

z

2

bklJkPl

Pc

,

(3.9)

tobecomparedwith(2.3)with

ω

=

0.

3.2. Atwisted

κ

-AdSωalgebra

Ifweadd tother-matrix(3.1)aReshetikhintwistofthetype

rϑ

=

ϑ

J3

∧

P0,weareledtoconsiderthetwo-parametricLie bial-gebrastructuregeneratedby

rz,ϑ

=

z

K1

∧

P1

+

K2

∧

P2

+

K3

∧

P3

+

√

ω

J1

∧

J2

+

ϑ

J3

∧

P0

,

(3.10) whichisjustther-matrixthatgeneratesaDrinfel’ddouble quan-tum deformation of the AdSω algebra in (3+1) dimensions that has been recently considered in [55]. The corresponding cocom-mutator map is the sum of two terms

δ

z,ϑ

=

δ

+

δ

ϑ where

δ

is

givenin(3.2)whilethetwistedcomponent

δ

ϑ reads

δ

ϑ

(

P0

)

=

0

,

δ

ϑ

(

J3

)

=

0

,

δ

ϑ

(

J1

)

= −

ϑ

J2

∧

P0

,

δ

ϑ

(

J2

)

=

ϑ

J1

∧

P0

,

δ

ϑ

(

P1

)

= −

ϑ (

P2

∧

P0

+

ω

J3

∧

K1

) ,

δ

ϑ

(

P2

)

=

ϑ (

P1

∧

P0

−

ω

J3

∧

K2

) ,

δ

ϑ

(

K1

)

= −

ϑ (

K2

∧

P0

−

J3

∧

P1

) ,

δ

ϑ

(

K2

)

=

ϑ (

K1

∧

P0

+

J3

∧

P2

) ,

δ

ϑ

(

K3

)

=

ϑ

J3

∧

P3

.

(3.11)

The Poisson version of the corresponding twisted

κ

-AdSω al-gebra can be obtained by following the same method as in the untwistedcase, andwill be presentedelsewhere.In fact, the ad-ditionaltwistcontribution(3.11) tothecocommutatorwouldlead to adual Poisson–Lie group G∗_ω,ϑ which wouldbe “lessAbelian” than G∗_ω, thus leading to a more complicated dual group law. Nevertheless,thePoisson–Liebracketscompatiblewithsuch two-parametric coproduct will be the same as in the non-twisted case.Again,theLiebialgebraunderlyingthecorrespondingPoisson twisted

κ

-Poincaré algebra isobtainedfrom(3.10) inthe vanish-ingcosmologicalconstantlimit

ω

→

0 and,itsfullHopfstructure hasbeenworkedoutin[35–37,43].

4. Concludingremarks

Severalfeaturesofthe

κ

-(A)dSPoisson–Hopfalgebrapresented in the previous section are worth to be commented. Firstly, it becomesapparentthattheexistenceofanon-vanishing cosmolog-icalconstantimpliesasignificantlymorecomplicatedHopfalgebra structure.Thiswillhavea deepimpactasfarasthequantization oftheHopfalgebraisconcerned,sincemanyorderingambiguities (whichareabsentinthe

κ

-Poincarécase)appear.Asitwasalready pointedoutin[43],such orderingissuescan beminimizedifwe considerabasisinwhichthedeformedcoproductisinvariant un-der thecomposition ofthe flip operator

σ

(

X

⊗

Y

)

=

Y

⊗

X with thechangez

→ −

zinthedeformationparameter.This “symmetri-cal”basis forthe

κ

-(A)dSω algebracanbefoundandturnsoutto be

Underthischangeofbasisfunctionswehave,forinstance,that

(

J

˜

1

)

= ˜

J1

⊗

e

All the remaining new coproducts in this basis can straightfor-wardlybewritten,butwe omitthemforthe sakeofbrevity.The

fullsolutionofthequantizationproblemiscurrentlyunder inves-tigation.

It is also worth mentioningthat expressions (4.1) reflect that the rotation sector of the (3

+

1)

κ

-(A)dS algebra is a quantum

so

(

3

)

subalgebra, which is generated by the

√

ω

J1

∧

J2 termin theclassicalr-matrix(3.1).Thisconstitutesanessential difference withrespecttothePoincarécase,sinceinthelimit

ω

→

0 this ro-tationsubalgebrabecomesanon-deformedone.Thesamehappens for

ω

=

0 inthe(2

+

1)

κ

-(A)dSdeformation,whichisgeneratedby theclassicalr-matrix

r

=

z

(

K1

∧

P1

+

K2

∧

P2

) ,

wherethecosmologicalconstantdoesnotappear.Moreover, both in the (2

+

1) and (3

+

1) cases the Lorentz sector is not a Hopf subalgebraforanyvalueof

ω

.

Ontheotherhand,itcanalsobeappreciatedthattheCasimirs (3.6) and(3.7) are profoundly modified by thecosmological con-stant when they are compared with their

κ

-Poincaré counter-parts (3.8) and (3.9). In particular, the quadratic one (3.8) giv-ing riseto the

κ

-Poincarédispersion relationisnow transformed into(3.6),wherebothboostandrotationgeneratorsdocontribute. The physical significance of thisfact deserves further study,that could be approached by takinginto account the resultsobtained in [49,50] for the (1

+

1)

κ

-dS case, and the common features withrespecttothe(2

+

1)

κ

-(A)dSCasimirsthatwerecommented in[33].

Anotherimportantaspecttobeconsideredisthe noncommuta-tive

κ

-(A)dSspacetimeassociatedwiththedeformationhere pre-sented.Thisnoncommutativespacetimewillbeanonlinearalgebra whose first-ordercan be directly obtainedasa Lie subalgebraof the dual Lie algebra (3.3). If we identify

ˆ

x0

≡

p0 and x

ˆ

a

≡

pa,

we find thatthe first-order

κ

-(A)dSω noncommutative spacetime reads

[ˆ

xa

,

x

ˆ

0

] =

z

ˆ

xa

,

[ˆ

xa

,

x

ˆ

b

] =

0

,

z

=

1

/

κ

,

a

,

b

=

1

,

2

,

3

.

(4.2)

This is just the well known (3

+

1)

κ

-Minkowski spacetime (see [9–12]), and the cosmological constant does not appear at this first-orderlevel.Higher-ordercontributionswillindeedcontain

ω

, and they will arise when the full Hopf algebra duality for the

κ

-(A)dScoproduct(3.4)iscomputed(see[59]).Alternatively,such all-orders (3

+

1)noncommutativespacetime canbeobtainedasa PoissonsubalgebrawithinthePoisson–Liestructuredefinedbythe Sklyaninbracketcomingfromtheclassicalr-matrix(3.1)(see[38] for the (2

+

1) case). In the same manner, the first-order twisted version ofthisnoncommutative spacetime is obtainedby adding to(4.2)thecontributionscomingfrom(3.11),andyields

[ˆ

x1

,

ˆ

x0

] =

zx

ˆ

1

+

ϑ

ˆ

x2

,

[ˆ

x2

,

x

ˆ

0

] =

z

ˆ

x2

−

ϑ

x

ˆ

1

,

[ˆ

x3

,

x

ˆ

0

] =

zx

ˆ

3

,

[ˆ

xa

,

ˆ

xb

] =

0

,

a

,

b

=

1

,

2

,

3

,

which is not isomorphic to the (3

+

1)

κ

-Minkowski spacetime (see[55]).Again,thecosmologicalconstantwillappearfor higher-orders within the full commutation rules. We recall that in the (2

+

1)casethetwistguaranteesthecompatibilitywiththeChern– Simonsapproachtogravity[33].Workonalltheseproblemsisin progressandwillbepresentedelsewhere.

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