Decoupling Braided Tensor Factors

a rX iv:mat h /12199v1[mat h.QA ]2Dec2Decoupling Braided Tensor Factors ∗Gaetano Fiore,1,2Harold Steinacker 3,Julius Wess 3,41Dip.di Matematica e Applicazioni,Fac.di Ingegneria Universit`a di Napoli,V.Claudio 21,80125Napoli 2I.N.F.N.,Sezione di Napoli,Complesso MSA,V.Cintia,80126Napoli 3Sektion Physik,Ludwig-Maximilian Universit¨a t,Theresienstraße 37,D-80333M¨u nchen 4Max-Planck-Institut f¨u r Physik F¨o hringer Ring 6,D-80805M¨u nchen Abstract We briefly report on our result [9]that the braided tensor product algebra of two module algebras A 1,A 2of a quasitriangular Hopf al-gebra H is equal to the ordinary tensor product algebra of A 1with a subalgebra isomorphic to A 2and commuting with A 1,provided thereexists a realization of H within A 1.As applications of the theorem we consider the braided tensor product algebras of two or more quantum group covariant quantum spaces or deformed Heisenberg algebras.1Introduction and main theoremAs is well known,given two associative unital algebras A1,A2(over thefield C,say),there is an obvious way to build a new algebra A which is as a vector space the tensor product A=A1⊗C A2of the two vector spaces(over the samefield)and has a product law such that A1⊗1and1⊗A2are subalgebrasisomorphic to A1and A2respectively:one just completes the product law by postulating the trivial commutation relations(1⊗a2)(a1⊗1)=(a1⊗1)(1⊗a2)(1) for any a1∈A1,a2∈A2.The resulting algebra is the ordinary tensor product algebra.With a standard abuse of notation we shall denote in the sequel a1⊗a2by a1a2for any a1∈A1,a2∈A2;consequently(1)becomesa2a1=a1a2.(2) If A1,A2are module algebras of a Lie algebra g,and we require A to be too,then(2)has no alternative,because any g∈g acts as a derivation on the (algebra as well as tensor)product of any two elements,or,in Hopf algebra language,because the coproduct∆(g)=g(1)⊗g(2)(at the rhs we have used Sweedler notation)of the Hopf algebra H≡U g is cocommutative.In this paper we shall work with right-module algebras(instead of left ones),and denote by⊳:(a i,g)∈A i×H→a i⊳g∈A i the right action;the reason is that they are equivalent to left comodule algebras,which are used in much of the literature.In Ref.[9]we give also the corresponding formulae for the left module algebras.We recall that a right action⊳:(a,g)∈A×H→a⊳g∈A by definition fulfillsa⊳(gg′)=(a⊳g)⊳g′,(3)(aa′)⊳g=(a⊳g(1))(a′⊳g(2)).(4) If we take as Hopf algebra H a quasitriangular noncocommutative one like the quantum group U q g,as A i some H-module algebras,and we require A to be a H-module algebra too,then(2)has to be replaced by one of the formulaea2a1=(a1⊳R(1))(a2⊳R(2)),(5)a2a1=(a1⊳R−1(2))(a2⊳R−1(1)).(6)2This yields instead of A two different braided tensor product algebras[10,11], which we shall call A+=A1⊗−A2respectively.Here R≡R(1)⊗R(2)∈H+⊗H−denotes the so-called universal R-matrix of H≡[6],R−1its inverse,and H±denote the Hopf positive and negative Borel subalgebras of H.If in particular H is triangular,then R−1=R21, A+=A−,and one has just one braided tensor product algebra.In any case, both A+and A−go to the ordinary tensor product algebra A in the limit q→1,because in this limit R→1⊗1.The braided tensor product is a particular example of a more general no-tion,that of a crossed(or twisted)tensor product[1]of two unital associative algebras.In view of(5)or(6)studying representations of A±is a more difficult task than just studying the representations of A1,A2and taking their tensor products.The degrees of freedom of A1,A2are so to say“coupled”.One might ask whether one can“decouple”them by a transformation of genera-tors.As shown in Ref.[9],the answer is positive if there respectively exists an algebra homomorphismϕ+1or an algebra homomorphismϕ−1ϕ±1:A1>⊳H±→A1(7) acting as the identity on A1,namely for any a1∈A1ϕ±1(a1)=a1.(8) (Here A1>⊳H±denotes the cross product between A1and H±).In other words,this amounts to assuming thatϕ+1(H+)[resp.ϕ−1(H−)]provides a realization of H+(resp.H−)within A1.In this report we summarize the main results of Ref.[9].The basic one isTheorem1[9].Let{H,R}be a quasitriangular Hopf algebra and H+,H−be Hopf subalgebras of H such that R∈H+⊗H−.Let A1,A2be respectively a H+-and a H−-module algebra,so that we can define A+as in(5),and ϕ+1be a homomorphism of the type(7),(8),so that we can define the map χ+:A2→A+byχ+(a2):=ϕ+1(R(1))(a2⊳R(2)).(9) Alternatively,let A1,A2be respectively a H−-and a H+-module algebra,so that we can define A−as in(6),andϕ−1be a homomorphism of the type(7), (8),so that we can define the mapχ−:A2→A−byχ−(a2):=ϕ−1(R−1(2))(a2⊳R−1(1)).(10)3In either caseχ±are then injective algebra homomorphisms and[χ±(a2),A1]=0,(11) namely the subalgebras˜A±2:=χ±(A2)≈A2commute with A1.Moreover A±=A1⊗˜A±2.The last equality means that A±are respectively equal to the ordinary tensor product algebra of A1with the subalgebras˜A±2⊂A±,which are isomorphic to A2!χ+,χ−will be called”unbraiding”maps.We recall the content of the hypotheses stated in the theorem.The alge-bra A1>⊳H±as a vector space is the tensor product A1⊗C H±,as an algebra it has subalgebras A1⊗1,1⊗H and has cross commutation relationsa1g=g(1)(a1⊳g(2)),(12) for any a1∈A1and g∈H±.ϕ±1being an algebra homomorphism means that for anyξ,ξ′∈A1>⊳H±ϕ±1(ξξ′)=ϕ±1(ξ)ϕ±1(ξ′).Applyingϕ±1to both sides of(12)wefind aϕ±(g)=ϕ±(g(1))(a⊳g(2)).Of course,we can use the above theorem iteratively to completely unbraid the braided tensor product algebra of an arbitrary number M of copies of A1.We end up withCorollary1If A1is a(right-)module algebra of the Hopf algebra H and there exists an algebra homomorphismϕ+1of the type(7),(8),then there is an algebra isomorphism.A1⊗+A1 M times ≈A1⊗...⊗A1M times.(13)An analogous claim holds for the second braided tensor product if there exists a mapϕ−1.2The unbraiding under the∗-structuresA+(as well as A−)is a∗-algebra if H is a Hopf∗-algebra,A1,A2are H-module∗-algebras(we shall use the same symbol∗for the∗-structure on all algebras H,A1,etc.),andR∗=R−1(14)4(here R∗means R(1)∗⊗R(2)∗).In the quantum group case(14)requires |q|=1.Under the same assumptions also A1>⊳H is a∗-algebra.Ifϕ±1exist settingϕ′1±:=∗◦ϕ±1◦∗we realize that alsoϕ′1±are algebra homomorphisms of the type(7),(8).If such homomorphisms are uniquely determined,we conclude thatϕ1±are∗-homomorphisms.More generally,one may be able to chooseϕ1±as∗-homomorphisms.How do the correspondingχ±behave under∗?Proposition1[9].Assume that the conditions of Theorem1for defin-ingχ+(resp.χ−)are fulfilled.If R∗=R−1andϕ+1(resp.ϕ−1)is a ∗-homomorphism thenχ+(resp.χ−)is,too.Consequently,A1,˜A±2are closed under∗.3ApplicationsIn this section we illustrate the application of Theorem1and Corollary1 to some algebras H,A i for which homomorphismsϕ±1are known.H will be the quantum group U q sl(N)or U q so(N),and A1is the U q sl(N)-or U q so(N)-covariant Heisenberg algebra(Section3.1.),the U q so(N)-covariant quantum space/sphere(Section3.2.).In Ref.[9]we have treated also the U q so(3)-covariant q-fuzzy sphere.As generators of H it will be convenient in either case to use the Faddeev-Reshetikhin-Takhtadjan(FRT)generators[7]L+a l∈H+and L−a l∈H−.They are related to R byL+a l:=R(1)ρa l(R(2))L−a l:=ρa l(R−1(1))R−1(2),(15) whereρa l(g)denote the matrix elements of g∈U q g in the fundamental N-dimensional representationρof U q g.In fact they provide,together with the square roots of the elements L±i i,a(overcomplete)set of generators of U q g.3.1.Unbraiding‘chains’of braided Heisenberg alge-brasIn this subsection we consider the braided tensor product of M≥2copies of the U q g-covariant deformed Heisenberg algebras Dǫ,g,g=sl(N),so(N). Such algebras have been introduced in Ref.[13,14,2].They are unital associative algebras generated by x i,∂j fulfilling the relationsP a ij hk x h x k=0,P a ij hk∂j∂i=0,∂i x j=δi j+(qγˆR)ǫjk ih x h∂k,(16)5whereγ=q12 denotes the rank of so(N).We shall enumerate the different copies of Dǫ,g by attachingto them an additional Greek index,e.g.α=1,2,...,M.The prescription(6)gives the following“cross”commutation relations between their respective generators(α<β).xα,i xβ,j=ˆR ijhkxβ,h xα,k,∂α,i∂β,j=ˆR kh ji∂β,h∂α,k,∂α,i xβ,j=ˆR−1jhik xβ,k∂α,h,∂β,i xα,j=ˆR jhikxα,k∂β,h.(18)Algebra homomorphismsϕ1:A1>⊳H→A1,for H=U q g and A1equal to(a suitable completion of)Dǫ,g have been constructed in Ref.[8,5].This is the q-analog of the well-known fact that the elements of g can be realized as“vectorfields”(first order differential operators)on the corresponding g-covariant(undeformed)space,e.g.ϕ1(E i j)=x i∂j−1ϕ1(L−i j)in terms of x1,i,∂1,a for U q sl(2),U q so(3)has been given in Ref.[9]. For different values of N it can be found from the results of Ref.[8,5]by passing from the generators adopted there to the FRT generators.By completely analogous arguments one determines the alternative un-braiding procedure for the braided tensor product stemming from prescrip-tion(5).A1>⊳H is a∗-algebra and the mapϕ1is a∗-homomorphism both for q real and|q|=1.Butϕ±1are∗-homomorphisms only for|q|=1.In the latter case the∗-structure of A1is(x i)∗=x i,(∂i)∗=−q±N g kh g ki∂h(22) Applying Proposition1in the latter case wefind that∗maps A1as well as each of the commuting subalgebras˜A±i into itself.3.2.Unbraiding‘chains’of braided quantum Euclideanspaces or spheresIn this section we consider the braided tensor product of M≥2copies of the quantum Euclidean space R N q[7](the U q so(N)-covariant quantum space),i.e.of the unital associative algebra generated by x i fulfilling the relations(16)1,or of the quotient space of R N q obtained by setting r2:=x i x i=1[the quantum(N−1)-dimensional sphere S N−1q ].(Thus,these willbe subalgebras of the Heisenberg algebras D+,so(N),D+,so(N)considered in the previous subsection).Again,the multiplet(x i)carries the fundamental N-dimensional representationρof U q so(N).As before,we shall enumerate the different copies of the quantum Euclidean space or sphere by attaching an additional Greek index to them,e.g.α=1,2,...,M.The prescription(6) gives the cross commutation relations(18)1.According to Ref.[3],to defineϕ±1(for q=1)one actually needs a slightlyenlarged version of R N q(or S N−1q ).One has to introduce some new generators√2,together with their inverses(√2 ,r2n coincides with r2,whereas for odd N r20=(x0)2,so we are adding also(x0)−1as a new generator).In fact,the com-mutation relations involving these new generators can befixed consistently,7and turn out to be simply q-commutation relations.r plays the role of‘de-formed Euclidean distance’of the generic‘point’(x i)of R N q from the‘origin’; r a is the‘projection’of r on the‘subspace’x i=0,|i|>a.In the previous equation g hk denotes the‘metric matrix’of SO q(N),g ij=g ij=q−ρiδi,−j, which is a SO q(N)-isotropic tensor and a deformation of the ordinary Eu-clidean metric.Here,(ρi):=(n−12,0,−12−n)for N odd,(ρi):=(n−1,...,0,0,...,1−n)for N even.g ij is related to the traceprojector appearing in(28)by P t ijkl =(g sm g sm)−1g ij g kl.The extension ofthe action of H to these extra generators is uniquely determined by the con-straints the latter fulfil.In the case of even N one needs to include also the FRT generators L+11,L−11(which are generators of H)among the generators of A1.In appendix3.2.we recall the explicit form ofϕ±1in the present case. Note that the mapsϕ±1have no analog in the“undeformed”case(q=1), because A1is abelian,whereas H is not.The unbraiding procedure is recursive.Thefirst step consists of using the homomorphismϕ±1found in Ref.[3]to unbraid thefirst copy from the others.Following Theorem1,we perform the change of generators(19)1, (20)in A−.In view of formula(29)we thusfindy1,i:=x1,i,yα,i:=g ih[µ1h,x1,k]q g kj xα,j,α>1.(24) The suffix1inµ1a means that the special elementsµa defined in(30)must be taken as elements of thefirst copy.In view of(30)we see that g ih[µ1h,x1,k]q g kj are rather simple polynomials in x i and r−1a,homogeneous of total degree1 in the coordinates x i and r ing the results given in the appendix we give now the explicit expression of(24)2for N=3:yα,−=−qhγ1rq(q+1)1q(q+1)hγ1rx+xα,0−1q−1/√The alternative unbraiding procedure for the braided tensor product al-gebra stemming from prescription(5)arises by iterating the change of gen-eratorsy′M,i:=x M,i y′α,i:=ϕM(L+i j)xα,j=g ih[¯µM h,x M,k]q−1g kj xα,j,(26)α<M.The special elements¯µa are defined in(30),and suffix M meansthat we must take¯µa as an element of the M-th copy of R N q(or S N−1q ).y M,i≡x M,i commutes with y1,i,...,y M−1,i.When|q|=1,by a suitable choice(32)ofγ1,¯γ1,as well as of the other free parametersγa,¯γa appearing in the definitions ofϕ±for N>3,one can makeϕ±into∗-homomorphisms.Applying Proposition1in the latter case wefind that∗maps A1as well as each of the commuting subalgebras˜A±i into itself.The braid matrixˆR is related to R byˆR ijhk≡R ji hk:=(ρj h⊗ρi k)R.With the indices’convention described in sections3.2.,3.1.ˆR is given byˆR=q−1AB −xBA .The images of ϕ−(resp.ϕ+)on the negative (resp.positive)FRT generators readϕ−(L −i j )=g ih [µh ,x k ]q g kj ,ϕ+(L +i j )=g ih [¯µh ,x k ]q −1g kj ,(29)whereµ0=γ0(x 0)−1¯µ0=¯γ0(x 0)−1for N odd,µ±1=γ±1(x ±1)−1L ∓11¯µ±1=¯γ±1(x ±1)−1L ±11for N even,µa =γa r −1|a |r −1|a |−1x −a ¯µa =¯γa r −1|a |r −1|a |−1x −a otherwise,(30)and γa ,¯γa ∈C are normalization constants fulfilling the conditions γ0=−q −12h −1for N odd,γ1γ−1= −q −1h −2k −2¯γ1¯γ−1= −qh −2k −2for N odd,for N even,γa γ−a =−q −1k −2ωa ωa −1¯γa ¯γ−a =−qk −2ωa ωa −1for a >1.(31)Here k :=q −1/q ,ωa :=(q ρa +q −ρa ).Incidentally,for odd N one can choose the free parameters γa ,¯γa in such a way that ϕ+,ϕ−can be ‘glued’into an algebra homomorphism ϕ:R N q >⊳U q so(N )→R N q [3].When |q |=1,the ∗-structure is given by (x i )∗=x i [see (22)].It turns out that ϕ±are ∗-homomorphisms if,in addition,γ∗±1=−γ±1if N even γ∗a =−γa 1if a <0q −2if a >0otherwise.(32)References[1]A.Van Daele and S.Van Keer,Compositio Mathematica 91,201(1994).A.Borowiec,W.Marcinek,“On crossed product of algebras”,math-ph/0007031,and references therein;“Hopf modules and their duals”,math.QA/0007151.[2]U.Carow-Watamura,M.Schlieker,S.Watamura,Z.Physik C 49(1991)439.10[3]B.L.Cerchiai,G.Fiore,J.Madore,“Geometrical Tools forQuantum Euclidean Spaces”,to appear in Commun.Math.Phys., math.QA/0002007[4]B.L.Cerchiai,J.Madore,S.Schraml,J.Wess,Eur.Phys.J.C16(2000),169-180.[5]C.-S.Chu,B.Zumino,“Realization of vectorfields fro quantum groupsas pseudodifferential operators on quantum spaces”,Proc.XX Int.Conf.on Group Theory Methods in Physics,Toyonaka(Japan),1995,and q-alg/9502005.[6]V.Drinfeld,“Quantum groups,”in I.C.M.Proceedings,Berkeley,(1986)p.798.[7]L.D.Faddeev,N.Y.Reshetikhin,L.Takhtadjan,Leningrad Math.J.1(1990),193.[8]G.Fiore,Commun.Math.Phys.169(1995),475-500.[9]G.Fiore,H.Steinacker,J.Wess“Unbraiding the braided tensor prod-uct”,Preprint00-30Dip.Matematica e Applicazioni,Universit`a di Napoli,math/0007174.[10]A.Joyal,R.Streat,Braided Monoidal Categories,Mathematics Reports86008,Macquarie University,1986.[11]S.Majid,Int.J.Mod.Phys.A5,1(1990);J.Algebra130,17(1990);Lett.Math.Phys.22,167(1991);J.Algebra163,191(1994).For a review:S.Majid,Foundations of Quantum Groups,Cambridge Univ.Press(1995);and references therein.[12]O.Ogievetsky,Lett.Math.Phys.24(1992),245.[13]W.Pusz,S.L.Woronowicz,Rep.Math.Phys.27(1989),231.[14]J.Wess,B.Zumino,Nucl.Phys.(Proc.Suppl.)18B(1990)302.11。