Quantum seaweed algebras and quantization of affine Cremmer-Gervais r-matrices

a r X i v :m a t h /0605236v 1 [m a t h .Q A ] 9 M a y 2006Quantum seaweed algebras and quantization ofaffine Cremmer–Gervais r -matricesM.E.Samsonov 1),A.A.Stolin 2)and V.N.Tolstoy 3)1)St.Petersburg State University,Institute of Physics,198504St.Petersburg,Russia;e-mail:samsonov@pink.phys.spbu.ru 2)Department of Mathematics,University of G¨o teborg,SE-41296G¨o teborg,Sweden;e-mail:astolin@math.chalmers.se 3)Institute of Nuclear Physics,Moscow State University ,119992Moscow,Russia;e-mail:tolstoy@nucl-th.sinp.msu.ru Abstract We propose a method of quantization of certain Lie bialgebra structures on the poly-nomial Lie algebras related to quasi-trigonometric solutions of the classical Yang–Baxter equation.The method is based on an affine realization of certain seaweed algebras and their quantum analogues.We also propose a method of ω-affinization,which enables us to quantize rational r -matrices of sl (3).1Introduction The aim of this paper is to propose a method of quantizating certain Lie bialgebras structures on polynomial Lie algebras.In the beginning of 90-th Drinfeld in [6]posed the following problem:can any Lie bialgebra be quantized?The problem was solved by Etingof and Kazhdan and the answer was positive.However,another problem of finding explicit quantization formulas remains open.First results in this direction were obtained in [15],[7]and [9],where the authorsquantized the so-called Belavin–Drinfeld list,the list of all quasi-triangular Lie bialgebra structures on finite dimensional simple Lie algebras.It should be noticed that first infinite-dimensional cases were considered in [11].However,a real break-through in the infinite dimensional case came in [12]and [17],where deformed versions of Yangians Y (sl N )and quantum affine algebras U q (ˆsl N )were constructed for N =2,3.In the present paper we solve the problem in the case U q (ˆsl N ).Our solution of the problem is based on a q -version of the so-called seaweed algebras.A seaweed subalgebra of sl N is an intersection of two parabolic subalgebras one of which containing the Borel subalgebra B +and another B −.The case when both parabolic subalgebras are maximal was studied in [19]in connection with the study of rational solutions of the classical Yang-Baxter equation.In particular,a complete answer to the question when such an algebra is Frobenius was obtained ter in [4]it was found out when an arbitrary seaweed algebra is Frobenius.In the present paper we quantize a Lie bialgebra,which as a Lie algebra is gl N[u] (sl N[u]).Its coalgebra structure is defined by a quasi-trigonometric solution of the classical Yang-Baxter equation.Quasi-trigonometric solutions were introduced in[10].We remind briefly some results obtained there.For convenience we consider the case g=gl N although the results are also valid for an arbitrary simple complex Lie algebra g.Let e ij,i,j=1,...,N,be the standard Cartan-Weyl basis of gl N:[e ij,e kl]=δjk e il−δil e kj.The element C2:=Ni,j=1e ij e ji∈U(gl N)isa gl N-scalar,i.e.[C2,x]=0for any x∈gl N,and it is called the second order Casimir element.The elementΩ:=12 1≤i≤N e ii⊗e ii+1≤i<j≤Ne ij⊗e ji andΩ−=1u−v+p(u,v),(1)where p(u,v)is a non-zero polynomial with coefficients in gl N⊗gl N.If p(u,v)=0 then X(u,v)is the simplest(standard)trigonometric r-matrix.Any quasi-trigonometric solution of the classical Yang-Baxter equation defines a Lie bialgebra structure on gl N[u] and the corresponding Lie cobracket on gl N[u]is given by the formula{A(u)∈gl N[u]}→{[X(u,v),A(u)⊗1+1⊗A(v)]∈gl N[u]⊗gl N[v]}.(2)It was proved in[10]that for g=sl N(gl N)there is a one-to-one correspondence between quasi-trigonometric r-matrices and Lagrangian subalgebras of g⊕g transversal to a certain Lagrangian subalgebra of g⊕g defined by a maximal parabolic subalgebra of g.Here we mean the Lagrangian space with respect to the following symmetric non-degenerate invariant bilinear form on g⊕g:Q((a,b),(c,d))=K(a,c)−K(b,d),(3)where K is the Killing form and a,b,c,d∈g.In their famous paper on the classical Yang–Baxter equation,Belavin and Drinfeld listed all the trigonometric r-matrices.In case sl3there exist4trigonometric r-matrices. Two of them relate to the quasi-triangular constant r-matrices and can be quantized using methods from[7],[9]and[15].In our paper we explain how to to quantize one of the two remaining trigonometric r-matrices found by Belavin and Drinfeld.We also notice that in cases sl2and sl3our methods lead to quantization of some rational r-matrices found in[19].As it was explained in[18],quantization of the rational r-matrix for sl2has close relations with the Rankin–Cohen brackets for modular forms(see[1]).Our method is based onfinding quantum twists,which are various solutions of the so-called cocycle equation for a number of Hopf algebras:F12(∆⊗id)(F)=F23(id⊗∆)(F).(4)We are thankful to S.M.Khoroshkin and I.Pop for fruitful discussions and suggestions.2A quantum seaweed algebra and its affine real-izationIt turns out that it is more convenient to use instead of the simple Lie algebra sl N its central extension gl N.The polynomial affine Lie algebra gl N[u]is generated by Cartan–Weyl basis e(n)ij:=e ij u n(i,j=1,2,...N,n=0,1,2,...)with the defining relations[e(n)ij,e(m)kl ]=δjk e(n+m)il−δil e(n+m)kj.(5)The total root systemΣof the Lie algebra gl N[u]with respect to an extended Cartan subalgebra generated by the Cartan elements e ii(i=1,2,...,N)and d=u(∂/∂u)is given byΣ(gl N[u])={ǫi−ǫj,nδ+ǫi−ǫj,nδ|i=j;i,j=1,2,...,N;n=1,2,...},(6) whereǫi(i=1,2,...,N)is an orthonormal basis of a N-dimensional Euclidean space R N: (ǫi,ǫj)=δij,and we have following correspondence between root vectors and generators of gl N[u]:e(n)ij=e nδ+ǫi−ǫjfor i=j,n=0,1,2,....It should be noted that the multiplicity of the roots nδis equal to N,Mult(nδ)=N,and the root vectors e(n)ii(i=1,2,...,N)”split”this multiplicity.We choose the following system of positive simple roots:Π(gl N[u])={αi:=ǫi−ǫi+1,α0:=δ+ǫN−ǫ1|i=1,2,...N−1}.(7) Let sw N+1be a subalgebra of gl N+1generated by the root vectors:e21,e i,i+1,e i+1,i for i=2,3,...,N and e N,N+1,and also by the Cartan elements:e11+e22,e ii for i= 2,3,...,N.It is easy to check that sw N+1has structure of a seaweed Lie algebra(see [4]).Letˆsw N be a subalgebra of gl N[u]generated by the root vectors:e(0)21,e(0)i,i+1,e(0)i+1,i for i=2,3,...,N and e(1)N,1,and also by the Cartan elements:e(0)ii for i=1,2,3,...,N.It is easy to check that the Lie algebrasˆsw N and sw N+1are isomorphic.This isomorphism is described by the following correspondence:e i+1,i↔e(0)i+1,i for i=1,2,...,N−1, e i,i+1↔e(0)i,i+1for i=2,3,...,N−1,e N,N+1↔e(1)N,1for i=2,3,...,N−1,and(e11+ e N+1,N+1)↔e(0)11,e ii↔e(0)ii for i=2,3,...,N.We will callˆsw N an affine realization of sw N+1.Now let us consider q-analogs of the previous Lie algebras.The quantum algebra U q(gl N)is generated by the Chevalley elements1e i,i+1,e i+1,i(i=1,2,...,N−1),q±e ii(i=1,2,...,N)with the defining relations:q e ii q−e ii=q−e ii q e ii=1,q e ii q e jj=q e jj q e ii,q e ii e jk q−e ii=qδij−δik e jk(|j−k|=1),e ii−e i+1,i+1−q e i+1,i+1−e ii[e i,i+1,e j+1,j]=δij qUsing these explicit constructions and the defining relations(8)for the Chevalley basis it is not hard to calculate the following relations between the Cartan–Weyl generators e ij (i,j=1,2,...,N):q e kk e ij q−e kk=qδki−δkj e ij(1≤i,j,k≤N),(16)q e ii−e jj−q e jj−e ii[e ij,e ji]=and e N,N+1,and also by the q-Cartan elements:q e11+e N+1,N+1,q e ii for i=2,3,...,N with the relations satisfying(8).Similarly,the quantum algebra U q(ˆsw N)is generated by the root vectors:e(0)21,e(0)i,i+1,e(0)i+1,i for i=2,3,...,N and e(1)N,1,and also by the q-Cartan elements:q e(0)ii for i=1,2,3,...,N with the relations satisfying(8)and(26).It is clear that the algebras U q(gl N+1)and U q(ˆsw N)are isomorphic as associative algebras but they are not isomorphic as Hopf algebras.However if we introduced a new coproduct in the Hopf algebra U q(gl N+1)∆(F1,N+1)q(x)=F1,N+1∆q(x)F−11,N+1(∀x∈U q(gl N+1)),(29) whereF1,N+1:=q−e11⊗e N+1,N+1,(30) we obtain an isomorphism of Hopf algebrasU(F1,N+1)q(sw N+1)≃U q(ˆsw N).(31)Here the symbol U(F1,N+1)q(sw N+1)denotes the quantum seaweed algebra U q(sw N+1)with the twisted coproduct(29).3Cartan part of Cremmer-Gervais r-matrixFirst of all we recall classification of quasi-triangular r-matrices for a simple Lie algebra g.The quasi-triangular r-matrices are solutions of the systemr12+r21=Ω,(32)[r12,r13]+[r12,r23]+[r13,r23]=0,whereΩis the quadratic the Casimir two-tensor in g⊗g.Belavin and Drinfeld proved that any solution of this system is defined by a triple(Γ1,Γ2,τ),whereΓ1,Γ2are subdiagrams of the Dynkin diagram of g andτis an isometry between these two subdiagrams.Further, eachΓi defines a reductive subalgebra of g,andτis extended to an isometry(with respect to the corresponding restrictions of the Killing form)between the corresponding reductive subalgebras of g.The following property ofτshould be satisfied:τk(α)∈Γ1for any α∈Γ1and some k.LetΩ0be the Cartan part ofΩ.Then one can construct a quasi-triangular r-matrix according to the followingTheorem1(Belavin–Drinfeld[2]).Let r0∈h⊗h satisfies the systemsr120+r210=Ω0,(33)(α⊗1+1⊗α)(r0)=hα(34)(τ(α)⊗1+1⊗α)(r0)=0(35) for anyα∈Γ1.Then the tensorr=r0+ α>0e−α⊗eα+ α>0;k≥1e−α∧eτk(α)(36)satisfies(32).Moreover,any solution of the system(32)is of the above form,for a suitable triangular decomposition of g and suitable choice of a basis{eα}.In what follows,for aim of quantization of algebra structures on the polynomial Lie algebra gl N [u ])we will use the twisted two-tensor q r 0(N )where r 0(N )is the Cartan part of the Cremmer–Gervais r -matrix for the Lie algebra gl N when Γ1={α1,α2,...,αN −2}Γ2={α2,α3,...,αN −1}and τ(αi )=αi +1.An explicit form of r 0(N )is defined by the following proposition (see [5]).Proposition1.TheCartanpartoftheCremmer–Gervaisr -matrix for gl N is given by the following expressionr 0(gl N )=12N e ii ∧e jj .(37)It is easy to check that the Cartan part (37),r 0(N ):=r 0(gl N ),satisfies the conditions (ǫk ⊗id +id ⊗ǫk )(r 0(N ))=e kk for k =1,2,...,N,(38)(ǫk ⊗id +id ⊗ǫk ′)(r 0(N ))=(k −k ′)C 1(N )−k −1 i =k ′+1e ii for 1≤k ′<k ≤N,(39)where C 1(N )is the normalized central element:C 1(N ):=1q −q −1.(45)It is not hard to check that the Chevalley elements e′i,i+1and e′i+1,i have the following coproducts after twisting by the two-tensor q r0(N):q r0(N)∆q(e′i,i+1)q−r0(N)=e′i,i+1⊗q2((ǫi−ǫi+1)⊗id)(r0(N))+1⊗e′i,i+1=e′i,i+1⊗q2e ii−2C1(N)+1⊗e′i,i+1,(46)q r0(N)∆q(e′i+1,i)q−r0(N)=e′i+1,i⊗1+q−2(id⊗(ǫi+1−ǫi))(r0(N))⊗e′i+1,i=e′i+1,i⊗1+q2e i+1,i+1−2C1(N)⊗e′i+1,i.(47) for1≤i<N.Since the quantum algebra U q(gl N)is a subalgebra of the quantum affinealgebra U q(gl N[u])let us introduce the new affine root vector e′(1)N1in accordance with(43):e′(1)N1=e N1q((ǫN−ǫ1)⊗id)(r0(N))=e N1q e NN−C1(N).(48)The coproduct of this element after twisting by the two-tensor q r0(N)has the formq r0(N)∆q(e′(1)N1)q−r0(N)=e′(1)N1⊗q2((ǫ1−ǫN)⊗id)(r0(N))+1⊗e′(1)N1=e′(1)N1⊗q2e NN−2C1(N)+1⊗e′(1)N1.(49)Consider the quantum seaweed algebra U q(sw N+1)after twisting by the two-tensor q r0(N+1).Its new Cartan–Weyl basis and the coproduct for the Chevalley generators are given by formulas(43),(44)and(46),(47),where N should be replaced by N+1and where i=1in(43)and(46),and j=N in(44),and i=N in(47).In particular,for the element e′N,N+1we havee′N,N+1=e N,N+1q e NN−C1(N+1).(50)q r0(N+1)∆q(e′N,N+1)q−r0(N+1)=e′N,N+1⊗q2e NN−2C1(N+1)+1⊗e′N,N+1.(51)Comparing the Hopf structure of the quantum seaweed algebra U q(sw N+1)after twisting by the two-tensor q r0(N+1)and its affine realization U q(ˆsw N)after twisting by the two-tensor q r0(N)we see that these algebras are isomorphic as Hopf algebras:q r0(N+1)∆q(U q(sw N+1))q−r0(N+1)≃q r0(N)∆q(U q(ˆsw N))q−r0(N).(52) In terms of new Cartan–Weyl bases this isomorphism,”ı”,is arranged as followsı(e′ij)=e′(0)ijfor2≤i<j≤N,(53)ı(e′ji)=e′(0)jifor1≤i<j≤N−1,(54)ı(e ii−C1(N+1))=e(0)ii−C1(N),for2≤i≤N,(55)ı(e′iN+1)=e′(1)i1=e(1)i1q((ǫi−ǫ1)⊗id)(r0(N))=e(1)i1qNk=iekk−(N+1−i)C1(N)(56)for2≤i≤N.Where the affine root vectors e(1)i1(2≤i<N)are defined by the formula (cf.14):e(1)i1=[e(0)iN,e(1)N1]q−1.(57)4Affine realization of a Cremmer-Gervais twist For construction of a twisting two-tensor corresponding to the Cremmer-Gervais r-matrix (37)we will follow to the papers[7,9].Let R be a universal R-matrix of the quantum algebra U q(gl N+1).According to[13] it has the following formR=R·K(58) where the factor K is a q-power of Cartan elements(see[13])and we do not need its explicit form.The factor R depends on the root vectors and it is given by the following formulaR=R12(R13R23)(R14R24R34)···(R1,N+1R2,N+1···R N,N+1)=↑N+1j=2↑j−1i=1R ij ,(59)whereR ij=exp q−2((q−q−1)e ij⊗e ji),(60) exp q(x):= n≥0x nN +1.From the results of the previous section it follows that we can immediately obtain an affine realization ˆF CG which twists the quantum affine algebra U q (gl N [u ]):ˆF CG =ˆF ·qr 0(N ),(68)ˆF:=(ı⊗ı)(F )=ˆR ′(N −1)ˆR ′(N −2)···ˆR ′(1)(69)ˆR′(k )=↑N +1−k j =2 ↑j −1 i =1ˆR ′(k )ij .(70)whereˆR ′(k )ij=exp q −2 (q −q −1)e ′(0)i +k,j +k ⊗e ′(0)ji for 1≤i <j ≤N −k ,(71)ˆR ′(k )i,N +1−k =exp q −2 (q −q −1)e ′(1)N,i +k ⊗e ′(0)N +1−k,ifor 1≤i <N +1−k .(72)5From constant to affine twists5.1Affine r -matrices of Cremmer-Gervais type and their quantizationIn this subsection we are going to describe two classes of the quasi-trigonometric r -matrices of Cremmer-Gervais type,which we call affine r -matrices of the Cremmer-Gervais type.We need some results proved in [10]for the case sl N .Let us consider the Lie algebra sl N ⊕sl N with the form Q ((a,b ),(c,d ))=K (a,c )−K (b,d ),where K is the Killing form on sl N .Let P i (resp.P −i )be the maximal parabolic subalgebra containing all positive (resp.negative)roots and not containing all negative roots which have the simple root αi in their decomposition.Assume we have two parabolic (or maybe sl N )subalgebras S 1,S 2of sl N such that their reductive parts R 1,R 2are isomorphic and let T :R 1→R 2be an isomorphism,which is an isometry with respect to the reductions of the Killing form K onto R 1and R 2.Then the triple (S 1,S 2,T )defines a Lagrangian subalgebra of sl N ⊕sl N (with respect to the form Q ).It was proved in [10]that any quasi-trigonometric solution is defined by a Lagrangian subalgebra W ⊂sl N ⊕sl N such that W is transversal to (P i ,P i ,id).Now we would like to present two types of W ,which define quasi-trigonometric r -matrices for sl N with i =N −1.Type 1,which we call affine Cremmer–Gervais I classical r -matrix :In this case S 1=S 2=sl N ,T =Ad(σN ),where σN is the permutation matrix,which represents the cycle {1,2,...,N }→{2,3,...,N,1}.Proposition 2.The triple (S 1,S 2,T )above defines a quasi-trigonometric r -matrix.Type 2,which we call affine Cremmer–Gervais II classical r -matrix :In this case S 3=P −1,S 4=P −N −1,T ′(e ij )=e i −1,j −1,where e ij are the matrix units.Proposition 3.The triple (S 3,S 4,T ′)above defines a quasi-trigonometric r -matrix.Both propositions can be proved directly.Theorem 2.Let ˆF ′CG be the twist (66)reduced to U q (ˆsl N ),and let ˆR be the universal R -matrix for U q (ˆsl N ).Then the R -matrix ˆF ′21CG ˆR ˆF ′−1CG quantizes the quasi-trigonometric r -matrix obtained in Proposition 2.Now we turn to quantization of the quasi-trigonometric r-matrix from Proposition 3.In order to do this we propose another method for construction of affine twists from constant ones.Let{α0,α1,...,αN−1}be the vertices of the Dynkin diagram of sl N.Then the mapτ:{α1,...,αN−1}to{α2,...,αN−1,α0}defines an embedding of Hopf algebras U q(sl N)֒→U q(ˆsl N).Abusing notations we denote this embedding byτ.The mapτsends each twist F of U q(sl N)to an affine twist(τ⊗τ)(F).Let us denote(τ⊗τ)(F′CG)byˆF(τ)CG. Theorem3.LetˆR′be the universal R-matrix for U q(ˆsl N).Then the R-matrixˆF(τ)21 CG ˆR′(ˆF(τ)CG)−1quantizes the quasi-trigonometric r-matrix obtained in Proposition3.Both theorems can be proved straightforward.5.2ω-affinization and quantization of rational r-matrices Aim of this section is to quantize certain rational r-matrices.Of course,we would like to use rational degeneration of the affine twists constructed above.Unfortunately,we cannot do this directly because in both cases lim q→1F=1⊗1.Therefore,propose a new method which we callω-affinization.We begin with the following result.Theorem4.Letπ:U q(sl N[u])→U q(sl N)be the canonical projection sending all the affine generators to zero.LetΨ∈U q(sl N[u])⊗U q(ˆsl N)be invertible and such thatΨ=Ψ1Ψ2,(73) whereΨ2=(π⊗id)(Ψ).Further,letΨ2be a twist and let the following two relations hold:Ψ231Ψ122=Ψ122Ψ231,(π⊗id)(∆⊗id)(Ψ1)=Ψ231(74) Finally,let there existω∈U q(sl N[u])[[ζ]]such thatΨω:=(ω⊗ω)Ψ∆(ω−1)∈U q(sl N)⊗U q(ˆsl N).(75) ThenΨis a twist.Proof.Let us consider the Drinfeld associatorAssoc(Ψ)=Ψ12(∆⊗id)(Ψ)(id⊗∆)(Ψ−1)(Ψ−1)23(76) and the following one equivalent to itAssoc(Ψω)=(ω⊗ω⊗ω)(Assoc(Ψ))(ω−1⊗ω−1⊗ω−1)=Ψ12ω(∆⊗id)(Ψω)(id⊗∆)(Ψ−1ω)(Ψ−1ω)23.(77)By(75)we haveAssoc(Ψω)=(π⊗id⊗id)(Assoc(Ψω)).(78) Further,we take into account the following considerations:(π⊗id)(Ψ2)=(π⊗id)(π⊗id)(Ψ1Ψ2)=(π⊗id)(Ψ1)(π⊗id)(Ψ2),(79) what implies that(π⊗id)(Ψ1)=1⊗1.Moreover,the latter also implies that (π⊗id⊗id)(id⊗∆)(Ψ−11)=(id⊗∆)(π⊗id)(Ψ−11)=1⊗1⊗1.(80)Thus,Assoc(Ψω)==Ad(π(ω)⊗ω⊗ω)(Ψ122Ψ231(∆⊗id)(Ψ2)(id⊗∆)(Ψ−12)(Ψ−12)23(Ψ−11)23)=Ad(π(ω)⊗ω⊗ω)(Ψ231Assoc(Ψ2)(Ψ231)−1).(81)SinceΨ2is a twist we deduce that Assoc(Ψω)=1⊗1⊗1and therefore Assoc(Ψ)=1⊗1⊗1 what proves the theorem.An elementωsatifying conditions of Theorem4we will call affinizator.Corollary1.LetΨ1,Ψ2,ωsatisfy the conditions of Theorem4.ThenΨ1=(ω−1π(ω)⊗id)Ψ2 (π⊗id)∆(ω−1) ∆(ω)Ψ−12.(82) Proof.We haveΨω=(π⊗id)(Ψω)=(π(ω)⊗ω)Ψ2(π⊗id)∆(ω−1),(83) becauseΨ2=(π⊗id)(Ψ).Now we see that(ω⊗ω)(Ψ1Ψ2)∆(ω−1)=(π(ω)⊗ω)Ψ2(π⊗id)∆(ω−1),(84) what yields the required expression forΨ1.Now we would like to explain howω-affinization can be used tofind a Yangian degen-eration of the affine Cremmer–Gervais twists.Let us consider the case sl3.We setΨ2:=F(τ)CG3:=exp q2(−(q−q−1)ζˆe(0)12⊗ˆe(0)32)·ˆK3,(85)whereˆK3=q 49h12⊗h23+59h23⊗h23(86)with h ij:=e ii−e jj.The twisting two-tensor(83)belongs to U q(sl3)⊗U q(sl3)[[ζ]].The following affinizatorωlong3was constructed in[17].It is given by the followingformulaωlong 3=exp q2(ζ(1−q2)2q2h⊥βˆe(1)31)×exp q−2(ζ21−q2ˆe(0)21)exp q−2(ζ2 3(e11+e22)−23e11−1Theorem5.The elementsωlong3,Ψ2=F(τ)CG3andΨ1constructed according to Corollary1satisfy the conditions of Theorem4and consequentlyΨω=(π⊗id)(ω⊗ω)Ψ2∆(ω−1) is a twist.Proof.Straigtforward.It turns out thatΨωhas a rational degeneration.To define this rational degeneration we have to introduce the so-called f-generators:f0=(q−q−1)ˆe(0)31,f1=q2h⊥βˆe(1)31+q−1ζˆe(0)31,f2=(1−q−2)ˆe(0)32,f3=q h⊥αˆe(1)32−ζˆe(0)32.(89) Let us consider the Hopf subalgebra of UˆK3q(ˆsl3)generated by{h12,h23,f0,f1,f2,f3,ˆe(0)12,ˆe(0)21}.When q→1we obtain the following Yangian twist(see[17]):f3−ζ2h⊥β⊗e(0)21)(−h⊥β⊗1)×exp(ζ2f0)exp(−ζf1)·exp(−ζf2)×(1⊗1−ζ1⊗f2)((h⊥β−h⊥α)⊗1),(90)where the overlined generators are the generators of Y(sl3).In the evaluation represen-tation we have:f1→u e31e21→e21,F1quantizes the following classical rational r−matrix r(u,v)=Ω3(hα1⊗hα1+hα2⊗hα2)+16hα1∧hα2.(93)We haveωshort3=exp q−2(ζˆe(0)21)exp q2 −ζ1−q2ˆe(0)32 ,(94) whereˆe(0)21=q−1We have to calculateAffωshort3(q r0(3)):=(π⊗id)◦ (ωshort3⊗ωshort3)q r0(3)∆(ωshort3)−1 .(95)Using standard commutation relations between q-exponents,the formula(76)can be brought to the following form:1⊗1+ζ1⊗q2h⊥αˆe(1)31+ζq−2h⊥α⊗(Ad exp q2(ζˆe(0)21))(ˆe(0)32) (−h⊥α1⊗1)q2× 1⊗1+ζ(1−q2)1⊗ˆe(0)21 (−1F2= 1⊗1+ζ1⊗(e(0)32)−ζ2h⊥α⊗g2)(−1F2quantizes the following rational r-matrix:r(u,v)=Ω3(h12−h23)∧e21.(99)Therefore we have quantized all non-trivial 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