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 of

a?ne Cremmer–Gervais r -matrices

M.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 a?ne realization of certain seaweed algebras and their quantum analogues.We also propose a method of ω-a?nization,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 ?nding explicit quantization formulas remains open.First results in this direction were obtained in [15],[7]and [9],where the authors

quantized the so-called Belavin–Drinfeld list,the list of all quasi-triangular Lie bialgebra structures on ?nite dimensional simple Lie algebras.It should be noticed that ?rst in?nite-dimensional cases were considered in [11].However,a real break-through in the in?nite dimensional case came in [12]and [17],where deformed versions of Yangians Y (sl N )and quantum a?ne algebras U q (?sl N )were constructed for N =2,3.In the present paper we solve the problem in the case U q (?sl 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 https://www.doczj.com/doc/7d15779446.html,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 de?ned by a quasi-trigonometric solution of the classical Yang-Baxter equation.Quasi-trigonometric solutions were introduced in[10].We remind brie?y 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:=

N

i,j=1

e ij e ji∈U(gl N)is

a gl N-scalar,i.e.[C2,x]=0for any x∈gl N,and it is called the second order Casimir element.The element?:=1

2 1≤i≤N e ii?e ii+

1≤i

e ij?e ji and??=1

u?v

+p(u,v),(1)

where p(u,v)is a non-zero polynomial with coe?cients 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 de?nes 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 de?ned 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 on?nding 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 a?ne real-ization

It turns out that it is more convenient to use instead of the simple Lie algebra sl N its central extension gl N.The polynomial a?ne 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 de?ning 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 by

Σ(gl N[u])={?i??j,nδ+?i??j,nδ|i=j;i,j=1,2,...,N;n=1,2,...},(6) where?i(i=1,2,...,N)is an orthonormal basis of a N-dimensional Euclidean space R N: (?i,?j)=δij,and we have following correspondence between root vectors and generators of gl N[u]:e(n)ij=e nδ+?

i??j

for 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:=?i??i+1,α0:=δ+?N??1|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]).

Let?sw 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 algebras?sw 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 call?sw N an a?ne 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 de?ning 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 q

Using these explicit constructions and the de?ning 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(?sw 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(?sw 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) where

F1,N+1:=q?e11?e N+1,N+1,(30) we obtain an isomorphism of Hopf algebras

U(F1,N+1)

q(sw N+1)?U q(?sw 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-matrix

First of all we recall classi?cation of quasi-triangular r-matrices for a simple Lie algebra g.The quasi-triangular r-matrices are solutions of the system

r12+r21=?,

(32)

[r12,r13]+[r12,r23]+[r13,r23]=0,

where?is the quadratic the Casimir two-tensor in g?g.Belavin and Drinfeld proved that any solution of this system is de?ned by a triple(Γ1,Γ2,τ),whereΓ1,Γ2are subdiagrams of the Dynkin diagram of g andτis an isometry between these two subdiagrams.Further, eachΓi de?nes 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 satis?ed:τk(α)∈Γ1for any α∈Γ1and some k.Let?0be the Cartan part of?.Then one can construct a quasi-triangular r-matrix according to the following

Theorem1(Belavin–Drinfeld[2]).Let r0∈h?h satis?es the systems

r120+r210=?0,(33)

(α?1+1?α)(r0)=hα(34)

(τ(α)?1+1?α)(r0)=0(35) for anyα∈Γ1.Then the tensor

r=r0+ α>0e?α?eα+ α>0;k≥1e?α∧eτk(α)(36)

satis?es(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 de?ned by the following proposition (see [5]).

Proposition

1.

The

Cartan

part

of

the

Cremmer–Gervais

r -matrix for gl N is given by the following expression

r 0(gl N )=1

2N e ii ∧e jj .(37)

It is easy to check that the Cartan part (37),r 0(N ):=r 0(gl N ),satis?es the conditions (?k ?id +id ??k )(r 0(N ))=e kk for k =1,2,...,N,

(38)(?k ?id +id ??k ′)(r 0(N ))=(k ?k ′)C 1(N )?

k ?1 i =k ′+1

e ii for 1≤k ′

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((?i??i+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?(?i+1??i))(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

algebra U q(gl N[u])let us introduce the new a?ne root vector e′(1)

N1

in accordance with(43):

e′(1)

N1

=e N1q((?N??1)?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 form

q r0(N)?q(e′(1)

N1)q?r0(N)=e′(1)

N1

?q2((?1??N)?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 have

e′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 a?ne realization U q(?sw 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(?sw N))q?r0(N).(52) In terms of new Cartan–Weyl bases this isomorphism,”?”,is arranged as follows

?(e′ij)=e′(0)

ij

for2≤i

?(e′ji)=e′(0)

ji

for1≤i

?(e′iN+1)=e′(1)

i1=e(1)i1q((?i??1)?id)(r0(N))=e(1)i1q

N

k=i

e

kk

?(N+1?i)C1(N)

(56)

for2≤i≤N.Where the a?ne root vectors e(1)i1(2≤i

e(1)i1=[e(0)iN,e(1)N1]q?1.(57)

4A?ne 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 form

R=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 formula

R=R12(R13R23)(R14R24R34)···(R1,N+1R2,N+1···R N,N+1)

=↑N+1

j=2

↑j?1

i=1

R ij ,(59)

where

R ij=exp q?2((q?q?1)e ij?e ji),(60) exp q(x):= n≥0x n

N +1.From the results of the previous section it follows that we can immediately obtain an a?ne realization ?F CG which twists the quantum a?ne algebra U q (gl N [u ]):

?F CG =?F ·q

r 0(N ),(68)?F

:=(???)(F )=?R ′(N ?1)?R ′(N ?2)···?R ′(1)(69)?R

′(k )=↑N +1?k j =2 ↑j ?1 i =1?R ′(k )ij .(70)where

?R ′(k )ij

=exp q ?2 (q ?q ?1)e ′(0)i +k,j +k ?e ′(0)ji for 1≤i

for 1≤i

5.1A?ne r -matrices of Cremmer-Gervais type and their quantization

In this subsection we are going to describe two classes of the quasi-trigonometric r -matrices of Cremmer-Gervais type,which we call a?ne 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 )de?nes 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 de?ned 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 de?ne quasi-trigonometric r -matrices for sl N with i =N ?1.

Type 1,which we call a?ne 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 de?nes a quasi-trigonometric r -matrix.

Type 2,which we call a?ne 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 de?nes a quasi-trigonometric r -matrix.

Both propositions can be proved directly.

Theorem 2.Let ?F ′CG be the twist (66)reduced to U q (?sl N ),and let ?R be the universal R -matrix for U q (?sl N ).Then the R -matrix ?F ′21CG ?R ?F ′?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 a?ne 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}de?nes an embedding of Hopf algebras U q(sl N)?→U q(?sl N).Abusing notations we denote this embedding byτ.The mapτsends each twist F of U q(sl N)to an a?ne twist(τ?τ)(F).Let us denote(τ?τ)(F′CG)by?F(τ)

CG

. Theorem3.Let?R′be the universal R-matrix for U q(?sl N).Then the R-matrix

?F(τ)21 CG ?R′(?F(τ)

CG

)?1quantizes the quasi-trigonometric r-matrix obtained in Proposition3.

Both theorems can be proved straightforward.

5.2ω-a?nization 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 a?ne 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-a?nization.We begin with the following result.

Theorem4.Letπ:U q(sl N[u])→U q(sl N)be the canonical projection sending all the a?ne generators to zero.LetΨ∈U q(sl N[u])?U q(?sl N)be invertible and such that

Ψ=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 that

Ψω:=(ω?ω)Ψ?(ω?1)∈U q(sl N)?U q(?sl N).(75) ThenΨis a twist.

Proof.Let us consider the Drinfeld associator

Assoc(Ψ)=Ψ12(??id)(Ψ)(id??)(Ψ?1)(Ψ?1)23(76) and the following one equivalent to it

Assoc(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 have

Assoc(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 a?nizator.

Corollary1.LetΨ1,Ψ2,ωsatisfy the conditions of Theorem4.Then

Ψ1=(ω?1π(ω)?id)Ψ2 (π?id)?(ω?1) ?(ω)Ψ?12.(82) Proof.We have

Ψω=(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-a?nization can be used to?nd a Yangian degen-eration of the a?ne Cremmer–Gervais twists.Let us consider the case sl3.We set

Ψ2:=F(τ)

CG3

:=exp q2(?(q?q?1)ζ?e(0)12??e(0)32)·?K3,(85)

where

?K

3=q 4

9

h12?h23+5

9

h23?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 a?nizatorωlong

3was constructed in[17].It is given by the following

formula

ωlong 3=exp q2(

ζ

(1?q2)2

q2h⊥β?e(1)31)×exp q?2(

ζ2

1?q2

?e(0)21)exp q?2(

ζ2 3

(e11+e22)?2

3

e11?1

Theorem5.The elementsωlong

3,Ψ2=F(τ)

CG3

andΨ1constructed according to Corollary

1satisfy 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 de?ne this rational degeneration we have to introduce the so-called f-generators:

f0=(q?q?1)?e(0)31,f1=q2h⊥β?e(1)31+q?1ζ?e(0)31,

f2=(1?q?2)?e(0)32,f3=q h⊥α?e(1)32?ζ?e(0)32.

(89) Let us consider the Hopf subalgebra of U?K3q(?sl3)generated by

{h12,h23,f0,f1,f2,f3,?e(0)12,?e(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 e31

e21→e21,

F1quantizes the following classical rational r?matrix r(u,v)=

?

3(hα

1

?hα

1

+hα

2

?hα

2

)+

1

6

hα

1

∧hα

2

.(93)

We have

ωshort

3

=exp q?2(ζ?e(0)21)exp q2 ?ζ1?q2?e(0)32 ,(94) where

?e(0)21=q?1

We have to calculate

A?ωshort

3

(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⊥α?e(1)31+ζq?2h⊥α?(Ad exp q2(ζ?e(0)21))(?e(0)32) (?h⊥α1?1)

q2

× 1?1+ζ(1?q2)1??e(0)21 (?1

F2= 1?1+ζ1?(e(0)32)?ζ2h⊥α?

g2)(?1

F2quantizes the following rational r-matrix:

r(u,v)=?

3

(h12?h23)∧e21.(99)

Therefore we have quantized all non-trivial rational r-matrices for sl3classi?ed in[19].

Acknowledgments

The paper has been partially supported by the Royal Swedish Academy of Sciences under the program”Cooperation between Sweden and former USSR”and the grants RFBR-05-01-01086,INTAS-OPEN-03-51-3350(V.N.T.).

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