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The Lie algebra of the sl(2, C)-valued automorphic functions on a

arXiv:hep-th/9302138v1 26 Feb 1993The Lie algebra of the sl(2, C)-valued automorphic functions on atorusD.B.Uglov 1Department of Physics, State University of New York at Stony BrookStony Brook, NY 11794-3800, USAFebruary 8 , 1993AbstractIt is shown that the Lie algebra of the automorphic, meromorphic sl(2, C)-valuedfunctions on a torus is a geometric realization of a certain infinite-dimensional finitelygenerated Lie algebra.In the trigonometric limit, when the modular parameter ofthe torus goes to zero, the former Lie algebra goes over into the sl(2, C)-valued loopalgebra, while the latter one - into the Lie algebra (A(1)1 )′/(centre).1e-mail: denis@max.physics.sunysb.edu

1IntroductionThe Inverse Scattering Transform Method has been a source of many important algebraicconstructions. The most spectacular of them are quantum groups associated with the quan-tum R-matrices, and their classical limits - the Lie bialgebras associated with the classicalr-matrices.

It is well-known, that the classical r-matrices can be classified into three cate-gories: rational, trigonometric and elliptic r-matrices [2]. While the Lie bialgebras, as wellas the quantum groups, associated with the rational ( Yangians ) and the trigonometric (quantized Kac-Moody affine Lie algebras ) solutions of the Yang-Baxter equation are wellunderstood by now [4,5], not so much is known about those associated with the ellipticR-matrices.

Important exceptions are the Lie algebras of the automorphic, meromorphicsl(n, C)-valued functions on a torus introduced by Reyman and Semenov-Tyan-Shanskii [1].These algebras are related to the elliptic solutions of the classical Yang-Baxter equation ,the simplest of whic h ( sl(2, C)-case ) is being the r-matrix of the classical Landau-Lifshitzmodel [8].The construction of Reyman and Semenov-Tyan-Shanskii is semi-geometric. Having inview the problem of quantization it is important to give a purely algebraic definition of theiralgebras in terms of finite number of generators and defining relations.

The example of whatwe are looking for is provided by the trigonometric case, where the loop algebras can beconsidered as the geometric realizations of the appropriate affine Kac-Moody Lie algebras[7].As it is shown is this letter, it indeed can be done, at least in the sl(2, C) case. We definean infinite-dimensional, finitely generated Lie algebra Ek,ν± , and show that the Lie algebraof automorphic, meromorphic sl(2, C)-valued functons on a torus provides a geometric re-alization of Ek,ν±.

It will be also shown that in the trigonometric limit, when the modularparameter of the torus goes to zero, the Lie algebra Ek,ν± goes over into the Lie algebraL = (A(1)1 )′/(centre).1

It turns out that the Lie (bi)algebra Ek,ν± can be quantized, the corresponding quantumgroup being related to the eight-vertex R-matrix.This will be the subject of the subsequentpaper [9].2Definition of the Lie algebra Ek,ν± and the main The-orem.Let k, k′ ∈C\{0, 1} , and k2 + k′2 = 1 . Let T = C/(Z4K + Z4iK′) be a torus with theperiods defined by the complete elliptic integrals K and K′ of the 1-st kind of the moduli kand k′ correspondingly.Introduce three meromorphic functions {wi}i=1,2,3 on T defined in the following way [6] :w1(u) =1sn(u), w2(u) = dn(u)sn(u), w3(u) = cn(u)sn(u); u ∈T(1)where the Jacobi elliptic functions are of the modulus k .

These functions obey the followingquadratic equations [6]:wi(u)2 −wj(u)2 = Jij, i, j ∈{1, 2, 3}, i ̸= j. (2)where J12 = k2, J23 = k′2, J31 = −1.Let ν+, ν−∈T and ν+ −ν−̸= n12K + n22iK′; (n1, n2) ∈Z22 = Z/2Z × Z/2Z.DefinitionEk,ν± is a complex Lie algebra generated by six generators {x±i }i=1,2,3which obey the following defining relations:[x±i , [x±j , x±k ]] = 0,(3)[x±i , [x±i , x±k ]] −[x±j , [x±j , x±k ]] = Jijx±k ,(4)[x+i , x−i ] = 0,(5)[x±i , x∓j ] =√−1(wi(ν∓−ν±)x∓k −wj(ν∓−ν±)x±k );(6)2

here, and througout the rest of the letter, {i, j, k} is a cyclic permutation of {1, 2, 3}.Remark : The defining relations which involve both x+i and x−i can be compactly writtenin the r-matrix form, namely, let:X± = x±3x±1 −ix±2x±1 + ix±2−x±3! ;X±1(2) = X± ⊗I(I ⊗X±); r12(u) = 123Xn=1wn(u)σn ⊗σn , u ∈T;(7)where I is an identity in Mat2 and {σn}n=1,2,3 are the Pauli matrices ; then (5,6) can bewritten as follows :[X±1 , X∓2 ]Ek,ν± = [r12(ν∓−ν±), X±1 + X∓2 ]Mat2(8)This representation turns out to be helpful when one is doing quantization of the Lie algebraEk,ν± [9].Now we formulate a theorem which describes the structure of the Lie algebra Ek,ν±.Theorem1.

Ek,ν± = E+ ⊕E−−Ek,ν± is, as a linear space, a direct sum of two Lie subalgebras E+and E−generated by {x+i }i=1,2,3 and {x−i }i=1,2,3 correspondingly.2. E± = ⊕n∈Z≥0E±n where dim(E±n ) = 3.

The elements {g(n),±i}i=1,2,3;n∈Z≥0 defined by therecurrent formula:g(0),±i= x±i ,(9)g(n),±k=1√−1nXm+l=n−1[g(m),±i, g(l),±j] , n ≥1, m, l ∈Z≥0,(10)form a basis in E±. The elements {g(n),±i}i=1,2,3 form a basis in E±n .3.The commutation relations of Ek,ν± in the basis {g(n),±i}i=1,2,3;n∈Z≥0 are given by theformulae:a).

[g(m),±i, g(n),±i] = 0,i ∈{1, 2, 3},(11)3

b).1√−1[g(m),±i, g(n),±j] = g(m+n+1),±k+ (−1)mm+nXr=m rm!wirg(m+n−r),±k−(−1)nm+nXr=n(−1)r rn!wjrg(m+n−r),±k,(12)c). [g(m),+i, g(n),−i] = 0,i ∈{1, 2, 3},(13)d).1√−1[g(m),±i, g(n),∓j] = (−1)mm+nXr=m rm!vir(±(ν−−ν+))g(m+n−r),∓k−(−1)nm+nXr=n(−1)r rn!vjr(±(ν−−ν+))g(m+n−r),±k,(14)m, n ∈Z≥0, and the coefficients wir, vir(ν) are defined by the formulae:wi(u) = 1u +Xr≥0wirur , vir(u) = 1n!dndunwi(ν).

(15)Proof of the theorem is given in the Appendix.Remark : Commutation relations in Ek,ν± can be compactly written down using thegenerating functions and the r-matrix representation. Let us define:G±i (α) =Xn≥0αng(n),±i, α ∈T;(16)G± = G±3G±1 −iG±2G±1 + iG±2−G±3!

;(17)then the commutation relations in the basis {g(n),±i} can be recovered from the followingcommutation relations for the generating functions:[G±1 (α), G±2 (β)]Ek,ν± = [r12(β −α), G±1 (α) + G±2 (β)]Mat2. (18)[G±1 (α), G∓2 (β)]Ek,ν± = [r12(β −α + (ν∓−ν±)), G±1 (α) + G∓2 (β)]Mat2.

(19)4

3Geometric realization of Ek,ν±In this section we describe a geometric realization of the Lie algebra Ek,ν± - an “ellipticanalog” of the loop Lie algebras. This turns out to be the Lie algebra of the sl(2, C)-valuedautomorphic meromorphic functions on T [1].Consider a set ˜F = {ϕ(n),±i}i=1,2,3, n ∈Z≥0 of meromorphic functions on T defined asfollows:ϕ(n),±i(u) = (−1)nn!dndunwi(u −ν±), u ∈T(20)Let G = sl(2, C).

Fix a basis {si}i=1,2,3 in G such that: [si, sj] = √−1sk. Let F be alinear span of ˜F over C .

Consider a tensor product A = F ⊗CG. Single out a linear subspaceof A spanned by the elements of the form ϕ(n),±i⊗si.

Denote this subspace by˜Ek,ν±. Theelements of this subspace satisfy the following automorphicity condition with respect to theaction of the group Z22 = Z/2Z × Z/2Z:ϕ(n),±i(u + n12K + n2i2K′) ⊗si = ϕ(n),±i(u) ⊗(T (n1,n2)si(T (n1,n2))−1) , (n1, n2) ∈Z22(21)where, in the 2×2 matrix realization of G , one has T (n1,n2) = σn13 σn21 .

Introduce a Lie bracketin˜Ek,ν± as follows: [ϕ(n),ǫi⊗si, ϕ(n),ǫ′j⊗sj] = ϕ(n),ǫiϕ(n),ǫ′j⊗[si, sj]; i, j = 1, 2, 3.; ǫ, ǫ′ ∈{+, −}.Then the map γ : Ek,ν± →˜Ek,ν±:γ(g(n),±i) = ϕ(n),±i⊗si is an isomorphism, as can bereadily checked using the relations :wi(u)wj(v) = wk(u −v)wi(v) −wk(u −v)wj(u) , u, v ∈T [6]. (22)Proposition 1 Ek,ν± is a Lie bialgebra [3,4] , cocommutator map δ : Ek,ν± →V2 Ek,ν± isbeing given by the following formulae :δ[g(n),±i] =√−1nXl=0g(n−l),±k^g(l),±j,(23)or, in terms of the generating functions:δ[G±i (·, α)](u, v) = [G±i (u, α) ⊗I + I ⊗G±i (v, α), r(u −v)]UG⊗UG(24)5

where G±i (u, α) = γ(G±i (α)) = wi(u −α −ν±) ⊗si, r(u) = 2 P3k=1 wk(u)sk ⊗sk∈UG ⊗UG, u ∈T , and I is an identity in UG-the universal enveloping algebra of G.4Trigonometric limit of Ek,ν±In this section it is shown that the trigonometric limit:k →0 of Ek,ν±, when ν+ =i32K′ , ν−= i12K′ coincides with the sl(2)-loop algebra : L = ( ˆG)′/(centre) , where ˆG = A(1)1 .Let us fix ν+ = i32K′ , ν−= i12K′ , let us also make a change of the generators of Ek,ν±: x±i →x′±i, where x′±1,2 =1√kx±1,2; x′±3= x±3 . Then the defining relations of Ek,ν± acquire aform (from now on we drop the prime):[x+1 , x−2 ] = x+3 ,[x−1 , x+2 ] = −x−3 ,[x+2 , x−3 ] = x+1 −kx−1 ,(25)[x−2 , x+3 ] = −x−1 + kx+1 ,[x+3 , x−1 ] = −x−2 ,[x−3 , x+1 ] = x+2 ;(26)[x±1 , [x±1 , x±3 ]] −[x±2 , [x±2 , x±3 ]] = kx±3 ,k[x±2 , [x±2 , x±1 ]] −[x±3 , [x±3 , x±1 ]] = k′2x±1 ,(27)[x±3 , [x±3 , x±2 ]] −k[x±1 , [x±1 , x±2 ]] = −x±2 ,[x±i , [x±j , x±k ]] = 0, [x+i , x−i ] = 0, i = 1, 2, 3.

(28)If k ̸= 0 , it follows from the above relations that[x±2 , [x±1 , [x±1 , x±2 ]]] = 0,and [x±2 , [x±2 , [x±2 , x±1 ]]] + [x±1 , [x±1 , [x±1 , x±2 ]]] = k[x±1 , x±2 ](29)Let us add these new ( non-independent only if k ̸= 0) relations to the relations (25-28)and then take a limit k →0. As a result Ek,ν± goes over into the Lie algebra ˜L generated bythe elements {y±i = limk→0x±i }i=1,2,3.

In the geometric realization this limit has the followingexplicit form:1√kw1(u −ν±) ⊗s1 =√ksn(u ∓i2K′) ⊗s1 →∓ie±iu ⊗s1,1√kw2(u −ν±) ⊗s2 = i√kcn(u ∓i2K′) ⊗s2 →ie±iu ⊗s2,w3(u −ν±) ⊗s3 = idn(u ∓i2K′) ⊗s3 →i ⊗s3.6

{y±i } obey the following defining relations:[y±1 , y∓2 ] = ±y±3 ,[y±2 , y∓3 ] = ±y±1 ,[y±3 , y∓1 ] = ∓y∓2 ,(30)[y±1 , [y±1 , y±3 ]] −[y±2 , [y±2 , y±3 ]] = 0,[y±3 , [y±3 , y±2 ]] = −y±2 ,[y±3 , [y±3 , y±1 ]] = −y±1 ,(31)[y±i , [y±j , y±k ]] = 0,[y+i , y−i ] = 0,[y±2 , [y±1 , [y±1 , y±2 ]]] = 0,(32)[y±2 , [y±2 , [y±2 , y±1 ]]] + [y±1 , [y±1 , [y±1 , y±2 ]]] = 0. (33)Proposition 2 ˜L contains an ideal ı generated by the elements:[y±2 , y±3 ] ∓y±1 ,[y±3 , y±1 ] ∓y±2 ,y+3 −y−3(34)Proposition 3 Lie algebra L = ˜L/ı is isomorphic to the sl(2, C)-loop Lie algebra ( ˆG)′/(centre), ˆG = A(1)1 .Remark: Non-independent relations (29) in Ek,ν± go over , as k →0 into the Serre relationsin L.5ConclusionThe fact, that the Lie algebra of the automorphic, meromorphic sl(2, C)-valued functions ona torus admits a purely algebraic description as the finitely generated infinite dimensionalLie algebra, suggests that it is possible to develop a theory of the algebras of such a typewhich would be parallel to the theory of the Kac-Moody affine Lie algebras [7].

Such adevelopement would be clearly important taking into account an abundance of physical andmathematical constructions related to the latter. Among the more immediate ramificationsof the result given in this letter, one can point out the problem of finding the quantum groupsrelated to the elliptic quantum R-matrices [9].7

AcknowlegementThe author is grateful to I.T.Ivanov and L.A.Takhtadjan for many helpful discussions.AppendixProof of the theorem . From the defining relations (3-6) and the Jacobi identity it followsthat Ek,ν± = E+ ∪E−, where E+ and E−are Lie subalgebras generated by {x+i }i=1,2,3 and{x−i }i=1,2,3 correspondingly.Consider E+ .

E+ = ∪n≥0E+n , where E+n is a linear subspace of E+ spanned by all theelements of the form [x+i1, . .

. , [x+in, x+in+1] .

. .

], ip ∈{1, 2, 3} if n ≥1, and E+0 is a linear spanof {x+i }i=1,2,3.In what follows g(n)idef= g(n),+i, i ∈{1, 2, 3}, n ≥0. We remind, that the indices {i, j, k}denote any cyclic permutation of {1, 2, 3}.

We will prove the statements 2 and 3 a).,b). ofthe theorem using induction.Fix n ∈Z, n ≥2 and assume, that:An.

The elements {g(s)i }i=1,2,3, 0 ≤s ≤n form a basis in ∪ns=0E+s . Denote by E+s a linearsubspace of ∪ns=0E+s spanned by the elements {g(s)i }i=1,2,3.

We have ∪nm=0E+m = ⊕nm=0E+m.Bn. [g(l)i , g(m)i] = 0, if 0 ≤l + m ≤n −1.Cn.

There exist the coefficients A(l,m)k,rsuch, that1√−1[g(l)i , g(m)j] = g(l+m+1)k+l+m−1Xr=0A(l,m)k,r g(r)k ,(35)if 0 ≤l + m ≤n −1 .The fact that these assumptions are true for n = 2 , and that A(0,1)k,0=12Jij = wi1 −wj1 , A(1,0)k,0= −12Jij, follows immediately from the defining relations (3,4). We will show thatunder the above assumptions the statements An+1, Bn+1, Cn+1 hold true .From An it follows that E+n+1 is a linear span of {[g(m)i, g(n−m)i′]}i,i′∈{1,2,3};0≤m≤n.

Let us8

show that among the elements {[g(m)i, g(n−m)j]}0≤m≤n only three are linearly independentmod(⊕ns=0E+s ). This fact follows from the:Lemma 1.

Under the assumptions An, Bn, Cn:1. [g(m+1)i, g(n−m−1)j] = [g(m)i, g(n−m)j] + η(n,m)k, 0 ≤m ≤n −1, whereη(n,m)k=√−1n−1Xs=0B(n,m)k,sg(s)k∈⊕n−1r=0E+r .

(36)2. The coefficients B(n,m)k,sare expressed in terms of the coefficients {A(l,m)k,s }0≤l+m≤n−1 bythe formulae:B(n,m)k,s= θ(s −m −1)A(n−2m−2,m+1)j,s−m−1−θ(s −m −2)A(n−2m−2,m)j,s−m−2+n−m−2Xr=0θ(m + r −1 −s)A(n−2m−2,m+1)j,rA(m,r)k,s−n−m−3Xr=0θ(m + r −s)A(n−2m−2,m)j,rA(m+1,r)k,s,if 0 ≤m ≤p −1, n = 2p or n = 2p + 1; (37)B(n,m)k,s= θ(s −n + m −1)A(n−m−1,2m−n)i,s−n+m−1−θ(s −n + m)A(n−m,2m−n)i,s−n+m+m−2Xr=0θ(r + n −m −1 −s)A(n−m−1,2m−n)i,rA(r,n−m)k,s−m−1Xr=0θ(r + n −m −2 −s)A(n−m,2m−n)i,rA(r,n−m−1)k,s,if p ≤m ≤n = 2p, or p + 1 ≤m ≤n = 2p + 1;(38)B(2p+1,p)k,s= θ(s −p −2)A(p−1,0)i,s−p−2 −θ(s −p)A(p+1,0)i,s−p+p−2Xr=0θ(r + p −s)A(p−1,0)i,rA(r,p+1)k,s−pXr=0θ(r + p −2 −s)A(p+1,0)i,rA(r,p−1)k,s−B(2p+1,p+1)k,s, where θ(x ≥(<)0) = 1(0).

(39)Proof. Let n = 2p or n = 2p + 1.Let 0 ≤m ≤p −1 .

According to the assumption Cn:[g(m)i, g(n−m)j] =√−1[g(m)i, [g(m+1)i, g(n−2m−2)k]] + ϑ(n,m)k, (40)−ϑ(n,m)k=√−1(n−m−1Xr=0A(n−2m−2,m+1)j,rg(m+r+1)k+n−m−2Xr=0A(n−2m−2,m+1)j,rm+r−1Xs=0A(m,r)k,sg(s)k ). (41)9

It is clear, that ϑ(n,m)k∈⊕n−1r=0E+r . [g(m+1)i, g(n−m−1)j] =√−1[g(m+1)i, [g(m)i, g(n−2m−2)k]] + ϑ(n,m+1)k,(42)−ϑ(n,m+1)k=√−1(n−m−3Xr=0A(n−2m−2,m)j,rg(m+r+2)k+n−m−3Xr=0A(n−2m−2,m)j,rm+rXs=0A(m+1,r)k,sg(s)k ) ∈⊕n−1r=0E+r .

(43)Using the Jacobi identity and the assumption Bn we get[g(m)i, [g(m+1)i, g(n−2m−2)j]] = [g(m+1)i, [g(m)i, g(n−2m−2)j]]. (44)Thus [g(m+1)i, g(n−m−1)j] = [g(m)i, g(n−m)j] + η(n,m)k, η(n,m)k= ϑ(n,m+1)k−ϑ(n,m)k∈⊕n−1r=0E+r .The statement 1. is proven in the case 0 ≤m ≤p −1 .

The statement 2. in this casefollows from the above formulae for η(n,m)k. The cases p ≤m ≤n = 2p ; p+1 ≤m ≤n = 2p+1and m = p, n = 2p + 1 are proven analogously ✷.From the Lemma 1. it follows that [g(m)i, g(n−m)j] = [g(0)i , g(n)j ] + Pm−1r=0 η(n,r)k, 1 ≤m ≤n.By the definition of g(n+1)kwe also have: √−1(n + 1)g(n+1)k= Pnm=0[g(m)i, g(n−m)j]. Hence weobtain the following expressions for the commutators [g(m)i, g(n−m)j]:[g(0)i , g(n)j ] =√−1g(n+1)k−1n + 1nXs=1s−1Xr=0η(n,r)k,(45)[g(m)i, g(n−m)j] =√−1g(n+1)k+m−1Xr=0η(n,r)k−1n + 1nXs=1s−1Xr=0η(n,r)k, 1 ≤m ≤n.

(46)From these expressions and the fact, that [g(m)i, g(n−m)i] = 0 , 0 ≤m ≤n; i ∈{1, 2, 3}, whichwill be proven in the Lemma 2., it follows, that {g(n+1)i}i=1,2,3 form a basis in E+n+1/∪0≤r≤nE+r .Thus we have proven that An+1 holds. From (45,46) and the second statement of the Lemma1.

it also follows, that Cn+1 is true . The coefficients A(m,n−m)k,rare expressed in terms of thecoefficients {A(l,m)k,r }0≤l+m≤n−1 by the following formulae:A(m,n−m)k,s= (1 −δ0m)m−1Xr=0B(n,r)k,s−1n + 1n−1Xr=0(n −r)B(n,r)k,s, 0 ≤m ≤n , 0 ≤s ≤n −1.

(47)Recall, that the coefficients B(n,r)k,sare expressed in terms of {A(l,m)i,s}i=1,2,3;0≤l+m≤n−1 as givenby the second statement of the Lemma 1. Combining the formulae (37-39) and (47) we obtain10

recurrent relations for A(l,m)k,s .From these relations it is evident, that all the coefficients A(l,m)k,sare completely and uniquely determined by A(0,1)k,0and A(1,0)k,0which enter into the definingrelations (3,4).Let us now prove Bn+1.Lemma 2. Under the assumptions An, Bn, Cn:[g(m)i, g(n−m)i] = 0 ; i ∈{1, 2, 3}, 0 ≤m ≤n.Proof.

First, let us prove, that [g(0)i , g(n)i] = 0. From the assumptions Bn and Cn we have:√−1[g(0)i , g(n)i] = [g(0)i , [g(0)j , g(n−1)k]] = [g(0)i , [g(1)j , g(n−2)k]] = .

. .

= [g(0)i , [g(n−1)j, g(0)k ]]. (48)By the Jacobi identity and the assumptions Bn , Cn we also have:[g(0)i , g(n)i] = −[g(m)j, g(n−m)j] −[g(n−m−1)k, g(m+1)k].

(49)From (48) and (49) we get:n[g(0)i , g(n)i] = −n−1Xm=0([g(m)j, g(n−m)j] + [g(n−m−1)k, g(m+1)k]) = −[g(0)j , g(n)j ] −[g(0)k , g(n)k ],(50)and(n −1)([g(0)i , g(n)i] −[g(0)j , g(n)j ]) = 0. (51)Recall, that we consider only n ≥2, so : [g(0)i , g(n)i] = [g(0)j , g(n)j ] = ξ .

From (50) we have(n + 2)ξ = 0 . Thus ξ = 0.Now let us prove, that [g(l)i , g(m)i] = 0 , l + m = n, l ≥1.From the Jacobi identity and the assumptions Bn and Cn we have:√−1[g(m)i, g(n−m)i] = [g(m)i, [g(0)j , g(n−1−m)k]] = −√−1[g(0)j , g(n)j ] −√−1[g(n−1−m)k, g(m+1)k] ,0 ≤m ≤n −1.

(52)hence [g(m)i, g(n−m)i] = [g(m+1)k, g(n−m−1)k], 0 ≤m ≤n −1. Taking into account that, as hasbeen proven, [g(0)i , g(n)i] = 0, we obtain the statement of the lemma ✷.11

Thus it is shown, that the statements An+1, Bn+1, Cn+1 are true if the An, Bn, Cn aretrue. From this the statements 2. and 3. a).

of the theorem follow for the subalgebra E+ .For the subalgebra E−which is isomorphic to E+ a proof is identical.By a direct calculation using the relations (22) one can verify that the structure constantsappearing in the RHS of the statement 3. b). satisfy the recurrent relations (47),(37-39) withthe initial conditions A(0,1)k,0= 12Jij = wi1 −wj1, A(1,0)k,0= −12Jij.

This proves the statement 3.b)..The statement 1. follows from the explicit form of the bases in E+ and E−.The proof of the statements 3 c).,d). is very similar to that one of the statements 3 a).,b)., for this reason we describe it only schematically.

Using induction one can derive recurrentrelations for the structure constants appearing in the decompositions of the commutators[g(l),±i, g(l),∓i′] in the basis {g(n),±i} . Then, we uniquely resolve these - the initial conditionsbeing provided by the defining relations (3-6).

The relation (22) again plays the crucial rolein this solution.To finish the proof of the theorem we need to check that the Lie bracket in Ek,ν± asgiven by the formulas (11-14) satisfies the Jacobi identity. This follows from the r-matrixrepresentation (18,19) of the commutation relations (11-14), and the well-known fact thatr(u) (see (7)) satisfies the Classical Yang-Baxter equation ✷.References1.

Reyman, A. G. and Semenov-Tyan-Shanskii, M. A., Journ. Sov.

Math. 46, 1631 (1989).2.

Belavin, A. A. and Drinfel’d V. G., Funct.

Anal. Appl.17, 220 (1984).3.

Drinfel’d V. G., Sov. Math.

Doklady 28, 667 (1983).4. Drinfel’d V. G., Proceedings of the ICM, Berkeley ,CA U.S.A., 798 (1986).5.

Chari, V. and Pressley, A., L’Enseignement Math´ematique 36, 267 (1990); Comm. Math.Phys.

142, 261 (1991).6. Sklyanin, E. K., Funct.

Anal. Appl.

16, 263 (1983).12

7. Kac, V. G., Infinite dimensional Lie Algebras, Cambridge University Press, Cambridge,1990.8.

Faddeev, L. D. and Takhtadjan, L. A., Hamiltonian Methods in the theory of solitons ,Springer-Verlag, Berlin, New-York, 1987.9. Uglov, D. B., in preparation.13

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