The quantum bialgebra associated with the eight-vertex R-matrix

arXiv:hep-th/9302139v1 26 Feb 1993The quantum bialgebra associated with the eight-vertex R-matrixD.B.Uglov 1Department of Physics, State University of New York at Stony BrookStony Brook, NY 11794-3800, USAFebruary 24, 1993AbstractThe quantum bialgebra related to the Baxter’s eight-vertex R-matrix is found asa quantum deformation of the Lie algebra of sl(2)-valued automorphic functions on acomplex torus.1e-mail: denis@max.physics.sunysb.edu

1. Recent years witnessed an extensive developement of the theory of quantum groupsand their quasiclassical limits - Lie bialgebras .

Although since the work of Belavin andDrinfel’d [3] it is known, that solutions of the Classical Yang-Baxter equation can be clas-sified into three categories: the rational, the trigonometric and the elliptic ones, the maindevelopement took place in the rational and the trigonometric cases. The underlying alge-braic structures in these cases were found to be the affine Kac-Moody Lie algebras (classicalcase)and the Yangians and the quantum affine Kac-Moody algebras (quantum case) [5,6].One of the major developements in the elliptic case was the discovery of the Sklyanin alge-bra [4].

This algebra, however, has no coproduct and thus it is not a bialgebra. Anotherimportant work had been done by Reyman and Semenov-Tyan-Shanskii [2], who found theLie bialgebras associated with the elliptic solutions of the Classical Yang-Baxter equation.The simplest example of their alge bras is the Lie algebra of sl(2)-valued automorphic mero-morphic functions on a complex torus.

The Lie bialgebra structure of this algebra is givenby the classical r-matrix of the Landau-Lifshitz model [2]. In the work [7] the generators andthe defining relations of this Lie algebra had been found.

In the present letter we addressthe problem of quantization of the above Lie bialgebra. The result is a quantum bialgebrarelated to the eight-vertex R-matrix.

We use the term “ quantum bialgebra“ to designate aHopf algebra without the antipode.2.In [7] it is shown, that the Lie algebra of sl(2)-valued automorphic meromorphicfunctions on a complex torus which we denote by Ek,ν± is a finitely generated (infinite-dimensional) C-Lie algebra defined upon the six generators {x±k }k=1,2,3 by the relations (i, j, k below is any cyclic permutation of 1,2,3 ):[x±i , [x±j , x±k ]] = 0,(1)[x±i , [x±i , x±k ]] −[x±j , [x±j , x±k ]] = Jijx±k ,(2)[x+i , x−i ] = 0,(3)1

[x±i , x∓j ] =√−1(wi(ν∓−ν±)x∓k −wj(ν∓−ν±)x±k . (4), where ν+, ν−∈T = C/(Z4K + Z4iK′), ν+ −ν−̸= Z2K + Z2iK′; K, K′-are thecomplete elliptic integrals of the moduli k and k′ correspondingly: k2 + k′2 = 1; J12 =k2, J23 = k′2, J31 = −1; and w1(u) =1sn(u,k); w2(u) = dnsn(u, k); w3(u) = cnsn(u, k), u ∈T.The structure of a Lie bialgebra upon Ek,ν± is defined by the classical r-matrix of theLandau-Lifshitz model [2].

A certain trigonometric limit k →0 of Ek,ν± coincides with theloop algebra A(1)′1/(centre) [7].3. To define a quantum deformation of the Lie algebra (1-4) we introduce a deformationparameter η ∈C and recall the definition of the Baxter’s eight-vertex R-matrix [1]: R(u) =I ⊗I +P3k=1wk(u+iη)wk(iη) σk ⊗σk ∈Mat2 ⊗Mat2, u ∈T , I is 2 × 2 identity matrix , and σkare the Pauli matrices.

Let H = R(0)R′(0).Introduce associative C-algebra Ek,ν±,η generated by the elements: 1 (identity) ,T ±ab, T±ab, a, b ∈{1, 2}. The defining relations of Ek,ν±,η have the following form:T±1 (H12 −H14)T ±1 + T±3 (H34 −H32)T ±3 + T ±2 (H23 −H21)T±2 + T ±4 (H41 −H43)T±4 = 0, (5)T±1 T ±1 = T ±1 T±1 = 1I1,(6)R12(ν∓−ν±)T ±1 T ∓2 = T ∓2 T ±1 R12(ν∓−ν±).

(7)We adopt the standard convention: T ±n means a matrix with Ek,ν±,η -valued entries , whichacts nontrivially only in the n−th factor of C⊗m ( m = 4 in (5) , m = 2 in (7) ) and coincidesthere with T ±ab.PROPOSITIONEk,ν±,η is a bialgebra, i.e.there exist a coproduct ∆: Ek,ν±,η →Ek,ν±,η ⊗Ek,ν±,η and a counit ε : Ek,ν±,η →C, such, that:∆(ab) = ∆(a)∆(b),a, b ∈Ek,ν±,η(8)(ε ⊗id)∆(a) = (id ⊗ε)∆(a) = a, a ∈Ek,ν±,η. (9)2

Explicit expressions for ∆, and ε are given by the following formulae:∆(T ±ab) =3Xc=1T ±ac ⊗T ±cb, ∆(T±ab) =3Xc=1T±cb ⊗T±ac, ∆(1) = 1 ⊗1. (10)ε(T ±ab) = δab, ε(T±ab) = δab, ε(1) = 1.

(11)In the quasiclassical limit η →0 , which is described as follows:T ± = I + 2iη3Xk=1x±k σk + O(η2), T± = I + 2iη3Xk=1x±k σk + O(η2),(12)one recovers from the defining relations (5-7) the defining relations (1-4) for the generators{x±k }k=1,2,3 in Ek,ν± . Note, that from (6) it follows that x±k = −x±k .The eight-vertex R-matrix R(u −v) appears again as an intertwiner between the tensorproducts πu ⊗πv and πv ⊗πu of the simplest 2-dimensional representations πu(v) of Ek,ν±,ηparametrized by u, v ∈T :πu(T ±) =1qD(u −ν±)R(u −ν±), πu(T±) =1qD(u −ν±)R(−u + ν±),(13)D(u) = 1 −3Xk=1wk(u + iη)wk(u −iη)w2k(iη).

(14)To establish a connection between the elliptic and the trigonometric cases we need toperform the trigonometric limit k →0 of Ek,ν±,η. By analogy with the classical case [7], wedo this limit at ν+ = i32K′, ν−= i12K′.

The author had succeeded to recover from (5-7)under this limit all the defining relations of Uq(A(1)′1/(centre)) at q = e2η except the quantumSerre relations.AcknowlegementThe author is grateful to I.T.Ivanov and L.A.Takhtadjan for helpful discussions.References1. Baxter, R. J., Ann.

Phys.76, 1 (1973).3

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

Sov. Math.

46, 1631 (1989).3. Belavin, A.

A. and Drinfel’d V. G., Funct. Anal.

Appl.17, 220 (1984).4. Sklyanin, E. K., Funct.

Anal. Appl.

16, 263 (1983).5. Drinfel’d V. G., Proceedings of the ICM, Berkeley ,CA U.S.A., 798 (1986).6.

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

142, 261 (1991).7. Uglov, D. B., to be published.4

출처: arXiv:9302.139 • 원문 보기