R-matrices for highest weight representations of sˆl (2,C) at roots of unity q 3 9 9 1 n T.S. Hakobyan a Yerevan State University, Manukyan J st. 9, Yerevan, Armenia 8 2 and v 5 2 A.G. Sedrakyan ∗ 1 L 1 Laboratoire de Physique Th´eorique ENS APP † 1 Chemin de Bellevue BP 110, F - 74941 Annecy-le-Vieux Cedex, France 2 9 / h t - p e Abstract h : v i We obtain the general formula for R-matrices of sˆlq(2,C) for the highest X weight representations both for general q and for q being a root of unity by gen- r a eralizing G.Gomez’s andSierra’s oneforsemiperiodic representations of sˆl (2,C) q at roots of unity. In a case of general q we obtain direct matrix form for the well known Jimbo’s R-matrix. In a case of qN = 1 for semiperiodic and spin j < N−1 2 representations we obtain the new R-matrices which together with Jimbo’s one obey the spectral parameter dependent Yang-Baxter equations. L ENS APP-A-406/92 October 1992 ∗permanent address: YerevanPhysics Institute, Br. Alikhanian, st.2, Yerevan36,Armenia †URA14-36duCNRS,associ´ee`al’E.N.S.deLyonetauL.A.P.P.(IN2P3-CNRS)d’Annecy- le-Vieux 0 Quantum groups had been introduced by Drinfeld [1] and Jimbo [2]. Shortly speaking, quantum group is some quasi-triangular Hopf algebra, parametrized by some parameter q [1]. They play an important role in two-dimensional integrable statistic models and conformal field theory. For general q irreducible representations of quantum algebras are in one to one correspondence with the representations of the classical algebras [2, 15]. For q being a root of 1 this is not true. Irreducible representations in this case had been considered in [7, 10, 11]. They are parametrized by some complex numbers. Due to Hopf algebra structure one can consider an intertwiner on tensor prod- ucts of two irreducible representations [1, 2]. In case of general q intertwiners satisfy Yang-Baxter equations [1, 3], that is the integrability condition for the two dimensional systems. For q being a root of unity the standard expression of the universal R-matrix [1] is ill defined because of singularities. In [9] Bazhanov and Stroganov showed howthechiralPottsmodel[17,18]isrelatedtothetheoryofcyclicrepresentations of the quantum algebra sˆl (2,C) at qN = 1. Intertwiners between semiperiodic q representations ofsl (2,C) hadbeen considered in[4]. In [12, 14] theintertwiners q between semiperiodic and spin j ≤ N−1 representations, where N is a minimal 2 integer, satisfying the condition qN = 1, had been considered. In [4] Gomez and Sierra found an interesting recursive formula for intertwiners of semiperiodic representations, having a reflection symmetry. Inthis articlewegeneralize thisformula forthecase ofaffinesˆl (2,C)algebra. q It canbeused forthehighest weight representations of ordinarysl (2,C)algebra. q In a particular case of general q we obtain well known Jimbo’s R-matrix [2], written in the matrix form. In case of q being an odd root of 1 by acting our general formula to tensor product of semiperiodic representation and spin j ≤ N−1 representation, we obtain baxterization of intertwiners, considered in [12]. 2 As a consequence we obtain new solutions of spectral parameter dependent Yang- Baxter equations. We show also that the intertwiners, considered in [4, 5], are particular cases of ones, obtained in [6] by a different approach. Let us recall the definition of the affine quantum algebra sˆl (2,C). It is q generated by generators e ,f ,h , satisfying the following relations [2]: i i i [h ,e ] = a e [h ,f ] = −a f [h ,h ] = 0 i j ij j i j ij j i j qhi −q−hi [e ,f ] = δ [h ] := δ (1) i j ij j q ij q −q−1 2 −2 (Here a = is Cartan matrix of sˆl (2,C), q is a parameter) ij −2 2 q ! 1−aij 1−a (−1)ν ij e1−aij−νe eν = 0 (i 6= j) ν i j i " # ν=0 q X 1 1−1aj 1−a (−1)ν ij f1−aij−νf fν = 0 (i 6= j) (2) ν i j i " # ν=0 q X m [m]! q (Here := , [m] ! := [1] [2] ..[m] ). " n #q [m−n]q![n]q! q q q q Following [6] we take the extension of sˆl (2,C) by central elements z±1(i = q i 0,1). There is the structure of Hopf algebra in this algebra: ∆(e ) = e ⊗k−1 +z ·k ⊗e ∆(k±1) = k±1 ⊗k±1 i i i i i i i i i (3) ∆(f ) = f ⊗k−1 +z−1 ·k ⊗f ∆(z±1) = z±1 ⊗z±1, i i i i i i i i i where ki := qh2i. Note, that in [6] another comultiplication had been considered: ∆˜(e ) = e ⊗k−ǫi +z ·qǫi ⊗e ∆˜(qǫi) = qǫi ⊗qǫi i i i i (4) ∆˜(f ) = f ⊗q−ǫi+1 +z−1 ·q−ǫi+1 ⊗f ∆˜(z±1) = z±1 ⊗z±1 i i i i i i i where ǫ (i = 0,1) are the orthonormal basis in Cartan subalgebra of sˆl (2,C) i q ǫi−ǫi+1 and ki = q 2 ,ǫι+2 = ǫi. The comultiplication (4) converts into (3) by changing: z ·e ⇒ e , i i z−1 ·fi ⇒ fi, z2zi ⇒ zi, where z := qǫ0+2ǫ1 lies in the center of sˆlq(2,C). From the representation of finite dimensional quantum algebra the parametri- zed representation of its affine extension can be constructed. Let’s regognize this method for sˆl (2,C). If e,f,k are the generators of sˆl (2,C), then there is the q q homomorphism ρ : sˆl (2,C) ⇒ sl (2,C) [2]: x q q ρ (e ) = x·f ρ (f ) = x−1 ·e ρ (k±1) = k∓1 x 0 x 0 x 0 (5) ρ (e ) = e ρ (f ) = f ρ (k±1) = k±1 x 1 x 1 x 1 This homomorphism converts representations of sl (2,C) to parametrized repre- q sentation of the affine quantum algebra sˆl (2,C) with the central charge equal to q zero. The Hopf algebra structure allows to consider the action of sˆl (2,C) on then- q sor products of representations. Let π and π be representations of sl (2,C) on 1 2 q V andV respectively. Then, asit was mentioned above, π (x ) := π ◦ρ ,i = 1,2 1 2 i i i xi are the representations of sˆl (2,C). The equivalence of tensor products π (x )⊗ q 1 1 π (x ) and π (x ) ⊗ π (x ) implies a relation between parameters of the repre- 2 2 2 2 1 1 sentations and means the existence of some intertwining operator Rˆ (x ,x ) π1π2 1 2 from V ⊗V into V ⊗V such that [1]): 1 2 2 1 Rˆ (x ,x )π (x )⊗π (x )(∆(g)) π1π2 1 2 1 1 2 2 = π (x )⊗π (x )(∆(g))Rˆ (x ,x ), (6) 2 2 1 1 π1π2 1 2 2 where g ∈ sˆl (2,C). We can take x = 1,x = x because Rˆ depends on x2 only. q 2 1 x1 Then the equations (6) for g = f ,f can be represented in the following form: 0 1 Rˆ(x)·(f ⊗k−1 +z−1 ·k ⊗f) = (f ⊗k−1 +z−1 ·k ⊗f)·Rˆ(x) 1 1 Rˆ(x)·(xf ⊗k +z ·k−1 ⊗f) = (f ⊗k +xz ·k−1 ⊗f)·Rˆ(x) (7) 0 0 From (7) we obtain: Rˆ(x)·(1⊗f) = [∆(f )·Rˆ(x)·x(1⊗k2)−∆(e )·Rˆ(x)] 1 0 ×(xz−1 ·k ⊗k2 −z ·k−1 ⊗1)−1 1 0 Rˆ(x)·(f ⊗1) = [∆(f )·Rˆ(x)·(z z ·k−2 ⊗1)−∆(e )·Rˆ(x)] 1 0 1 0 ×(z z ·k−2 ⊗k−1 −x(1⊗k))−1, (8) 0 1 where for simplicity we use the notations: ∆(f ) := π (1)⊗π (x)(∆(f )),∆(e ) := π (1)⊗π (x)(∆(e )). i 2 1 i i 2 1 i If π and π are the highest weight representations with highest vectors v(1) 1 2 c andv(2) respectively andV ⊗V decomposesintodirect sumofpairwisenonequiv- c 1 2 alent irreducible representations, then Rˆ(x)(v(1) ⊗v(2)) = α·(v(2) ⊗v(1)), where c c c c α ∈ C can be equated to 1. In this case we can use (8) to obtain the recursive formula for Rˆ(x)(fr1v(1) ⊗fr2v(2)). In fact, using the equalities: c c 1⊗k2 fr1v(1) ⊗fr2v(2) = λ q−2r2 fr1v(1) ⊗fr2v(2) , c c (2) c c k(cid:16)−2 ⊗1(cid:17)(cid:16)fr1v(1) ⊗fr2v(2)(cid:17) = λ−1 q2r2 f(cid:16)r1v(1) ⊗fr2v(2) (cid:17), c c (1) c c (cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17) we obtain from (8): Rˆ(x)(fr1 ⊗fr2)v(1) ⊗v(2) = xλ2 q−2(r2−1)∆(f )−∆(e ) Rˆ(x) c c (2) 1 0 ×(fr1 ⊗fr2−1)(q−2r2−r1−2xz−h1k ⊗k2 −qr1z k−1 ⊗1)−1v(1i) ⊗v(2) 1 0 c c (9) Rˆ(x)(fr1 ⊗fr2)v(1) ⊗v(2) = λ−2 q2(r1−1)(z z ⊗1)∆(f )−∆(e ) c c (1) 0 1 1 0 ×Rˆ(x)(fr1−1 ⊗fr2)(q2r1+r2−2hz z k−2 ⊗k−1 −q−r2x·1⊗k)−1v(1)i⊗v(2) 0 1 c c Here λ (i = 1,2) are values of k on highest vectors v(i). Using this and denoting (i) c by z(i) the values of central elements z (j = 0,1) on V , we obtain by induction j j i from (10): (z(1))−r1 R(x)(fr1 ⊗ fr2)v(1) ⊗v(2) = 1 c c r1+r2−1 qι(λ λ )−1z(1) −q−ι(λ λ )xz(1)−1 ι=0 1 2 0 1 2 c r1−1 Q (cid:16) (cid:17) × λ−1qι1z(2)z(2)∆(f )−λ q−ι1∆(e ) (1) c 1 1 (1) 0 ιY1=0h i r2−1 × λ−1qι2∆(e )−xλ q−ι2∆(f ) (v(2) ⊗v(1)) (10) (2) 0 (2) 1 c c ιY2=0h i 3 It can be proved that this R-matrix commutes also with ∆(f ) and ∆(e ) 0 1 (As in example 3 the condition on parameters of representations follows from the equality of central elements in tensor products). The formula (10) generalizes the results of [4] in case of spectral parameter depending R-matrix and any highest weight representation of quantum algebra sl (2,C). In the particular case of x = 1,z = 1, the formula (10) determines the q i reflection symmetric intertwiner for sl (2,C), constructed in [4]. Note, however, q that we used generators, slightly differing by ones of sl (2,C), so there are some q differences in formulas. We kept track by the method of [4] in our derivation. Consider now some examples. Example 1. First we consider the case of general q. We put z = 1. The i irreducible representations of quantum algebra for general q are the deformations of representations of corresponding classic algebra and characterized by half in- teger highest weight. As it was proved by in ref. [3], Rˆ(x) exists and is unique. It also satisfies Yang-Baxter equations on π (xy)⊗π (x)⊗π (1) [1, 2]: 1 2 3 (Rˆ(x)⊗id)(id⊗Rˆ(xy)(Rˆ(y)⊗id) = (id⊗Rˆ(y))(Rˆ(xy)⊗id)(id⊗Rˆ(x)) (11) From the uniqueness of R-matrix it follows that constructed above operator (10) coincides with Jimbo’s R-matrix in [2]. The later, which had been repre- sented by means of projecting operators, can be also represented in the explicit form: Rˆ(x) v(j1) ⊗v(j2) = r1+r2−1 qι−j1−j2 −xq−(ι−j1−j2) −1 r1 r2 ιY=0 (cid:16) (cid:17) r1−1 × qι1−j1∆(f )−q−(ι1−j1)∆(e ) 1 0 ιY1=0h i r2−1 × qι2−j2∆(e )−xq−(ι2−j2)∆(f ) v(j2) ⊗v(j1) , (12) 0 1 c c ιY2=0h i(cid:16) (cid:17) where v(j) := fkv(j). k c Example 2. Let q be a root of 1 and N be the minimal integer, such that qN = 1. For simplicity we consider the case of odd N only. Then eN,fN,kN lie in the center of sl (2,C) [10, 11]. It follows from this that in this case appear new q irreducible representations that are called periodic or cyclic [10, 11]. They are N- dimensional and parametrized by 3 complex numbers. The factorized S-matrix of sl (N,C)-generalized chiral Potts model has been found first by Bazhanov et q all in ref.[16]. In the article [6] Date, Jimbo, Miki and Miwa by means of [16] constructed intertwining operators (6) for the minimal cyclic representation of the gˆl (N,C) affine algebra with central extension. They also proved the Yang- q Baxter equations between them for parameters of corresponding representations lying on some algebraic curve. 4 Recall that the minimal cyclic representation of sl (2,C) has the following q form: π (e)w = yqµ1+µ0[2µ −m] w π (k)w = qµ1−µ0−m−1w ξ m 1 q m−1 ξ m m (13) π (f)w = y−1q−µ1−µ0[2µ +m+2] w π (z ) = ci (i = 0,1), ξ m 0 q m+1 ξ i ci+1 wherec := c ,w := w ,m = 0,1...N−1,andξ := qµ0,qµ1,y, c1 isaparameter 2 0 N 0 c2 of the representation. (cid:16) (cid:17) After affinization of(13) andimposing comultiplication (3)we obtainminimal representation of sˆl (2,C), considered in [6]. (Here we use slightly different basis q and comultiplication). Recall that inertwining operator: V ⊗V′ ⇒ V′⊗V exist ξ ξ ξ ξ if ξ and ξ′ lie on algebraic curve, which is parametrized by 2 complex varieties s and s′: uN = s−λ vN = s−µ i i i i (14) u′N = s′ −λ v′N = s′ −µ (i = 0,1), i i i i i where: u q2µi v′ 1 u′ i = , i−1 = q2µic , x = ωN i, u′ c v i i i v′ i i i−1 i (Here x := y,x := x). The moduli parameters γ := (ω ,λ ,µ ) are fixed for 1 0 y i i i different representations (µ is not the same variable as in (13)). i Ifµ = 0thenπ (e)w = 0,andperiodicrepresentationconvertsintosemiperi- 1 ξ 0 odic one. It follows from (14) that µ = λ in γ in this case. So, the R-matrix, 0 1 which intertwines semiperiodic representations, is a particular case of R-matrix, intertwining periodic ones. 1 (For µ = λ , as it can be verified, there are no 0 1 singularities in R-matrix.) Thus the formula (10) for R matrix can be used for semiperiodic representations. In the case of x = c = z = 1 we obtain R-matrix, i i considered in [4]. Example 3. Consider now the tensor product V ⊗V , where by V we denote ξ j ξ the semiperiodic representation with parameter ξ = (qµ,λ) and by V the spin j j one, 2j +1 < N. We take the following basis for V [12]: ξ π (e)w(ξ) = [2µ−m+1] [m] w π (k)w(ξ) = qµ−mw ξ m q q m−1 ξ m m (15) π (f)w(ξ) = w ,m = 0,..N −2 π (f)w(ξ) = λw ξ m m+1 ξ N−1 0 We shall show that Rˆ(x), obtained using (10) for all z(i) = 1, is an intertwiner j from V ⊗V into V ⊗V of sˆl (2,C) (see (6)). ξ j j ξ q 1We use the unicity arguments of R-matrix which follows from the irreducibility of tensor product in sˆl (2,C) q 5 First we note that Rˆ(x) commute with ∆(f ) and ∆(e ). To see this it’s 1 0 enough to check: N−1 qι−µ∆(f )−q−(ι−µ∆(e ) (v(j) ⊗v(µ)) ι=0 1 0 c c = λ(v(j) ⊗v(µ)) (16) Q h N−1(qι−µ−j −xq−(ι−µi−j)) c c ι=0 Q 2j qι−j∆(e )−xq−(ι−j)∆(f ) (v(j) ⊗v(µ)) ι=0 0 1 c c = 0 (17) Q h 2j (qι−µ−j −xq−(ι−µ−ij)) ι=0 (16) can be proven usingQcommutativity of ∆(f ) and ∆(e ), Gauss binomial 1 0 formula: N−1 N N (1−q2ιa) = (−1)ι qι(ι−1)aι (= 1+aNfor our q), 1 " # ι=0 ι=0 q Y X the equations ∆(f )N = fN ⊗k−N +kN ⊗FN,∆(e )N = fN ⊗kN +xNk−N ⊗fN, 1 0 the fact that fN| ≡ 0, k2N| ≡ id and (14). To prove (17) we consider it for Vj Vj general q. Because of existence and uniqueness of an R-matrix for half integer values of µ the equation (17) is valid for such µ. It is evident, that the numerator in (17) can be represented as a linear combination of v(j) ⊗v(µ),p+m = 2j +1, p m with the coefficients, which are µ-independent polynomials on qµ. Because of vanishing such polynomials on infinite numbers of points qhalfinteger they are vanished trivially. So, (17) is valid. Now recall the fact that V ⊗V as a sl (2,C) module is a fully reducible. It ξ j q decomposes into 2j + 1 semiperiodic representations V (i),i = 1..2j + 1 ([12]). ξ From this it follows that ∆(e ) and ∆(f ) also commute with Rˆ(x). Indeed, if 1 0 vµi) is highest weight of V (i) then considering the half integer values of µ as c ξ above it can be proven that ∆(f )Rˆ(x)v(µi) = ∆(e )Rˆ(x)v(µi) = 0. Then: 0 c 1 c ∆(e )Rˆ(x)∆(fk)v(µi) = ∆(e )∆(fk)Rˆ(x)v(µi) = Rˆ(x)∆(e )∆(f )v(µi) 1 1 c 1 1 c 1 1 c ∆(f )Rˆ(x)∆(ek)v(µi) = ∆(f )∆(ek)Rˆ(x)v(µi) = Rˆ(x)∆(f )∆(e )v(µi) 0 0 c 0 0 c 0 0 c So, we proved the Rˆ(x) is an intertwiner from V ⊗V to V ⊗V of affine sˆl (2,C). ξ j j ξ q The Yang-Baxter equations on V ⊗V ⊗V can be proven using the irreducibility ξ j j of tensor product as in [6]. At the end we write the components of Rˆ(x) : V ⊗ V ⇒ V ⊗ V for N = ξ j j ξ 3,j = 1. 2 [2µ] (−µ+ 1) Rˆ00(x) = 1 Rˆ10(x) = − Rˆ01(x) = 2 x 00 10 (−µ− 1) 10 (−µ− 1) 2 x 2 x (µ− 1) x Rˆ10(x) = 2 x Rˆ01(x) = Rˆ02(x) = − 01 (−µ− 1) 01 11 (−µ− 1) 2 x 2 x 6 (µ− 3) x(1) Rˆ11(x) = 2 x Rˆ02(x) = x 11 (−µ− 1) 20 (−µ− 1) (−µ+ 3) 2 x 2 x 2 x x[µ− 3]+(µ− 1) (1−x) αx Rˆ11(x) = 2 2 x Rˆ00(x) = 20 (−µ− 1) (−µ+ 1) 21 (−µ− 1) 2 x 2 x 2 x x[2µ−3]+(µ− 3) (−µ+ 1) Rˆ12(x) = 2 x 2 x 21 (−µ− 1)(−µ+ 3) 2 2 x Here we omitted the index q in [n] and used the notation: q qn −xq−n (n) := x q −q−1 We would like to thank V.Bazhanov for a comments. Note added. When this work was finished and submited for a publication B.M.McCoy kindly informed us that similar results was obtained by I.T.Ivanov and D.B.Uglov in the article Phys.Lett. A 167 (1992) 459, for which we express our gratitudes to him. References [1] V.G. Drinfeld, Quantum Groups, ICM proceedings, p.798-820, New York: Berkeley 1986. [2] M. 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