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SPhT-94-146 On the universal R-matrix of U sl q 2 at roots of unity 1 c 5 T.Hakobyan 2 and A.Sedrakyan 3 9 9 1 Service de Physique Theorique de Saclay, n F-91191, Gif-sur-Yvette, France. a J 4 1 1 v 6 1 Abstract 0 1 WeshowthattheactionofuniversalR-matrixofaffineU sl quan- 0 q 2 tum algebra, when q is a root of unity, can be renormalized by some 5 9 scalar factor to give a well defined nonsingular expression, satisfying / g Yang-Baxter equation. It reduced to intertwining operators of all rep- l resentations, correspondingto ChiralPotts, if the parameters of these a - representations lie on well known algebraic curve. q : We also show that affine Uqsl2 for q is a root of unity form the v autoquasitriangular Hopf algebra in the sence of Reshetikhin. i X r a 1This work is supported by NATO linkage grant LG 9303057 2Permanent address: Yerevan Physics Institute, 375036 Yerevan, Armenia, e-mail: [email protected] [email protected] 3Permanent address: Yerevan Physics Institute, 375036 Yerevan, Armenia, e-mail: [email protected] [email protected] 1 1 Introduction The intertwining operators of quantum groups ([1, 2, 3, 4]) lead to solutions of Yang-Baxter equation, which play the crucial role in two dimensional field theory and integrable statistical systems ([4, 5]). It is well known that the most of them can be obtained from the universal R-matrix ([1]) for a given quantum group: the solutions of spectral parameter dependent Yang-Baxter equation can be obtained from the universal R-matrix of affine quantum groups ([6]) and the solutions of non-spectral parameter dependant Yang-Baxter equations can be obtained from the universal R-matrix of finite quantum groups. The situation is not the same for the case, when the parameter q of quantum group is a root of unity. In this case the center of quantum group is larger and new type of representations appear, which have noaclassical analog([5,7,8,9]). It wasshown in [10, 11] that the cyclic representations lead to solutions of Yang-Baxter equation with a spectral parameter, lying on some algebraic curve. These solutions correspond toChiral Potts Model([12,13,14]) andits generalizations (for quantum groups U sl ). q n The formal expression of the universal R-matrix fails in this case: it have b a singularities at q is a root of unity. Recently in [15] Reshetikhin introduced the notion of autoquasitriangular Hopf algebra to avoid these singularities. He treated the U sl case. q 2 The main goal of this paper is to show that after suitable renormalization by scalar factor the universal R-matrix produces R-matrices for concrete representations. In section 2, we consider the universal R-matrix on Verma modules of U sl for q is a root of unity. We prove that it is well defined and make a q 2 connection with the R-matrix of autoquasitriangular Hopf algebra, founded by Reshetikhin. In section 3 we consider the algebra U sl at roots of unity. We found q 2 the central elements of its Poincaré-Birkoff-Witt (PBW) basis, generalizing b the results of [9] for affine case. It appears that new type of central elements appear for some imaginary roots, which have no analog for finite quantum groups. After this we prove the autoquasitriangularity of U sl , generalizing q 2 the results of [15]for affine case. Then we consider the actionof affine univer- b sal R-matrix on U sl - and U sl -Verma modules. On U sl -Verma modules q 2 q 2 q 2 b b 2 it is well defined. For U sl -Verma modules (evaluation representation) we q 2 renormalize its expression by scalar factor to exclude the singularities. The rest part lead to solutions of infinite dimensional spectral parameter dependent Yang-Baxter equation. We showed that under the certain condition this R-matrix can be restricted to semicyclic representations, giving the Bolts- mann weights of Chiral Potts model, corresponding to such type representations, which was considered in [16, 17, 18, 19]. The condition, mentioned above, is on the parameters of representations: they must lie on well known algebraic curve. It is integrability condition of Chiral Potts model. In the last section we made same type suggestion for the cyclic representations. 2 The U sl case q 2 2.1 The Universal R-matrix on Verma Modules at root of unity The quantum group U sl is a [q,q−1]-algebra, generated by the elements q 2 E, F, K with the following relations between them K −K−1 [K,K−1] = 0 [E,F] = (1) q −q−1 KEK−1 = q2E KFK−1 = q−2F, On U sl there is a Hopf algebra structure with comultiplication ∆ : q 2 U sl → U sl ⊗U sl defined by q 2 q 2 q 2 ∆(K) = K ⊗K ∆(E) = E ⊗1+K−1 ⊗E ∆(F) = F ⊗K +1⊗F We denote K = qH, q = e¯h, as usually, and consider the [[h]]-algebra U sl with the same defining relations. U sl is a quasitriangular Hopf al- h 2 h 2 gebra, i.e. it possess the universal R-matrix R ∈ U sl ⊗ U sl connecting h 2 h 2 the comultiplication ∆ with the opposite comultiplication ∆′ = σ◦∆, where σ(x⊗y) := y ⊗x: ∆′(a) = R∆(a)R−1, ∀a ∈ U sl (2) q 2 3 It satisfies the quasitrialgularity relations (∆⊗1)R = R R (1⊗∆)R = R R (3) 13 23 13 12 and Yang-Baxter equation ([1]) R R R = R R R (4) 12 13 23 23 13 12 Here we used the usual notation: if R = a ⊗b , a ,b ∈ U sl , then i i i i i q 2 P R = a ⊗b ⊗1 R = a ⊗1⊗b R = 1⊗a ⊗b 12 i i 13 i i 23 i i i i i X X X The explicit expression of R in terms of formal power series is R = expq−2((q −q−1)(E ⊗F))q21H⊗H (5) where the q-exponent is defined by exp (z) = zn , (z) := 1−qn. q n≥0 (z)q! q 1−q Note, that to be precise, U sl is not a quasitriangular Hopf algebra, q 2 P because the term qH⊗H in (5) do not belong to U sl ⊗ U sl , but it is an q 2 q 2 autoquasitriangular Hopf algebra ([15]). The latter is a Hopf algebra A, where the condition (2) is generalised by ∆′ = Rˆ(∆), where Rˆ is an automorphism of A⊗A (not inner, in general). So, although (5) is ill defined on U sl , but the action q 2 Rˆ(a) = RaR−1, (6) where a ∈ U sl ⊗U sl is till well defined. q 2 q 2 For two representations of U sl V and V one can consider two U sl - q 2 1 2 q 2 actions on V ⊗ V by means of both comultiplications ∆ and ∆′. If R is 1 2 defined on V ⊗ V , then both ∆- and ∆′-actions are equivalent via inter- 1 2 twining operator R = R| . For general q the restriction of (5) on V1⊗V2 V1⊗V2 tensor product of two irreducible representations (in general, of any highest weight representations) is well defined. And all solutions of Yang-Baxter equation (4), having U sl - symmetry in sense of (2) can be obtained from q 2 the universal R-matrix (5) in such way. 4 The situation is different for q being a root of unity. In this case the singularities appear in the formal expression of R. Recall that for q = exp 2πi the elements FN, FN, KN, where N = N′ N′ for odd N′ and N = N′ (cid:16)for (cid:17)even N′, belong to the center of U sl . In 2 q 2 irreducible representations they are multiples of identity. Recall that every N-dimensionalirreducible representation ischaracterized bythevaluesx,y,z of these central elements (and also by the value of q-deformed Casimir c = Kq+K−1q−1 + EF, which for the fixed x,y,z can have in general N discrete q−q−1 values ([9])). Although the expression of R-matrix (5) of U sl has singularities for q 2 qN′ → 1 in all terms 1 En ⊗ Fn for n ≥ N, its restriction on tensor (n)q−2! product of Verma modules M ⊗M is well defined. λ1 λ2 Recall that M is formed by the basic vectors vλ, m = 0,1,..., satisfying λ m Evλ = 0 Fv = vλ Hv = λv , λ ∈ C 0 m m+1 0 0 To consider the action of R on M ⊗M we use the formula, which can λ1 λ2 be obtained from the defining relations (1) ([9]): min(n,s) j n s [En,Fs] = [j]!Fs−j [H +j −n−s+r] En−j, " j #" j #   j=1 r=1 X Y   where a [a]! qn −q−n = and [n] = [n] = . " b # [b]![a−b]! q q −q−1 So, for n > s En vλ = 0 and for n ≤ s: (n)q−2! s (nE)qn−2!vsλ = qn(n2−1) " ns #r=n1[λ−s+r]vsλ−n Y s The q-binomial has a non-infinity limit for qN′ → 1. So, n " # R(vsλ1 ⊗vsλ′2) = s q(λ1−2s)2(λ2−2s′)qn(n2−1)(q −q−1)n ns " # n=0 X n × [λ −s+r]vλ1 ⊗vλ2 (7) 1 s−n s′+n r=1 Y 5 is well defined for q is a root of unity. 2.2 The connection with Reshetikhins R-matrix of autoquasitriangular Hopf algebra This R-matrix can be presented in another form by using resent results of Reshetikhin ([15]). Heusedanasymptoticformulaforq-exponent inthelimit ′ qN → 1 to bring out multiplicatively singularities fromexp ((q−q−1)E⊗F). q The expression of universal R-matrix in this limit then acquires the form: 1 R = exp 2N2h¯Li2(EN ⊗FN) (1−EN ⊗FN)−21 (cid:18) (cid:19) N−1 × (1−εmE ⊗F)−Nmq12H⊗H ·O(h¯) m=0 Y x Here q = exp(h¯)ε, ε = exp(2πi) and Li (x) = − ln(1−y)dy is a dilogarit- N′ 2 y 0 mic function. R Recall that although the elements EN FN , and H (N) ! (N) ! q−2 q−2 don’t belong to U sl for qN′ → 1, but their adjoint actions q 2 ad(x)a = [x,a], Ad(exp(x))a = exp(x)aexp(−x) = exp(ad(x))a on U sl are well defined in this limit and give rise to some derivations ([9]). q 2 Let’s denote them by e, f and h correspondingly. The element 1 Li (EN ⊗ FN) in the exponent of (8) in the adjoint 2¯hN2 2 representation also acts on U sl ⊗U sl as a derivation in the limit h¯ → 0 . q 2 q 2 It can be expressed by means of the derivations e and f as follows: 1 ln(1−EN ⊗FN) limad Li (EN ⊗FN) = c ¯h→0 2h¯N2 2 N′ EN ⊗FN (cid:18) (cid:19) × (e⊗FN +EN ⊗f), 6 where −(1−ε−2)−N for odd N (N′ = N) c = (8) N′ (−1)N(1−ε−2)−N for even N (N′ = 2N) ( Note, that Ad(ε21H⊗H) = ε12(h⊗H+H⊗h) = K1⊗21h ⊗K12h⊗1 is well defined in the adjoint representation. ˆ So, one can write down the automorphism R (6) in the limit h¯ → 0, obtained in [15], in the following form 4 N−1 Rˆ = Ad (1−εmE ⊗F)−Nm mY=0 (cid:16) (cid:17) ln(1−EN ⊗FN) × exp −(1−ε−2)−N (e⊗FN +EN ⊗f) (9) EN ⊗FN ! × K1⊗21h ⊗K21h⊗1 Let us nowconsider the restriction of(9)onthe quotient algebraobtained from U sl by factorisation on the ideal, generated by EN, i.e. impose EN = q 2 0. Although this ideal is not stable with respect to derivations e,f,h, it is ˆ easy to see that it is stable with respect to R. Moreover, the left U sl ⊗U sl -module q 2 q 2 I = (U sl ⊗I ) (I ⊗U sl ) λ1,λ2 q 2 λ2 λ1 q 2 M is also stable with respect to Rˆ. Here we denoted by I the left U sl - λ q 2 module, generated by E and (K − ελ). This fact allows to restrict (9) on Verma modules, because we have the left U sl -module equivalence q 2 ∼ (U sl ⊗U sl )/I = M ⊗M q 2 q 2 λ1,lt λ1 λ2 So, one can derive from (9) the restriction of R on this factormodule is given by the multiplication on 5 4For quantum groups one can introduce 4 equivalent comultiplications: ∆q, ∆′q, ∆q−1, ∆′ [6]. In [15] the comultiplication ∆′ had been used as a basic one. So, R-matrix, q−1 q−1 used there, is R−1 in our notations anddiffers from∆ -case used here by permutation of q−1 q q-exponent and q21H⊗H. 5Note that both h and e are well defined on M in contrast to f λ 7 N−1 R = (1−εmE ⊗F)−mN mY=0(cid:16) (cid:17) × exp (1−ε−2)−N(e⊗FN) K1⊗21h ⊗K12h⊗1 (10) (cid:16) (cid:17) This expression is another form of expression of universal R-matrix (5) on Verma modules and coincide with (7). 3 The case of affine U sl q 2 c 3.1 The PBW basis and the universal R-matrix The affine quantum universal enveloping algebra U sl is a [q,q−1]-Hopf al- q 2 gebra, generated by elements E := E , F = F , K = qHi, i = 0,1 and qd i αi i αi bi with defining relations ([2]): [qHi,qHj] = 0 qdqHi = qHiqd [E ,F ] = δ [H ] i j ij i q qHiE q−Hi = qaijE qHiF q−Hi = q−aij qdE q−d = qE (11) j j j 1 1 qdF q−d = q−1E qdE q−d = E qdF q−d = F 1 1 0 0 0 0 (ad E )1−aijE = 0 (ad F )1−aijF = 0 q i j q i j and comultiplication ∆(qHi) = qHi ⊗qHi ∆(qd) = qd ⊗qd ∆(E ) = E ⊗1+q−Hi ⊗E ∆(F ) = F ⊗qHi +1⊗F i i i i i i Here we use the q-deformed adjoint action (ad x)y := x ys(xi), where q i i ∆(x) = x ⊗xi and s : U sl → U sl is antipode of U sl , defined by i i q 2 q 2 qP 2 P s(E ) = −K E bs(F ) =b−F K−1 s(K ) = Kb−1 i i i i i i i i ˆ Also we denoted by a the Cartan matrix of affine sl(2) Lie algebra ij 2 −2 a = ij −2 2 ! 8 Let’s denote by c the central element c = H +H . 1 2 Define on U sl an antiinvolution ι by q 2 ι(Kb) = K−1 ι(E ) = F ι(F ) = E ι(q) = q−1 i i i i i i As above, denote by U sl the [[h]]-algebra with the same relations but h 2 the elements H instead of qHi. i The PBW basis of U sl bis formed by elements H , d, E , F , E′ h 2 i αi+nδ αi+nδ nδ and F′ , which are inductively defined by the relations nδ b E = (−1)n(ad )nE E = (ad )nE α0+nδ Eδ′ 0 α1+nδ Eδ′ 1 E′ = [2]−1(E E −q−2E E ) (12) nδ α0+(n−1)δ 1 1 α0+(n−1)δ F = ι(E ) F′ = ι(E′ ) αi+nδ αi+nδ nδ nδ Theexpression oftheuniversalR-matrixofU sl issimpler ifoneredefine h 2 E′ and F′ by means of Schur polynomials ([6]): nδ nδ b E′ = (q2−q−2) pi−1(E )p1...(E )pm nδ 0<k1<...<km p1!...pPm! k1δ kmδ k1p1+...P+kmpm=n F′ = ι(E′ ) nδ nδ In order to rewrite all the relations between (12) in compact form it is suitable to change slightly the basis as follows: E = (−1)nq−2nx− k−1 E = (−1)nq−(c+2)nx+ (13) α0+nδ n+1 α1+nδ n E′ = (−1)nq−2cn−2nψ k−1 E = (−1)nq−(2c+2)na nδ q2 −q−2 n nδ [2] n Then the elements x±,(n ∈ Z), a ,(k ∈ Z, k 6= 0), ψ ,ϕ , (m ≥ 1) n k m −m and ψ = ϕ−1 = k satisfy the following relations 0 0 [a ,a ] = δ [2m][mc] [a ,k] = 0 m n m,−n m m kx±k−1 = q±2x± [a ,x±] = ±[2m]q∓|m|cx± m m m n m 2 m+n 9 x± x± −q±2x±x± = q±2x±x± −x± x± (14) m+1 n n m+1 m n+1 n+1 m [x+,x−] = 1 (qc(m−n)ψ −q−c(m−n)ϕ ) m n q−q−1 2 m+n 2 m+n ∞ ψ z−m = kexp((q −q−1) ∞ a z−m) m=0 m m=1 m ∞ ϕ zm = k−1exp(−(q −q−1) ∞ a zm) Pm=0 −m P m=1 −m P P These relations had been introduced by Drinfeld in [20] and define another realization of affine algebra U sl . The antiinvolution ι in this notations is q 2 ι(x±) = x∓ ι(ψ ) =b ϕ ι(a ) = a ι(q) = q−1 n −n n −n n −n We choose the normal ordering of positive root system ∆ of U sl as + q 2 follows: b α ,α +δ,...,α +nδ,...δ,2δ,...,nδ,...,α +nδ,...,α +δ,...,α (15) 0 0 0 1 1 1 Then the universal R-matrix has the form [6]: R = exp ((q −q−1)(E ⊗F ))  q−2 α0+nδ α0+nδ  n≥0 Y  nE ⊗F  nδ nδ × exp (16) q2n −q−2n ! n>0 X ×  expq−2((q −q−1)(Eα1+nδ ⊗Fα1+nδ))q21H0⊗H0+c⊗d+d⊗c, n≥0 Y   where the product is given according to the normal order (15). 3.2 U sl at roots of unity q 2 For q beingca root of unity (q = ε, ε = e2Nπ′i) the center of Uqsl2 is enlarged by the N-th power of the root vectors, as for finite quantum groups: b [EN,x] = 0 [FN,x] = 0 [KN,x] = 0, (17) γ γ i where γ ∈ ∆ := {α +nδ,mδ|n ≥ 0,m > 0} and x ∈ U sl . + i ε 2 b 10