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A New Solution to the Star–Triangle

arXiv:hep-th/9108017v1 23 Aug 1991UGVA–DPT 1991/08–739CERN–TH.6200/91A New Solution to the Star–TriangleEquation Based on Uq(sl(2)) atRoots of UnitC´esar G´omez∗†D´ept. Physique Th´eorique, Universit´e de Gen`eve, CH–1211 Gen`eve 4Germ´an Sierra†Theory Division, CERN, CH–1211 Gen`eve 23ABSTRACTWe find new solutions to the Yang–Baxter equation in terms of the intertwinermatrix for semi-cyclic representations of the quantum group Uq(sℓ(2)) with q =e2πi/N.

These intertwiners serve to define the Boltzmann weights of a lattice model,which shares some similarities with the chiral Potts model. An alternative interpre-tation of these Boltzmann weights is as scattering matrices of solitonic structureswhose kinematics is entirely governed by the quantum group.

Finally, we considerthe limit N →∞where we find an infinite–dimensional representation of the braidgroup, which may give rise to an invariant of knots and links.∗Supported in part by the Fonds National Suisse pour la Recherche Scientifique.†Permanent address: Instituto de F´ısica Fundamental, CSIC, Serrano 123, Madrid 28006.

1. IntroductionThe N–chiral Potts model [1, 2] is a solvable lattice model satisfying the star–trianglerelation.Their Boltzmann weights are meromorphic functions on an algebraic curve ofgenus N3 −2N2 + 1.

These models are the first example of solutions to the Yang–Baxterequation with the spectral parameters living on a curve of genus greater than one. Recentdevelopments [3, 4] strongly indicate that quantum groups at roots of unit [5] characterizethe underlying symmetry of these models.

More precisely, it was shown in reference [4] thatthe intertwiner R matrix for cyclic representations of the affine Hopf algebra Uq( ˆsℓ(2)) at qan N-th root of unit, admit a complete factorization in terms of the Boltzmann weights ofthe N–chiral Potts model. In this more abstract approach, the spectral parameters of thePotts model are represented in terms of the eigenvalues of the central Hopf subalgebra ofUq( ˆsℓ(2)) (q = e2πi/N) with the algebraic curve fixed by the intertwining condition.In a recent letter [6], we have discovered a new solution to the Yang–Baxter equation,with the spectral parameters living on an algebraic curve.

This solution was obtained asthe intertwiner for semi-cyclic representations of the Hopf algebra Uq(sℓ(2)) for q a thirdroot of unit and it shares some of the generic properties of factorizable S–matrices. In thispaper, we generalize the results of [6] for qN = 1, N ≥5.

The case N = 4 is special and isstudied separately in [7]. Inspired by the chiral Potts model, we propose a solvable latticemodel whose Boltzmann weights are identified with the Uq(sℓ(2)) intertwiners for semi-cyclicrepresentations.

Since we are working with Uq(sℓ(2)) and not its affine extension, and basedon the very simple structure of the spectral manifold, we conjecture that the models wedescribe correspond to critical points.The organization of the paper is as follows. In section 2 we review the chiral Potts modelfrom the point of view of quantum groups.

In section 3 we give the intertwiner for semi-cyclicrepresentations of Uq(sℓ(2)) for N ≥3 (qN = 1). In section 4 we construct a solvable latticemodel whose Boltzmann weights are given by the previous intertwiner.In section 5 westudy the decomposition rules of tensor products of semi-cyclic representations, and finallyin section 6 we consider the N →∞limit of the intertwiners of semi-cyclic representationswhich leads to an infinite–dimensional representation of the braid group.

This representationalso satisfies the Turaev condition for defining an invariant of knots and links.2. Chiral Potts as a model for affine Uq( ˆsℓ(2)) intertwinersTo define the N–chiral Potts model we assign to the sites of a square lattice two differentkinds of state variables: a ZN variable m (= 0, 1, .

. .

, N −1) and a neutral variable ∗. On

these variables one defines the action of the group ZN as σ(m) = m + 1 and σ(∗) = ∗. Theallowed configurations for two adjacent sites of the lattice are of the kind (∗, m), i.e.

withone of the site variables the ZN neutral element ∗. The rapidities p, q, .

. .

are associatedwith each line of the dual lattice (figure 1).Figure 1.1. The chiral Potts model: pi and qi are rapidities, whereasm, n, .

. .

∈ZN.Correspondingly, there are two types of Boltzmann weights represented graphically asWpq(m −n) =W pq(m −n) =(1)The star–triangle relation of the model isNXd=1W qr(b −d)Wpr(a −d)W pq(d −c) = RpqrWpq(a −b)W pr(b −c)Wqr(a −c)(2)Representing the vector rapidity by (ap, bp, cp, dp) ∈C, it is found that the star trianglerelations of the model restrict the rapidities to lie on the intersection of two Fermat curves:aNp + k′bNp = kdNpk′aNp + bNp = kcNpk2 + k′2 = 1(3)In reference [8] the previous model has been interpreted as describing a scattering ofkinks defined as follows. To each link of the lattice one associates a kink operator which2

can be of the type K∗,n(p) or Kn,∗(p).This kink operators can be interpreted, in thecontinuum, as configurations which interpolate between two extremal of some potential `a laLandau–Ginzburg, and which move with a rapidity p.The Boltzmann weights of the lattice model are used in this interpretation to define theS–matrix for these kinks:S |Kn∗(q)K∗m(p)⟩= Wpq(m −n) |Kn∗(p)K∗m(q)⟩S |K∗m(q)Km∗(p)⟩= W pq(m −n) |K∗n(p)Kn∗(q)⟩(4)These models can be characterized very nicely in connection with the quantum affine exten-sion of sℓ(2), namely Uǫ(ˆsℓ(2)) with ǫ = e2πi/N. First of all, the finite–dimensional irreps ofUǫ(ˆsℓ(2)) are parametrized by the eigenvalues of the central subalgebra which is generated,in addition to the Casimir, by EN ′i, F N ′iand KN ′i, where i = 0, 1 and N′ = N if N is odd,and N′ = N/2 if N is even.

The representations where xi = EN ′i, yi = F N ′iand zi = KN ′iare all different from zero, have dimension N′ and are called cyclic [4] or periodic [9] repre-sentations. The intertwining condition for the tensor product of cyclic representations forcethese parameters xi, yi, zi to lie on an algebraic curve which factorizes into two copies of thecurve (3).

The corresponding intertwiner R–matrix admits a representation as the productof four Boltzmann weights of the chiral Potts model, actually two of them are chiral (W)and the other two antichiral (W) (see figure 2 and reference [4] for details).Figure 1.2. The R–matrix for Uq( ˆsℓ(2)) and its chiral Potts interpretationas a product of four Boltzmann weights.The need of four Boltzmann weights can be intuitively understood comparing the struc-ture of indices of a generic R–matrix Rr′1r′2r1r2(ξ1, ξ2) with the Boltzmann weights (1).

In fact,using only four Boltzmann weights and therefore two independent rapidities one has enoughdegrees of freedom to match the indices of the affine R–matrix in terms of lattice variables.This is the formal reason for using the affine Hopf algebra to describe the chiral Potts model.Returning to the kink interpretation, one is associating to each N′–irrep of Uǫ(ˆsℓ(2)) atwo–kink state with two different rapidities.3

3. Intertwiners for semi-cyclic irreps of Uǫ(sℓ(2)) with ǫ = e2πi/N, N ≥3In reference [6] the intertwiner for semi-cyclic representations in the case N = 3 wasconsidered.

We shall give now the result for N ≥3. The case N = 4, i.e.

q2 = 1, has morestructure and is analyzed separately [7].Let us first fix notation, essentially as in [5]. We consider the Hopf algebra Uǫ(sℓ(2))with ǫ = e2πi/N (we use the letter ǫ instead of q in order to distinguish the case of q a rootof unit), generated by E, F and K subject to the relationsEF −ǫ2FE = 1 −K2KE = ǫ−2EKKF = ǫ2FK(5)and co-multiplication∆E = E ⊗1 + K ⊗E∆F = F ⊗1 + K ⊗F∆K = K ⊗K(6)Notice that we do not include in the commutator between E and F the usual denominator1 −ǫ−2.When ǫ = e2πi/N the central Hopf subalgebra Zǫ is generated by x = EN ′, z = F N ′ andz = KN ′ where N′ = N (N odd) or N′ = N/2 (N even).

We shall be interested in the specialclass of representations for which x = 0 but y and z = λN ′ ̸= ±1 are arbitrary non–zerocomplex numbers. These are the so–called semi-cyclic or semi-periodic representations [6].Denoting by ξ the couple of values (y, λ) which characterizes a semi-cyclic representation,then the problem is to find a matrix R(ξ1, ξ2) which intertwines between the tensor productsξ1 ⊗ξ2 and ξ2 ⊗ξ1.

Let Vξ be the representation space associated with the semi-cyclicrepresentation ξ, which is spanned by a basis {er(ξ)}N ′−1r=0 , then the intertwiner R–matrix isan operator R : Vξ1 ⊗Vξ2 →Vξ2 ⊗Vξ1 :R(ξ1, ξ2)er1(ξ1) ⊗er2(ξ2) = Rr′1r′2r1r2er′1(ξ2) ⊗er′2(ξ1)(7)which satisfies the equationR(ξ1, ξ2)∆ξ1ξ2(g) = ∆ξ2ξ1(g)R(ξ1, ξ2)[∀g ∈Uǫ(sℓ(2)](8)where ∆ξ1ξ2(g) reflects the action of the quantum operator g on Vξ1 ⊗Vξ2:∆ξ1ξ2(g) (er1(ξ1) ⊗er2(ξ2)) = ∆ξ1ξ2(g)r′1r′2r1r2er′1(ξ1) ⊗er′2(ξ2)(9)4

Equation (8) then reads explicitely asRs1s2r′1r′2(ξ1, ξ2)∆ξ1ξ2(g)r′1r′2r1r2 = ∆ξ2ξ1(g)s1s2r′1r′2Rr′1r′2r1r2(ξ1, ξ2)(10)Hence our convention is that contracted indices are summed up in the SW–NE direction.An alternative way to write eq. (8) is in terms of the matrix R(ξ1, ξ2) = PR(ξ1, ξ2) whereP : Vξ1 ⊗Vξ2 →Vξ2 ⊗Vξ1 is the permutation map.

Note that Rr′1r′2r1r2 = Rr′2r′1r1r2. Equation (8)reads thenR(ξ1, ξ2)∆ξ1ξ2(g) =σ ◦∆ξ1ξ2(g)R(ξ1, ξ2)(11)where σ is the permutation map σ(a ⊗b) = b ⊗a of the Hopf algebra.

The universal form of(11) is in fact one of the defining relations of a quantum group, but as will be clear soon, weshall be working at the representation level, without assuming the existence of a universalR–matrix.After these preliminaries, the first important result that one derives from (8), whenapplied to an element g belonging to the center Zǫ of the algebra, is the following constrainton the possible values of ξ1 and ξ2:y11 −λN ′1=y21 −λN ′2= k(12)with k an arbitrary complex number different from zero. When N is odd, an explicit formof a semi-cyclic representation satisfying y = k(1 −λN) is given in the basis {er}N−1r=0 by∗Fer = k1/N(1 −λǫr)er+1Eer =1k1/N [r](1 + λǫr−1)er−1Ker = λǫ2rer(13)where [r] is the q–number[r] = 1 −ǫ2r1 −ǫ2(14)Notice that this representation is highest weight, with e0 the highest weight vector.∗In reference [6] we used another basis ˜er which is related to the one we use now by˜er = kr/Nr−1Yℓ=0(1 −λǫℓ)erwhere the generator F acts as F ˜er = ˜er+1 for 0 ≤r ≤N −2 and F ˜eN−1 = k˜e0.

In goingto the new basis er, we want to exploit the cyclicity of the generator F: in fact, we mayidentify er with er+N.5

It will be convenient for the rest of our computations to introduce the following numbers(λ)r1,r2 =Qr2−1ℓ=r1(1 −λǫℓ)if r1 < r2[r1]! [r2]!Qr1−1ℓ=r2(1 + λǫℓ)if r1 > r21if r1 = r2(15)in terms of whichF ner = (λ)r,r+n er+nEner = (λ)r,r−n er−n(16)They satisfy the obvious relation(λ)r1,r2(λ)r2,r3 = (λ)r1,r3r1 ≥r2 ≥r3r1 ≤r2 ≤r3(17)and the less obvious one which reflects the quantum algebra (5):(λ)r,r+1(λ)r+1,n −ǫ2(r+1−n)(λ)r,n(λ)n,n−1(λ)n−1,n= [r −n + 1][r + n + 1](λ)r,n(λ)r+n+1,r+n(λ)r+n,r+n+1(0 ≤n ≤r)(18)Using now the representation (13) we shall look for intertwiners R(ξ1, ξ2) satisfying theYang–Baxter equation(1 ⊗R(ξ1, ξ2)) (R(ξ1, ξ3) ⊗1) (1 ⊗R(ξ2, ξ3)) = (R(ξ2, ξ3) ⊗1) (1 ⊗R(ξ1, ξ3)) (R(ξ1, ξ2) ⊗1)(19)which guarantees a unique intertwiner between the representations ξ1⊗ξ2⊗ξ3 and ξ3⊗ξ2⊗ξ1.In reference [6], it was found (when N = 3) that an intertwiner R matrix satisfying theYang–Baxter equation (19) exists and is unique, and that in addition it satisfies the followingthree properties:i Normalization.R(ξ, ξ) = 1 ⊗1(20)ii Unitarity.R(ξ1, ξ2)R(ξ2, ξ1) = 1 ⊗1(21)iii Reflection symmetryR(ξ1, ξ2) = PR(ξ2, ξ1)P(22)Among these properties, the last one is the most important and, as we shall see, it isthe key to find solutions for N ≥3 (the case N = 4, thus N′ = 2, is singular in this crucialpoint, see [7]).

Explicitly, condition (22) readsRr′1r′2r1r2(ξ1, ξ2) = Rr′2r′1r2r1(ξ2, ξ1)(23)6

The strategy for finding solutions to the Yang–Baxter equation (19) is to look for in-tertwiners satisfying (8) and having the reflection symmetry (22). We also normalize the Rmatrix to be one when acting on the vector e0 ⊗e0 (i.e.

, R0000(ξ1, ξ2) = 1):R(ξ1, ξ2)e0(ξ1) ⊗e0(ξ2) = e0(ξ2) ⊗e0(ξ1) = Pe0(ξ1) ⊗e0(ξ2)(24)The procedure we follow consists of finding, from the intertwiner condition (8) and thereflection symmetry (22), a set of recursive equations for the R–matrix which can be finallysolved using the equation (24). Indeed, introducing (22) into (8), we obtainR(ξ1, ξ2)P∆ξ2ξ1(g)P = P∆ξ1ξ2(g)PR(ξ1, ξ2)(25)Specializing (8) and (25) to the case g = F, we obtain a set of equations which can be solvedto yield the following recursion formulae:R(ξ1, ξ2)(F1 ⊗1) = [AR(ξ1, ξ2) −BR(ξ1, ξ2)(K1 ⊗1)]11 −K1 ⊗K2R(ξ1, ξ2)(1 ⊗F2) = [BR(ξ1, ξ2) −AR(ξ1, ξ2)(1 ⊗K2)]11 −K1 ⊗K2(26)where A and B are two commuting operators given byA = ∆ξ2,ξ1(F) = F2 ⊗1 + K2 ⊗F1B = (σ ◦∆)ξ2,ξ1(F) = F2 ⊗K1 + 1 ⊗F1(27)The subindices in F1, F2, etc.

are in fact unnecessary if we recall that the whole equation isdefined acting on the space Vξ1 ⊗Vξ2.Iterating eqs. (26) we findR(ξ1, ξ2)(F r11 ⊗F r22 ) =r1Xs1=0(−1)s1ǫs1(s1−1)r1s1Ar1−s1Bs1r2Xs2=0(−1)s2ǫs2(s2−1)r2s2Br2−s2As2× R(ξ1, ξ2)(Ks11 ⊗Ks22 )1Qr1+r2−1ℓ=0(1 −ǫ2ℓK1 ⊗K2)(28)The final result is obtained by applying the above equation to the vector e0(ξ1)⊗e0(ξ2) andmaking use of (24):R(ξ1, ξ2)(F r11 e0(ξ1) ⊗F r22 e0(ξ2)) =1Qr1+r2−1ℓ=0(1 −ǫ2ℓλ1λ2)r1−1Yℓ1=0(A −λ1ǫ2ℓ1B)r2−1Yℓ2=0(B −λ2ǫ2ℓ2A)e0(ξ2) ⊗e0(ξ1)(29)7

In deriving (29), we have used the Gauss binomial formulanXν=0(−1)νnνzνǫν(ν−1) =n−1Yν=0(1 −zǫ2ν)(30)Letting r1 or r2 be N′ in (29), one derives the intertwining conditions (12) which meansthat the spectral manifold for the solution (29) is fiven by the genus zero algebraic curvey = k(1 −λN ′)(31)The general solution (29) is valid for N odd or even and it coincides, up to a change ofbasis, with the one presented in [6] for the particular case ǫ3 = 1.Expanding (29) one can find the entries Rr′1r′2r1r2 as functions of the spectral variables ξ1and ξ2. Some of them are easy to compute since they are products of monomials, for example(0 ≤n ≤r)Rr−n,nr,0(ξ1, ξ2) = (λ1)r,n(λ2)0,r−n[r −n]!Qn−1ℓ=0 (λ2 −λ1ǫ2ℓ)Qr−1ℓ=0(1 −λ1λ2ǫ2ℓ)(32)but in general they have a polynomial structure as inR1111(ξ1, ξ2) = 1 −[2](λ1 −λ2)2(1 −λ1λ2)(1 −ǫ2λ2λ2)(33)We shall give a general formula of Rr′1r′2r1r2 when we consider the limit N →∞.Another property of the R matrix is the conservation of the quantum number r moduloN′, i.e.Rr′1r′2r1r2 = 0unlessr1 + r2 = r′1 + r′2 mod N′(34)The previous construction reveals that the intertwining condition, when supplemented withthe reflection relation (22), are enough data to produce solutions to the Yang–Baxter equa-tion.The general solution also satisfies the normalization (20) and unitarity conditions(21).4.

A solvable lattice model for Uǫ(sℓ(2)) intertwinersOur next task will be to define a lattice model whose Boltzmann weights admit a directinterpretation in terms of intertwiners for semi-cyclic representations. For these models, theR–matrix given in (29) is the solution to the associated star–triangle relation.

To define themodel we associate to each site of the lattice a ZN variable m such that σ(m) = m + 1 with8

σ the generator of ZN transformations. In the same way as for chiral Potts, we associatewith each line of the dual lattice a rapidity which we denote by ξ.

The Boltzmann weightfor the plaquette is then defined byWξ1ξ2(m1, m2, m3, m4) = Rr′1r′2r1r2(ξ1, ξ2) =(35)r1 = m1 −m2,r2 = m2 −m3,r′1 = m1 −m4,r′2 = m4 −m3The R–matrix in (35) is the intertwiner for two semi-cyclic representations ξ1 , ξ2. Therapidities are now two–vectors ξ = (y, λ) and they are forced to live on the algebraic curve(31).

The definition (35) is manifestly ZN invariant: mi →σ(mi). From (35) it is also easyto see that the star–triangle relation for the W’s becomes the Yang–Baxter equation for theR’s.The equivalent here to the kink interpretation can be easily done substituting the kinksby some solitonic–like structure.

In fact, if we interpret the latice variables m as labellingdifferent vacua connected by ZN transformations, we can interpret in the continuum each linkas representing an interpolating configuration between two different vacua. er is identifiedwith a soliton configuration connecting the two vacua m and n with r = m −n.

In theseconditions the Boltzmann weights (35) become the scattering S–matrix for two of theseconfigurations with rapidities ξ1 and ξ2 (see fig. 3).Figure 1.3.

S–matrix interpretation of the Uq(sℓ(2)) R–matrix (29).Notice that the conditions we have used in the previous section for solving the Yang–Baxter equation are very natural in this solitonic interpretation.In fact, the reflectionrelations (22) simply mean that the scattering is invariant under parity transformations.The condition R0000 = 1 can now be interpreted as some kind of vacuum stability. Notice9

that this condition is very dependent on the highest weight vector nature of semi-cyclicrepresentations.Moreover, the normalization condition (20) and the unitarity property(21) strongly support the S–matrix interpretation. The main difference between the model(35) and chiral Potts is that for (35) we allow for all sites in the same plaquette arbitraryZN variables.

In this way we loose the chiral difference between vertical and horizontalinteractions (W, W). What we gain is the possibility to connect the Uǫ(sℓ(2)) intertwinerswith Boltzmann weights without passing through the affine extension.5.

Decomposition rules for semi-cyclic representations: the bootstrap propertyOne of the main ingredients used in ref. [4] to obtain solutions of the Yang–Baxterequation was the fact that generic cyclic representations of Uǫ(ˆsℓ(2)) are indecomposable.This is not the case for semi-cyclic representations of Uǫ(sℓ(2)) [6, 9].

The decompositionrules are given by(λ1, y1) ⊗(λ2, y2) =N ′−1Mℓ=0(ǫ2ℓλ1λ2, y1 + λN ′1 y2)(36)If we consider the tensor product in the reverse order,(λ2, y2) ⊗(λ1, y1) =N ′−1Mℓ=0(ǫ2ℓλ1λ2, y2 + λN ′2 y1)(37)we deduce that the irreps appearing in ξ1 ⊗ξ2 and ξ2 ⊗ξ1 are the same provided the spectralcondition (12) is satisfied. This is of course related to the existence of an intertwiner betweenthe two tensor products.

We also observe that the tensor product of irreps on the algebraicvariety (31) decomposes into irreps belonging to the same variety. To obtain these rules, wesimply need to use the co-multiplication laws (6) and the relation∆ξ1ξ2(g)Kξξ1ξ2 = Kξξ1ξ2ρξ(g)∀g ∈Uǫ(sℓ(2))(38)with Kξξ1ξ2 the Clebsch–Gordan projector Kξξ1ξ2 : Vξ →Vξ1 ⊗Vξ2 which exists wheneverξ ⊂ξ1 ⊗ξ2.

Writing nower(ξ) = Kr1r2ξξ1ξ2r er1(ξ1) ⊗er2(ξ2)(39)we obtain for the CG coefficients Kr1r2ξ(ℓ)ξ1ξ20, which give the highest weight vector e0(ξ) ofthe irrep ξ(ℓ) = (ǫ2ℓλ1λ2, y12) in the semi-cyclic basis (13), the following expression withℓ= r1 + r2:Kr1r2ξ(ℓ)ξ1ξ20= (−1)r1ǫr1(r1−1)ℓr1λr11ℓ−1Yr1(1 + λ1ǫν)ℓ−1Yr2(1 −λ2ǫν)= (−1)r1[ℓ]!ǫr1(r1−1)λr11 (λ1)ℓ,r1 (λ2)ℓ,r2(40)10

Coming back to the S–matrix picture the decomposition rules (36) should be interpreted asreflecting some kind of bootstrap property. This naive interpretation is not quite correct.In fact, the decomposition rule (36) implies a strictly quantum composition of the rapiditiesdetermined by the Hopf algebra structure of the center Zǫ of Uǫ(sℓ(2)) at ǫ a root of unit.This is a new phenomenon derived from the fact that we are considering something thatlooks like a factorized S–matrix for particles, but where the kinematical properties of theseparticles, the rapidities, are quantum group eigenvalues and therefore their compositionrules are not classical.

Moreover, from the explicit expression of the Clebsch–Gordan (40)we observe another interesting phenomenon of mixing between what we should considerinternal quantum numbers, those labelling the basis for the representation (i.e. , the r’s)with the kinematical ones, i.e.

the ones labelling the irreps (see fig. 4).

If this physicalpicture is correct this is the first case where the quantum group appears not only at the levelof internal symmetries but also determines the kinematics.Figure 1.4. Decomposition rules derived from the Clebsch–Gordan coef-ficients (40) with the quantum group decomposition of rapidities.From a strictly quantum group point of view we can, using the intertwiner solution (29)and the CG coefficients (40), check some of the standard results in representation theory ofquantum groups.

As an interesting example we consider the relationR(ξ1, ξ2)Kξ1ξ2ξ= φ(ξ1, ξ2, ξ)Kξ2ξ1ξ(41)which is true for regular representations of spin j with φ(j1, j2, j) = (−1)j1+j2−jǫCj−Cj1−Cj2and Cj = j(j + 1) the classical Casimir. For semi-cyclic representations and the case N = 3we have found that the factors φ(ξ1, ξ2, ξ) in (41) are equal to one, and we presume that thisfact persists for all N.6.

The limit N →∞In this section we shall study the limit N →∞of the R–matrix found in section 3. Thequantum deformation parameter ǫ goes in this limit to 1, so one could expect that the Hopf11

algebra Uǫ(sℓ(2)) becomes the classical universal enveloping algebra of sℓ(2). This is nothowever what is happening here as can be seen by taking ǫ →1 in eqs.

(5):[E, F] = 1 −K2[K, E] = [K, F] = 0(42)Let us recall that in the quantum group relations (5) we have not included the denominator1 −ǫ−2, hence there is no need to apply the L’Hˆopital rule. What we obtain rather in thelimit N →∞is the Heisenberg algebra of a harmonic oscillator.

Inded, defining a and a† asa =11 + K Ea† =11 −K F(43)we find that (42) amount to[a, a†] = 1[K, a] = [K, a†] = 0(44)The role of the operator K is therefore to produce non–trivial co-multiplications preservingthe algebra (42) or (44), and this is why we may have non–trivial R–matrices. The represen-tation spaces of the algebra (42) are now infinite–dimensional and are labelled by the valueof K; we shall call them Hλ.

In a basis {er}∞r=0 of Hλ we haveFer = (1 −λ)er+1Eer = r(1 + λ)er−1Ker = λer(45)Hence we see that er can be identified with the r–th level of a harmonic oscillator. Of cousethe value λ = 1 in (45) has to be treated with care.The R–matrix is now an operator R(λ1, λ2) : Hλ1 ⊗Hλ2 →Hλ2 ⊗Hλ1 which in thebasis (45) has the following non–vanishing entries:Rr1+r2−ℓ,ℓr1,r2(λ1, λ2) =1(1 −λ1λ2)r1+r2Xℓ1+ℓ2=ℓr1ℓ1r2ℓ2[(1 + λ1)(1 −λ2)]r1−ℓ1[(1 −λ1)(1 + λ2)]ℓ2(λ2 −λ1)ℓ1(λ1 −λ2)r2−ℓ2(46)where 0 ≤ℓ≤r1 + r2.

This expression has been obtained from (29) taking the limit ǫ →1and using (45).A first observation is that R(λ1, λ2) depends only on the following harmonic ratioη12 = z12z34z13z24= 2λ1 −λ2(1 + λ1)(1 −λ2)(47)12

which corresponds to a sphere with 4 punctures at the points z1 = λ1, z2 = λ2, z3 = −1and z4 = 1. In these new variables we haveRr1+r2−ℓ,ℓr1,r2(η12) =1(1 −η12/2)r1+r2Xℓ1+ℓ2=ℓ(−1)ℓ1r1ℓ1r2ℓ2(η12/2)r2−ℓ2+ℓ1(1 −η12)ℓ2(48)A reflection transformation η12 →η21 =η12η12−1 corresponds to a M¨obius transformation.It is however more convenient to use another variable u defined asu12 = λ1 −λ21 −λ1λ2=η122 −η12(49)which changes sign under a reflection symmetry.

The R–matrix reads in the u–variable asRr1+r2−ℓ,ℓr1,r2(u12) = (1 + u12)r1ur212Xℓ1+ℓ2=ℓ(−1)ℓ1r1ℓ1r2ℓ2 u121 + u12ℓ1 1 −u12u12ℓ2(50)Recalling the definition of the Jacobi polynomials P (α,β)n(x) (n = 0, 1, . .

. ):P (α,β)n(x) = 12nnXm=0n + αmn + βn −m(x −1)n−m(x + 1)m(51)we see finally that R(u) can be written in the formRr1+r2−ℓ,ℓr1,r2(u) = (1 + u)r1−ℓur2−ℓP (r2−ℓ,r1−ℓ)ℓ(1 −2u2)(52)As an application of this formula we may derive the classical limit of eq.

(32) knowing thatP (−ℓ,r−ℓ)ℓ(x) = (−1)ℓ2ℓrℓ(1 −x)ℓ(53)The reflection symmetry of RRr1+r2−ℓ,ℓr1,r2(u) = Rℓ,r1+r2−ℓr2,r1(−u)(54)implies the following identity between Jacobi polynomials:P (−α,−β)α+β+n (x) =x −12α x + 12βP (α,β)n(x)(55)whenever n+α, n+β and n+α+β are all non–negative integers. The Yang–Baxter equationof R(u) implies a cubic equation for the Jacobi polynomials that we shall not write down.The fact that the R–matrix can be written entirely as a function of a single variableu has some interesting consequences.

First of all, we see from the definition of u12 that13

interpreting λi as the velocity of the i-th soliton, then uij is nothing but the relative speedof the i-th soliton with respect to the j-th soliton in a 1 + 1 relativistic world. Moreover,the YB equation (19) when written in the u variables has a relativistic look:(1 ⊗R(u))R u + v1 + uv⊗1(1 ⊗R(v)) = (R(v) ⊗1)1 ⊗R u + v1 + uv(R(u) ⊗1)(56)and in particular the scattering matrices R(u) of these relativistic solitons give us a repre-sentation of the braid group providedu = v = u + v1 + uv(57)a situation which happens when u = 0, ±1.

The case u = 0 is trivial, since R(u = 0) = 1,but the other two yield the following braiding matrices:Rr′1r′2r1r2(+) = limu→1 Rr′1r′2r1r2(u) = δr1+r2,r′1+r′2(−1)r′22r1−r′2r1r′2Rr′1r′2r1r2(−) = limu→−1 Rr′1r′2r1r2(u) = δr1+r2,r′1+r′2(−1)r′12r2−r′1r2r′1(58)which satisfy, in addition to the YB equation, the relationR(+)R(−) = 1(59)which is a consequence of the unitarity condition (21).This means that we have obtained an infinite–dimensional representation π of the braidgroup Bn given byπ : Bn →End(H⊗n)σ±1i7→1 ⊗· · · ⊗Ri,i+1(±) ⊗· · · ⊗1(60)where H is isomorphic to the Hilbert space of a harmonic oscillator.In fact, using theuniversal matrix R = PR ∈End(H⊗2) we can write eqs. (58) asR(+) = (eiπN ⊗1)e2a⊗a†R(−) = (1 ⊗eiπN )e2a†⊗a(61)where N = a†a is the number operator anda†er = er+1aer = rer−1(62)The Yang–Baxter relation (56) in the limit u →±1 can be most easily proved in terms ofthe YB solution for the R matrix which readsR12R13R23 = R23R13R12(63)14

where R12 = (eiπN ⊗1 ⊗1)e2a⊗a†⊗1 etc.We have thus seen that the limit N →∞of our construction is well defined, and hassome interesting structure. It describes essentially the scattering of relativistic solitons whosespectrum is that of a harmonic oscillator.

Interestingly enough, in the limit of very deepscattering one obtains an infinite–dimensional representation of the braid group. One maywonder whether this representation provides us with new invariants of knots and links, justas the usual finite–dimensional R matrices from quantum groups do.With this in mind, we shall propose a slight generalization of the R matrices (61) givenbyR(x, y; +) = (xN ⊗y−N)e(y−x)a⊗a†R(x, y; −) = e(x−y)a†⊗a(yN ⊗x−N )(64)where x and y are two independent complex numbers.

It is easy to see that these new Rmatrices also satisfy the YB equation, yielding a representation πx,y : Bn →End(H⊗n) ofthe braid group. The previous case is recoverd with x = −1, y = 1.

The braiding matricesthat follow from (64) areRr′1r′2r1r2(x, y) = δr1+r2,r′1+r′2r1r′2(y −x)r1−r′2xr′2y−r′1R−1r′1r′2r1r2(x, y) = δr1+r2,r′1+r′2r2r′1(x −y)r2−r′1x−r2yr1(65)This braid group representation admits an extension `a la Turaev [10], i.e. there existsan isomorphism µ : H →H satisfying the following three conditions:i)(µiµj −µkµℓ) Rkℓij = 0ii)XjRkjij µj = δki abiii)XjR−1kjij µj = δki a−1b(66)for some constants a and b.

For the R matrix (65), the Turaev conditions hold ifµ = 1a = b−1 =p(y/x)(67)The invariant of knots and links that one would get is thusTx,y(α) = (x/y)12[w(α)−n]tr πx,y(α)(68)where α ∈Bn and w(α) is the writhe of α. The trace in (68) is defined on the n-th tensorproduct of Hilbert spaces, therefore to make sense of Tx,y(α) one should regularize this trace15

without losing the invariance under the Markov moves.We leave the identification andproper definition of the invariant (68) for a future publication.7. Final CommentsA possible framework where to study in more detail the physical meaning of our resultscould be the one recently developed by Zamolodchikov [11] in connection with the analysis ofintegrable deformations of conformal field theories.

The moral we obtain from our analysisis that a unique mathematical structure, the quantum group, can describe at the same timeconformal field theories [12] and integrable models. The dynamics that fixes what of the twokinds of physical systems is described is the way the central subalgebra Zǫ, for q a root of unit,is realized.

In the conformal case, the central subalgebra is realized trivially with vanishingeigenvalues which correspond to the regular representations. The quantum group symmetryis defined in this case by the Hopf algebra quotiented by its central subalgebra.

From theresults in [3, 4] and the ones described here it seems that when the center is realized in anon–trivial way the system we describe is an integrable model. The star–triangle solution(29) for the lattice model we have defined in section 4 has good chances of describing aself–dual point.

The heuristic reason for this is that the algebraic curve (31) on which thespectral parameters live is of genus zero.We want to mention also the possible physical implications in this context of the recentmathematical results of reference [5]. In fact, these authors have defined a quantum co-adjoint action on the space of irreps.

This action divides the finite–dimensional irreps intoorbits each one containing a semi-cyclic representation.The co-adjoint action does notpreserve the spectral manifold (31) but if we maintain the interpretation of the irrep labelsas rapidities then it acts on them, opening in this way the door to new kinematics completelybased on quantum group properties.Finally, we summarize the non–trivial results we have obtained in the limit N →∞.First of all, the intertwiner matrix for semi-cyclic representations can be interpreted asdescribing the scattering of solitons in a 1 + 1 relativistic world. The Hilbert space of thesesolitons is isomorphic to that of the harmonic oscillator.

Moreover, in the limit when thesolitons become relativistic, the scattering matrices provide us with an infinite–dimensionalrepresentation of the braid group. This representation seems to have a Markov trace whichwould allow us to find an invariant of links and knots.Acknowledgements.

We would like to thank M. Ruiz–Altaba for continuing discus-sions on all aspects of this work and for sharing with us his insights and results. This workis partially supported by the Fonds National Suisse pour la Recherche Scientifique.16

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