Dynamical R Matrices of Elliptic Quantum Groups and Connection Matrices for the q-KZ Equations

For any affine Lie algebra ${\mathfrak g}$, we show that any finite dimensional representation of the universal dynamical $R$ matrix ${\cal R}(\lambda)$ of the elliptic quantum group ${\cal B}_{q,\lambda}({\mathfrak g})$ coincides with a corresponding connection matrix for the solutions of the $q$-KZ equation associated with $U_q({\mathfrak g})$. This provides a general connection between ${\cal B}_{q,\lambda}({\mathfrak g})$ and the elliptic face (IRF or SOS) models. In particular, we construct vector representations of ${\cal R}(\lambda)$ for ${\mathfrak g}=A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, $D_n^{(1)}$, and show that they coincide with the face weights derived by Jimbo, Miwa and Okado. We hence confirm the conjecture by Frenkel and Reshetikhin.


Introduction
The quantum group U q (g) is one of the fundamental structures appearing in the wide class of trigonometric quantum integrable systems. Among others, we remark the following two facts. 1) For g being affine Lie algebra, finite dimensional representations of U q (g) allow a systematic derivation of trigonometric solutions of the Yang-Baxter equation (YBE) [1,2].
2) A combined use of finite and infinite dimensional representations allows us to formulate trigonometric vertex models and calculate correlation functions [3].
To extend this success to elliptic systems is our basic aim. In this paper, we consider a problem analogous to 1). As for developments in the direction 2), we refer the reader to the papers [4,5,6,7,8]. We are especially interested in the two dimensional exactly solvable lattice models. There are two types of elliptic solvable lattice models. The vertex type and the face (IRF or SOS) type. The vertex type elliptic solutions to the YBE were found by Baxter [9] and Belavin [10]. These are classified as the elliptic R matrices of the type A (1) n . The face type elliptic Boltzmann weights associated with A (1) 1 were first constructed by Andrews-Baxter-Forrester [11], and extended to A  n by Jimbo-Miwa-Okado [12,13], to A (2) 2n , A (2) 2n−1 by Kuniba [14], and to G by Kuniba-Suzuki [15].
Concerning the elliptic face weights, Frenkel and Reshetikhin made an interesting observation [16] that the connection matrices for the solution of the q-KZ equation associated with U q (g) (g: affine Lie algebra) provide elliptic solutions to the face type YBE. They also conjectured that the connection matrices in the vector representation are equal to Jimbo-Miwa-Okado's face weights for g = A (1) n , B (1) n , C (1) n , D (1) n . In order to confirm this conjecture, one needs to solve the q-KZ equation of general level and find connection matrices. Within our knowledge, no one has yet confirmed it. Instead of doing this, Date, Jimbo and Okado [17] considered the face models defined by taking the connection matrices as Boltzmann weights. They showed that the one-point function of such models is given by the branching function associated with g. The same property of the one-point function had been discovered in Jimbo-Miwa-Okado's A (1) n , B (1) n , D (1) n face models. An attempt to formulate elliptic algebras was first made by Sklyanin [18]. He considered an algebra defined by the RLL-relation associated with Baxter's elliptic R-matrix. It was extended to the elliptic algebra A q,p ( sl 2 ) by Foda et al. [19], based on a central extension of Sklyanin's RLL-relation. In the same year, Felder proposed a face type elliptic algebra E τ,η (g) associated with the dynamical RLL-relation [20]. Jimbo-Miwa-Okado's elliptic solutions to the face type YBE were interpreted there as the dynamical R matrices. We classify the former elliptic algebra the vertex-type and the latter the face-type. Another formulation of the face type elliptic algebra was discovered by the author [4]. It is based on an elliptic deformation of the Drinfel'd currents.
A coalgebra structure of these elliptic algebras was clarified in the works by Frønsdal [21], Enriquez-Felder [22] and Jimbo-Konno-Odake-Shiraishi [23]. It is based on an idea of quasi-Hopf deformation [24] by using the twistor operators satisfying the shifted cocycle condition [25]. In this formulation, we regard the coalgebra structures of the vertex and the face type elliptic algebras as two different quasi-Hopf deformation of the corresponding affine quantum group U q (g). We call the resultant quasi-Hopf algebras the elliptic quantum groups of the vertex type A q,p ( sl N ) and the face type B q,λ (g). A detailed description for the face type case is reviewed in Section 2. For the face type, a different coalgebra structure as a h-Hopf algebroid was developed by Felder, Etingof and Varchenko [26,27,28], Koelink-van Norden-Rosengren [29].
One of the advantages of the quasi-Hopf formulation is that it allows a natural derivation of the universal dynamical R matrix from one of U q (g) as a twist. However, a disadvantage is that there are no a priori reasons for the resultant universal R matrix to yield elliptic R matrices. One needs to check this point in all representations. We have done this for the vector representations of A q,p ( sl 2 ) and B q,λ ( sl 2 ), which led to Baxter's elliptic R matrix and Andrews-Baxter-Forrester's elliptic face weights, respectively [23]. The same checks for the face weights were also done in the cases g = A (1) n , A 2 [6,7]. The aim of this paper is to overcome this disadvantage by clarifying the following point concerning the face type.
i) Any representations of the universal dynamical R matrix of B q,λ (g) are equivalent to the corresponding connection matrices for the q-KZ equation of U q (g).
The connection matrices are known to be elliptic. See Theorem 3.5. In addition, we show n , the vector representation of the the universal dynamical R matrix of B q,λ (g) is equivalent to Jimbo-Miwa-Okado's elliptic face weight up to a gauge transformation.
Combining i) and ii), we confirm Frenkel-Reshetikhin's conjecture on the equivalence between the connection matrices and Jimbo-Miwa-Okado's face weights. For the purpose of showing i), we follow the idea by Etingof and Varchenko [28], and give an exact relation between the face type twistors and the highest to highest expectation values of the composed vertex operators (fusion matrices) of U q (g). To show ii), we solve the difference equation for the face type twistor, which is equivalent to the q-KZ equation of general level. This paper is organized as follows. In the next section, we summarize some basic facts on the affine quantum groups U q (g) and the face type elliptic quantum groups B q,λ (g). In Section 3, we introduce the vertex operators of U q (g) and fusion matrices. We discuss equivalence between the face type twistors and the fusion matrices. Then in Section 4, we show equivalence between the dynamical R matrices of B q,λ (g) and the connection matrices for the q-KZ equation of U q (g) in general finite dimensional representation. Section 5 is devoted to a discussion on an equivalence between the vector representations of the the universal dynamical R matrix of B q,λ (g) and Jimbo-Miwa-Okado's elliptic face weights for g = A (1) n , B (1) n , C (1) n , D (1) n .
2 Af f ine quantum groups U q (g) and elliptic quantum groups B q,λ (g) 2.1 Affine quantum groups U q (g) Let g be an affine Lie algebra associated with a generalized Cartan matrix A = (a ij ), i, j ∈ I = {0, 1, . . . , n}. We fix an invariant inner product (·|·) on the Cartan subalgebra h and identify h * with h through (·|·). We follow the notations and conventions in [30] except for A (2) 2n , which we define in such a way that the order of the vertices of the Dynkin diagram is reversed from the one in [30]. We hence have a 0 = 1 for all g. Let {α i } i∈I be a set of simple roots and set We denote the canonical central element by c = i∈I a ∨ i h i and the null root by δ = i∈I a i α i . We set and impose the pairings The Λ j are the fundamental weights. We also use P cl = P/Zδ, (P cl ) * = ⊕ n i=0 Zh i ⊂ P * . Let cl : P → P cl denote the canonical map and define af : P cl → P by af (cl(α i )) = α i (i = 0) and af (cl(Λ 0 )) = Λ 0 so that cl • af = id and af (cl(α 0 )) = α 0 − δ.
Definition 2.1. The quantum affine algebra U q = U q (g) is an associative algebra over C(q 1/2 ) with 1 generated by the elements e i , f i (i ∈ I) and q h (h ∈ P * ) satisfying the following relations Here

H. Konno
The algebra U q has a Hopf algebra structure with comultiplication ∆, counit ǫ and antipode S defined by U q is a quasi-triangular Hopf algebra with the universal R matrix R satisfying Here ∆ op denotes the opposite comultiplication, ∆ op = σ • ∆ with σ being the flip of the tensor Proposition 2.2.
Let {h l } be a basis of h and {h l } be its dual basis. We denote by U + (resp. U − ) the subalgebra of U q generated by e i (resp. f i ) i ∈ I and set for β ∈ Q + . The universal R matrix has the form [31] where {u β,j } and {u j −β } are bases of U + β and U − −β , respectively. Note that T is the canonical element of h ⊗ h w.r.t (·|·) and (R 0 ) β is the canonical element of U + β ⊗ U − −β w.r.t a certain Hopf paring.
We write U ′ q = U ′ q (g) for the subalgebra of U q generated by e i , f i (i ∈ I) and h ∈ (P cl ) * . Let (π V , V ) be a finite dimensional module over U ′ q . We have the evaluation representation (π V,z , V z ) of U q by V z = C(q 1/2 )[z, z −1 ] ⊗ C(q 1/2 ) V and where n ∈ Z, and v ∈ V denotes a weight vector whose weight is wt(v). We write vz n = z n ⊗ v (n ∈ Z). For generic λ ∈ h * , let M λ denote the irreducible Verma module with the highest weight λ. We have the weight space decomposition M λ = ν∈λ−Q + (M λ ) ν . We write wt(u) = ν for u ∈ (M λ ) ν .

Elliptic quantum groups B q,λ (g)
Let ρ ∈ h be an element satisfying (ρ|α i ) = d i for all i ∈ I. For generic λ ∈ h, let us consider an automorphism of U q given by where Ad(x)y = xyx −1 . We define the face type twistor F (λ) ∈ U q ⊗U q as follows.
Note that the k-th factor in the product (2.4) is a formal power series in x k i = q 2k α i ,λ (i ∈ I) with leading term 1.
We hence call R(λ) the universal dynamical R matrix. Now let us parametrize λ in the following way.
whereλ stands for the classical part of λ, and h ∨ denotes the dual Coxeter number of g. Note also ρ = h ∨ Λ 0 +ρ and d = a ∨ 0 Λ 0 . Let {h j } and {h j (=Λ j )} denote the classical part of the basis and its dual of h. We then have Here we set p = q 2r . Set further Then R(z) and F (z, λ) are formal power series in z, whereas R(z, λ) contains both positive and negative powers of z. From the definition (2.4) of F (λ), one can easily derive the following difference equation for the twistor.
Theorem 2.7 (Difference equation [23]). (2.17) Furthermore, noting Ad(z d )(e i ) = z δ i,0 e i , one can drop all the e 0 dependent terms in R(z) and F (z, λ) by taking the limit z → 0. We thus obtain where Rḡ and Fḡ(λ) are the universal R matrix and the twistor of U q (ḡ). Then from (2.17), we obtain the following equation for Fḡ(λ).
Remark. (2.20) corresponds to (18) in [32], where the comultiplication and the universal R matrix are our ∆ op and R −1 g , respectively.
3 Vertex operators and fusion matrices

The vertex operators of U q (g)
Let V and W be finite dimensional irreducible modules of U ′ q . Let λ, µ ∈ h * be level-k generic elements such that c, λ = c, µ = k. We denote by M λ and M µ the two irreducible Verma modules with highest weights λ and µ, respectively.
1 , let us consider the formal series given by Here {v j } denotes a weight basis of V . The coefficients ( Ψ µ λ ) j,n are the maps Here ⊗ denotes a formal completion We call Ψ µ λ (z) the vertex operator (VO) of U q .
Remark. Ψ µ λ (z) is the type II VO in the terminology of [3].
Note that there is a bijective correspondence between Ψ µ λ (z) and Ψ µ λ . Let u λ and u µ denote the highest weight vectors of M λ and M µ , respectively. Let us write the image of u λ by the VO as This theorem tells that Ψ µ λ is determined by its leading term. Namely, for given v 0 ∈ V µ λ , there exists a unique VO satisfying
Remark [16]. A solution of the q-KZ equation (3.6) is a function of z = z 1 /z 2 and has a form G(z)f (z 1 , z 2 ). Here G(z) is a meromorphic function multiplied by a fractional power of z determined from the normalization function of the R matrix R W V (z), while f (z 1 , z 2 ) is an analytic function in |z 1 | > |z 2 | and can be continued meromorphically to (C × ) 2 . Hence a solution of the q-KZ equation is uniquely determined, if one fixes the normalization. It also follows that the composition of the VOs is well defined in the region |z 1 | > |z 2 | and can be continued meromorphically to (C × ) 2 apart from an overall fractional power of z.
The following commutation relation holds in the sense of analytic continuation.
where v i ′ and w j ′ are base vectors of V µ ′ λ and W ν µ ′ , respectively.
The matrix C W V is called the connection matrix. The following theorem states basic properties of the connection matrix. 1) The matrix elements of C W V are given by a ratio of elliptic theta functions.
2) The matrix C W V satisfies i) the face type YBE and ii) the unitarity condition (the first inversion relation): Here ξ = q th ∨ (t = (long root) 2 /2), and G λ and α V V (z) are given by (5.16) and (5.13) in [17], respectively.
Furthermore, if V z is self dual i.e. there exists an isomorphism of U q -modules Q : V ξ −1 z ≃ V * a z , we have the following crossing symmetry.
where g νjj ′ µ denotes a certain matrix element appearing in the inversion relation of the VO's, and γ ν µ is its inverse matrix such that j g νjj µ γ νjj ′ µ = δ jj ′ .
In Section 5, we will discuss the evaluation molude V z with V being the vector representation n . There V z is self dual except for A (1) n (n > 1), and is multiplicity free. Therefore, dim V µ λ = 1 etc. Hence the matrices g ν µ , γ ν µ are scalars satisfying γ ν µ = 1/g ν µ . In this case, let us consider the gauge transformation Then we can change the crossing symmetry relation (3.7) tõ The same gauge transformation makes the face type YBE i), the unitarity condition ii) and the second inversion relation 3) in Theorem 3.5 unchanged except for the factor α V V (z) in the RHS of 3), which is changed to 1.

Fusion matrices
Note that from (3.2), Noting (3.5) and the intertwining property of the vertex operators, we have where wt(v l ) < wt(v i ), and C l (λ) is a function of λ. Hence J V W (λ) is an upper triangular matrix with all the diagonal components being 1. Therefore The definition of J W V (λ) indicates that the leading term of the intertwiner where u l ∈ U λ ω .
Theorem 3.8. The fusion matrix satisfies the shifted cocycle condition Proof . Consider the composition and express the highest to highest expectation value of it in two ways, and use cl(λ) = cl(ω) − wt(u l ).
From the remark below Theorem 3.3, J W V (z 1 , z 2 ; λ) is a function of the ratio z = z 1 /z 2 . Let us parameterize a level-k λ ∈ h * as (2.12). Comparing Theorem 2.7 and Lemma 3.9, we find that the difference equation for F W V (z, −λ) −1 = (π W ⊗ π V )F (z, −λ) −1 coincides with the q-KZ equation (3.11) for J W V (z 1 , z 2 ; λ) on w j ⊗ v i under the identification r = −(k + h ∨ ). Hence the uniqueness of the solution to the q-KZ equation yields the following theorem.
Theorem 3.10. For a level-k λ ∈ h * in the parameterization (2.12), Remark. Relation between the twistors and the fusion matrices was first discussed by Etingof and Varchenko for the case g being simple Lie algebra (Appendix 9 in [28]). Their coproduct and the universal R matrix correspond to our ∆ op and R −1 g , respectively, and the twistor is identified with the two point function of the U q (ḡ)-analogue of the type I VOs.

Dynamical R matrices and connection matrices
Let (π V , V ), (π W , W ) be finite dimensional representations of U ′ q (g), and {v i }, {w j } be their weight bases, respectively. Now we consider the dynamical R matrices given as the images of the universal R matrix R(λ) (4.1) Note that F (21) V W (z −1 , λ) = P F V W (z −1 , λ)P . By using Theorems 3.10 and 3.4, we show that the dynamical R matrix R W V (z, λ) coincides with the corresponding connection matrix for the q-KZ equation of U q (g) associated with the representations (π V , V ) and (π W , W ).
Proof . From Theorem 3.10 and (4.1), we have where we set z = z 1 /z 2 . Then from Theorem 3.4, we obtain Note that in view of Theorem 3.5, this theorem indicates that the dynamical R matrices of B q,λ (g) are elliptic.

Vector representations
In this section, we consider the vector representation of the universal R matrix R(λ) of B q,λ (g) for g = A
We also use the usual order < in J. LetΛ j (1 ≤ j ≤ n) be the fundamental weights ofḡ. Following Bourbaki [34] we introduce orthonormal vectors {ε 1 , . . . , ε n } with the bilinear form (ε i |ε j ) = δ i,j . Then one has the following expression ofΛ j as well as the set A of weights belonging to the vector representation ofḡ.
C n : A = {±ε 1 , . . . , ±ε n }, Now for µ ∈ J, let us defineμ ∈ A bŷ We then define a dynamical variable a ∈ h * of the face model of type g as follows.
Then Jimbo-Miwa-Okado's solutions to the face type YBE are given as follows.
where η = −th ∨ /2 (t = (long root) 2 /2) is the crossing parameter, and the symbol [u] denotes the Jacobi elliptic theta function The κ(u) denotes a function satisfying the following relations Ga denotes a ratio of the principally specialized character G a for the dual affine Lie algebra g ∨ [13] Here ε(a) denotes a sign factor such that ε(a+μ)/ε(a) = s. s and h(a) are listed in the following Table  A (1) n Remark. Our normalization of the weights W and some notations are different from those in [13]. Their relations are given as follows Here the symbols with subindex JM O denote the ones in [13].
The following theorem states basic properties of the face weights.
In addition, we have the crossing symmetry except for g = A (1) The following theorem is communicated by Jimbo and Okado and is not written explicitly in [13].
n , the weights listed in the part (II) of (5.1) is determined uniquely from those in (I) by requiring the face type Yang-Baxter equation and the crossing symmetry relations.
Sketch of proof. It is easy to see that the first type of weights in (II) is determined by those in (I) by using the crossing symmetry relation. Then the second type of weights in (II) is determined by solving the system of two linear equations (YBE) shown in Fig. 1. Here unknowns are the weights W a a + µ a + µ a and W a + ν a + µ + ν a + µ + ν a + ν , and the other weights are in (I).

The difference equations for the twistor
We here solve the difference equation for the twistor. Then using Theorem 4.1, we derive the dynamical R matrix R V V (z, λ) as the connection matrix in the vector representation (π V , V ), and argue that it coincides with Jimbo-Miwa-Okado's solution up to a gauge transformation.
Let us consider the difference equation (2.17) in the vector representation.
where λ is parameterized as (2.12),θ(λ) is given by (2.13), Let {v j |j ∈ J} be a basis of V and E i,j be the matrix unit defined by E i,j v k = δ j,k v i . The action of the generators on V is given by [35,36,37] (for C (1) n , the conventions used here are slightly different from [37]) and π V (f i ) = π V (e i ) t . A basis {h i } ofh and its dual basis {h i } w.r.t (·|·) are given as follows Then one can easily verify the following.
Proposition 5.5. If we parameterizeλ such thatλ = n i=1 (s i + 1)h i , we have where a j (j ∈ J) is given by Proposition 5.1.
The R matrix R(z) of U q (g) in the vector representation is well known [2,35,36, 37] (for C (1) n , we modified the R matrix in [37] according to the convention used here) Here ξ = q th ∨ , and ǫ j = 1 (j > 0), −1 (j < 0) for g = C (1) n and ǫ j = 1 (j ∈ J) for the other cases. The symbolj is defined bȳ Then due to the formula (2.4), we make the following ansatz for the twistor F (z, λ) in the vector representation.
where X kl ij denote unknown functions to be determined. From the from of R(z) and F (z, λ) in (5.4) and (5.5), one finds that the difference equation (5.3) consists of 1 × 1, 2 × 2 and N × N blocks. The numbers of blocks of each size contained in the equation are listed as follows 2n(n − 1) 1 By using Propositions 5.4, 5.5 and (5.4), (5.5), we obtain the following equations. 1 × 1 blocks: = qρ(pz)f (z) for B n , C n , D n .
By setting z = q 2u and using (5.2), we can reexpress these matrix elements in terms of the theta functions with some extra factors including q with fractional power and infinite products. Then making an appropriate gauge transformation, we can sweep away all the extra factors and find that the 1 × 1 and 2 × 2 block parts coincide with the part (I) of Jimbo-Miwa-Okado's solution, i.e.
R kl ij (z, w ij ) ⇔ W a a +k a +î a +î +ĵ z in (I).
From 2) of Theorem 3.5, (3.2) and Theorem 5.3, the remaining part (II) is determined uniquely from the part (I). We hence obtain the following theorem.
n , the vector representation of the universal dynamical R matrix R(λ) coincides with Jimbo-Miwa-Okado's elliptic solutions to the face type YBE.
To solve the difference equation in the N × N block directly is an open problem. For the cases g being the twisted affine Lie algebras A (2) 2n and A (2) 2n−1 , Kuniba derived elliptic solutions to the face type YBE. His construction is based on a common structure of the R matrices of the twisted U q (g) to those of the B 2 case was investigated in [7]. In view of these facts, we expect the same statement as Theorem 5.3 is valid in the twisted cases, too. A Proof of Lemma 3.9 We here give a direct proof of Lemma 3.9 and leave a derivation of the q-KZ Equation(3.6) from it as an exercise.
Let R be the universal R matrix of U q (g) and write R = j a j ⊗ b j , (A.1) and set U = j S(b j )a j = j b j S −1 (a j ) and Z = q 2ρ U . (1) U xU −1 = S 2 (x) ∀ x ∈ U q , (2) ZxZ −1 = x, Proof . The statement follows (id ⊗ ∆)R = R (13) R (12) .