Dynamical R matrices of elliptic quantum groups and connection matrices for the q-KZ equations
For any affine Lie algebra g, we show that any finite dimensional representation of the universal dynamical R matrix R(λ) of the elliptic quantum group Bq,λ(g) coincides with a corresponding connection matrix for the solutions of the q-KZ equation associated with Uq(g). This provides a general conne...
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Konno, H. 2019-02-06T17:13:12Z 2019-02-06T17:13:12Z 2006 Dynamical R matrices of elliptic quantum groups and connection matrices for the q-KZ equations / H. Konno // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 38 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 33D15; 81R50; 82B23 https://nasplib.isofts.kiev.ua/handle/123456789/146062 For any affine Lie algebra g, we show that any finite dimensional representation of the universal dynamical R matrix R(λ) of the elliptic quantum group Bq,λ(g) coincides with a corresponding connection matrix for the solutions of the q-KZ equation associated with Uq(g). This provides a general connection between Bq,l(g) and the elliptic face (IRF or SOS) models. In particular, we construct vector representations of R(λ) for g = An⁽¹⁾, Bn⁽¹⁾, Cn⁽¹⁾, Dn⁽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. This paper is a contribution to the Proceedings of the O’Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 22–24, 2006, Budapest, Hungary). The author would like to thank Michio Jimbo and Masato Okado for stimulating discussions and valuable suggestions. He also thanks Atsuo Kuniba and Atsushi Nakayashiki for discussions. He is also grateful to the organizers of O’Raifeartaigh Symposium, Janos Balog, Laszlo Feher and Zalan Horvath, for their kind invitation and hospitality during his stay in Budapest. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Dynamical R matrices of elliptic quantum groups and connection matrices for the q-KZ equations Article published earlier |
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Dynamical R matrices of elliptic quantum groups and connection matrices for the q-KZ equations |
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Dynamical R matrices of elliptic quantum groups and connection matrices for the q-KZ equations Konno, H. |
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Dynamical R matrices of elliptic quantum groups and connection matrices for the q-KZ equations |
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Dynamical R matrices of elliptic quantum groups and connection matrices for the q-KZ equations |
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dynamical r matrices of elliptic quantum groups and connection matrices for the q-kz equations |
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For any affine Lie algebra g, we show that any finite dimensional representation of the universal dynamical R matrix R(λ) of the elliptic quantum group Bq,λ(g) coincides with a corresponding connection matrix for the solutions of the q-KZ equation associated with Uq(g). This provides a general connection between Bq,l(g) and the elliptic face (IRF or SOS) models. In particular, we construct vector representations of R(λ) for g = An⁽¹⁾, Bn⁽¹⁾, Cn⁽¹⁾, Dn⁽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.
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Dynamical R matrices of elliptic quantum groups and connection matrices for the q-KZ equations / H. Konno // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 38 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 091, 25 pages
Dynamical R Matrices of Elliptic Quantum Groups
and Connection Matrices for the q-KZ Equations?
Hitoshi KONNO
Department of Mathematics, Graduate School of Science,
Hiroshima University, Higashi-Hiroshima 739-8521, Japan
E-mail: konno@mis.hiroshima-u.ac.jp
Received October 02, 2006, in final form November 28, 2006; Published online December 19, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper091/
Abstract. For any affine Lie algebra g, we show that any finite dimensional representation
of the universal dynamical R matrix R(λ) of the elliptic quantum group Bq,λ(g) coincides
with a corresponding connection matrix for the solutions of the q-KZ equation associated
with Uq(g). This provides a general connection between Bq,λ(g) and the elliptic face (IRF
or SOS) models. In particular, we construct vector representations of R(λ) for g = A
(1)
n ,
B
(1)
n , C
(1)
n , D
(1)
n , and show that they coincide with the face weights derived by Jimbo, Miwa
and Okado. We hence confirm the conjecture by Frenkel and Reshetikhin.
Key words: elliptic quantum group; quasi-Hopf algebra
2000 Mathematics Subject Classification: 33D15; 81R50; 82B23
1 Introduction
The quantum group Uq(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 Uq(g) allow a syste-
matic 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
(1)
n , B
(1)
n , C
(1)
n , D
(1)
n by Jimbo–Miwa–Okado [12, 13], to A
(2)
2n , A
(2)
2n−1 by Kuniba [14], and to G
(1)
2
by Kuniba–Suzuki [15].
Concerning the elliptic face weights, Frenkel and Reshetikhin made an interesting observa-
tion [16] that the connection matrices for the solution of the q-KZ equation associated with Uq(g)
(g: affine Lie algebra) provide elliptic solutions to the face type YBE. They also conjectured
?This paper is a contribution to the Proceedings of the O’Raifeartaigh Symposium on Non-Perturbative and
Symmetry Methods in Field Theory (June 22–24, 2006, Budapest, Hungary). The full collection is available at
http://www.emis.de/journals/SIGMA/LOR2006.html
mailto:konno@mis.hiroshima-u.ac.jp
http://www.emis.de/journals/SIGMA/2006/Paper091/
http://www.emis.de/journals/SIGMA/LOR2006.html
2 H. Konno
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 Aq,p(ŝl2) 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 Eτ,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 al-
gebras as two different quasi-Hopf deformation of the corresponding affine quantum group Uq(g).
We call the resultant quasi-Hopf algebras the elliptic quantum groups of the vertex typeAq,p(ŝlN )
and the face type Bq,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 Uq(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 Aq,p(ŝl2) and Bq,λ(ŝl2), 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)
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 Bq,λ(g) are equivalent to the
corresponding connection matrices for the q-KZ equation of Uq(g).
The connection matrices are known to be elliptic. See Theorem 3.5. In addition, we show
ii) For g = A
(1)
n , B
(1)
n , C
(1)
n , D
(1)
n , the vector representation of the the universal dynamical R
matrix of Bq,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 Uq(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 Uq(g) and the face type elliptic quantum groups Bq,λ(g). In Section 3, we
Dynamical R Matrices of Elliptic Quantum Groups 3
introduce the vertex operators of Uq(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 Bq,λ(g) and the connection matrices for the q-KZ equation of Uq(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 Bq,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 Affine quantum groups Uq(g)
and elliptic quantum groups Bq,λ(g)
2.1 Affine quantum groups Uq(g)
Let g be an affine Lie algebra associated with a generalized Cartan matrix A = (aij), 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 a0 = 1 for all g. Let {αi}i∈I be a set of simple roots and set hi = α∨i .
We have aij = 〈αj , hi〉 = 2(αi|αj)
(αi|αi)
and diaij = aijdj with di = 1
2(αi|αi). We denote the canonical
central element by c =
∑
i∈I a∨i hi and the null root by δ =
∑
i∈I aiαi. We set
Q = Zα0 ⊕ · · · ⊕ Zαn, Q+ = Z≥0α0 ⊕ · · · ⊕ Z≥0αn,
P = ZΛ0 ⊕ · · · ⊕ ZΛn ⊕ Zδ, P ∗ = Zh0 ⊕ · · · ⊕ Zhn ⊕ Zd
and impose the pairings
〈αi, d〉 = a∨0 δi,0, 〈Λj , hi〉 =
1
a∨0
δij , 〈Λj , d〉 = 0, (i, j ∈ I).
The Λj are the fundamental weights. We also use Pcl = P/Zδ, (Pcl)∗ = ⊕n
i=0Zhi ⊂ P ∗. Let
cl : P → Pcl denote the canonical map and define af : Pcl → P by af(cl(αi)) = αi (i 6= 0) and
af(cl(Λ0)) = Λ0 so that cl ◦ af = id and af(cl(α0)) = α0 − δ.
Definition 2.1. The quantum affine algebra Uq = Uq(g) is an associative algebra over C(q1/2)
with 1 generated by the elements ei, fi (i ∈ I) and qh (h ∈ P ∗) satisfying the following relations
q0 = 1, qhqh′ = qh+h′ (h, h′ ∈ P ∗),
qheiq
−h = q〈αi,h〉ei, qhfiq
−h = q−〈αi,h〉fi,
eifj − fjei = δij
ti − t−1
i
qi − q−1
i
,
1−aij∑
m=0
(−1)m
[
1− aij
m
]
qi
e
1−aij−m
i eje
m
i = 0 (i 6= j),
1−aij∑
m=0
(−1)m
[
1− aij
m
]
qi
f
1−aij−m
i fjf
m
i = 0 (i 6= j).
Here qi = qdi, ti = qhi
i , and
[n]x =
xn − x−n
x− x−1
, [n]x! = [n]x[n− 1]x · · · [1]x,
[
n
m
]
x
=
[n]x!
[m]x![n−m]x!
.
4 H. Konno
The algebra Uq has a Hopf algebra structure with comultiplication ∆, counit ε and antipode S
defined by
∆(qh) = qh ⊗ qh,
∆(ei) = ei ⊗ 1 + ti ⊗ ei,
∆(fi) = fi ⊗ t−1
i + 1⊗ fi, (2.1)
ε(qh) = 1, ε(ei) = ε(fi) = 0,
S(qh) = q−h, S(ei) = −t−1
i ei, S(fi) = −fiti.
Uq is a quasi-triangular Hopf algebra with the universal R matrix R satisfying
∆op(x) = R∆(x)R−1 ∀x ∈ Uq,
(∆⊗ id)R = R13R23, (id⊗∆)R = R13R12.
Here ∆op denotes the opposite comultiplication, ∆op = σ ◦∆ with σ being the flip of the tensor
components; σ(a⊗ b) = b⊗ a.
Proposition 2.2.
R(12)R(13)R(23) = R(23)R(13)R(12), (2.2)
(ε⊗ id)R = (id⊗ ε)R = 1,
(S ⊗ id)R = (id⊗ S−1)R = R−1,
(S ⊗ S)R = R.
Let {hl} be a basis of h and {hl} be its dual basis. We denote by U+ (resp. U−) the
subalgebra of Uq generated by ei (resp. fi) i ∈ I and set
U+
β = {x ∈ U+| qhxq−h = q〈β,h〉x (h ∈ h)},
U−
−β = {x ∈ U−| qhxq−h = q−〈β,h〉x (h ∈ h)}
for β ∈ Q+. The universal R matrix has the form [31]
R = q−TR0, T =
∑
l
hl ⊗ hl,
R0 =
∑
β∈Q+
q(β,β)
(
q−β ⊗ qβ
)
(R0)β = 1−
∑
i∈I
(qi − q−1
i )eit
−1
i ⊗ tifi + · · · ,
(R0)β =
∑
j
uβ,j ⊗ uj
−β ∈ U+
β ⊗ U−
−β, (2.3)
where {uβ,j} and {uj
−β} are bases of U+
β and U−
−β , respectively. Note that T is the canonical
element of h⊗ h w.r.t (·|·) and (R0)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 Uq generated by ei, fi (i ∈ I) and h ∈ (Pcl)∗.
Let (πV , V ) be a finite dimensional module over U ′
q. We have the evaluation representation
(πV,z, Vz) of Uq by Vz = C(q1/2)[z, z−1]⊗C(q1/2) V and
πV,z(ei)(zn ⊗ v) = zδi0+n ⊗ π(ei)v, πV,z(fi)(zn ⊗ v) = z−δi0+n ⊗ π(fi)v,
πV,z(ti)(zn ⊗ v) = zn ⊗ π(ti)v, πV,z(qd)(zn ⊗ v) = (qz)n ⊗ v,
wt(zn ⊗ v) = nδ + af(wt(v)),
Dynamical R Matrices of Elliptic Quantum Groups 5
where n ∈ Z, and v ∈ V denotes a weight vector whose weight is wt(v). We write vzn = zn ⊗ v
(n ∈ Z).
For generic λ ∈ h∗, let Mλ denote the irreducible Verma module with the highest weight λ. We
have the weight space decomposition Mλ =
⊕
ν∈λ−Q+
(Mλ)ν . We write wt(u) = ν for u ∈ (Mλ)ν .
2.2 Elliptic quantum groups Bq,λ(g)
Let ρ ∈ h be an element satisfying (ρ|αi) = di for all i ∈ I. For generic λ ∈ h, let us consider an
automorphism of Uq given by
ϕλ = Ad(q−2θ(λ)), θ(λ) = −λ + ρ− 1
2
∑
l
hlh
l,
where Ad(x)y = xyx−1. We define the face type twistor F (λ) ∈ Uq⊗̂Uq as follows.
Definition 2.3 (Face type twistor).
F (λ) = · · ·
(
(ϕλ)2 ⊗ id
)
R−1
0
(
ϕλ ⊗ id
)
R−1
0
=
x∏
k≥1
(
(ϕλ)k ⊗ id
)
R−1
0 , (2.4)
where
x∏
k≥1
Ak = · · ·A3A2A1.
Note that the k-th factor in the product (2.4) is a formal power series in xk
i = q2k〈αi,λ〉 (i ∈ I)
with leading term 1.
Theorem 2.4 ([23]). The twistor F (λ) satisfies the shifted cocycle condition and the normali-
zation condition given by
1) F (12)(λ)(∆⊗ id)F (λ) = F (23)(λ + h(1))(id⊗∆)F (λ), (2.5)
2) (ε⊗ id) F (λ) = (id⊗ ε) F (λ) = 1. (2.6)
In (2.5), λ and h(1) means λ =
∑
l λlh
l and h(1) =
∑
l h
(1)
l hl, h
(1)
l = hl ⊗ 1⊗ 1, respectively.
Note that from (2.3), one has
[h⊗ 1 + 1⊗ h, F (λ)] = 0 ∀h ∈ h.
Now let us define ∆λ, R(λ), Φ(λ) and αλ, βλ by
∆λ(a) = F (12)(λ) ∆(a) F (12)(λ)−1, (2.7)
R(λ) = F (21)(λ)RF (12)(λ)−1, (2.8)
Φ(λ) = F (23)(λ)F (23)(λ + h(1))−1, (2.9)
αλ =
∑
i
S(di)li, βλ =
∑
i
miS(gi) (2.10)
for
∑
i ki ⊗ li = F (λ)−1,
∑
i mi ⊗ ni = F (λ).
Definition 2.5 (Face type elliptic quantum group). With S and ε defined by (2.1), the
set (Uq(g),∆λ, S, ε, αλ, βλ,Φ(λ),R(λ)) forms a quasi-Hopf algebra [23]. We call it the face type
elliptic quantum group Bq,λ(g).
6 H. Konno
From (2.2), (2.5) and (2.8), one can show that R(λ) satisfies the dynamical YBE.
Theorem 2.6 (Dynamical Yang–Baxter equation).
R(12)(λ + h(3))R(13)(λ)R(23)(λ + h(1)) = R(23)(λ)R(13)(λ + h(2))R(12)(λ). (2.11)
We hence call R(λ) the universal dynamical R matrix.
Now let us parametrize λ in the following way.
λ =
(
r +
h∨
a∨0
)
d + sc + λ̄ (r, s ∈ C), (2.12)
where λ̄ stands for the classical part of λ, and h∨ denotes the dual Coxeter number of g. Note
also ρ = h∨Λ0 + ρ̄ and d = a∨0 Λ0. Let {h̄j} and {h̄j(= Λ̄j)} denote the classical part of the basis
and its dual of h. We then have
ϕλ = Ad(pdq2cdq−2θ̄(λ)), θ̄(λ) = −λ̄ + ρ̄− 1
2
∑
h̄j h̄
j . (2.13)
Here we set p = q2r. Set further
R(z) = Ad(zd ⊗ 1)(R), (2.14)
F (z, λ) = Ad(zd ⊗ 1)(F (λ)), (2.15)
R(z, λ) = Ad(zd ⊗ 1)(R(λ)) = σ(F (z−1, λ))R(z)F (z, λ)−1. (2.16)
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]).
F (pq2c(1)z, λ) = Ad(q2θ̄(λ) ⊗ id)
(
F (z, λ)
)
· qTR(pq2c(1)z). (2.17)
Furthermore, noting Ad(zd)(ei) = zδi,0ei, one can drop all the e0 dependent terms in R(z)
and F (z, λ) by taking the limit z → 0. We thus obtain
lim
z→0
qc⊗d+d⊗cR(z) = Rḡ, (2.18)
lim
z→0
F (z, λ) = Fḡ(λ̄), (2.19)
where Rḡ and Fḡ(λ̄) are the universal R matrix and the twistor of Uq(ḡ). Then from (2.17), we
obtain the following equation for Fḡ(λ̄).
Lemma 2.8.
Fḡ(λ) = Ad(q2θ̄(λ) ⊗ id)
(
Fḡ(λ̄)
)
· qT̄Rḡ, (2.20)
where T̄ =
n∑
i=1
h̄i ⊗ h̄i.
Remark. (2.20) corresponds to (18) in [32], where the comultiplication and the universal R
matrix are our ∆op and R−1
ḡ , respectively.
Dynamical R Matrices of Elliptic Quantum Groups 7
Lemma 2.9 ([32]). The equation (2.20) has the unique solution Fḡ(λ̄) ∈ Uq(b̄+)⊗̂Uq(b̄−) in
the form Fḡ(λ̄) = 1 + · · · . Here Uq(b̄+) (resp. Uq(b̄−)) is the subalgebra of Uq(ḡ) generated by
ei, ti (i = 1, 2, . . . , n) (resp. fi, ti (i = 1, 2, . . . , n)).
Theorem 2.10. For λ ∈ h given by (2.12), the difference equation (2.17) has a unique solution.
Proof. Let us set ϕ̄λ = Ad(q−2θ̄(λ)). Iterating (2.17), N times, we obtain
F (z, λ) =
(
ϕ̄N
λ ⊗ id
)
F ((pq2c(1))Nz, λ)
x∏
N≥k≥1
(
(ϕ̄λ)k ⊗ id
)
R0((pq2c(1))kz)−1.
Taking the limit N →∞, one obtains
F (z, λ) = A
x∏
k≥1
(
(ϕ̄λ)k ⊗ id
)
R0((pq2c(1))kz)−1,
where we set
A = lim
N→∞
(
ϕ̄N
λ ⊗ id
)
F ((pq2c(1))Nz, λ)
= lim
N→∞
(
ϕ̄N
λ ⊗ id
)
Fḡ(λ̄).
Then the statement follows from Lemma 2.9. �
3 Vertex operators and fusion matrices
3.1 The vertex operators of Uq(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 Mλ and Mµ the two irreducible Verma
modules with highest weights λ and µ, respectively.
Definition 3.1 (Vertex operator). Writing 4λ = (λ|λ+2ρ)
2(k+h∨)
1, let us consider the formal series
given by
Ψµ
λ(z) = z4µ−4λΨ̃µ
λ(z), Ψ̃µ
λ(z) =
∑
j
∑
n∈Z
vjz
−n ⊗ (Ψ̃µ
λ)j,n. (3.1)
Here {vj} denotes a weight basis of V . The coefficients (Ψ̃µ
λ)j,n are the maps
(Ψ̃µ
λ)j,n : (Mλ)ξ → (Mµ)ξ−wt(vj)+nδ, (3.2)
such that Ψ̃µ
λ(z) is the Uq-module intertwiners
Ψ̃µ
λ(z) : Mλ → Vz⊗̂ Mµ,
Ψ̃µ
λ(z) x = ∆(x) Ψ̃µ
λ(z) ∀x ∈ Uq. (3.3)
Here ⊗̂ denotes a formal completion
M⊗̂ N =
⊕
ν
∏
ξ
Mξ⊗ Nξ−ν .
We call Ψµ
λ(z) the vertex operator (VO) of Uq.
1Hopefully, there is no confusion of 4λ with ∆λ in (2.7).
8 H. Konno
Remark. Ψµ
λ(z) is the type II VO in the terminology of [3].
We also define U ′
q-module intertwiners Ψµ
λ : Mλ → V ⊗ M̂µ by
Ψµ
λ =
∑
j
vj ⊗
(∑
n∈Z
(Ψ̃µ
λ)j,n
)
. (3.4)
Here M̂µ =
∏
ν(Mµ)ν . Note that there is a bijective correspondence between Ψµ
λ(z) and Ψµ
λ.
Let uλ and uµ denote the highest weight vectors of Mλ and Mµ, respectively. Let us write
the image of uλ by the VO as
Ψµ
λ uλ = v ⊗ uµ +
∑
i′
vi′ ⊗ ui′ , (3.5)
where ui′ ∈ Mµ, wt(ui′) < µ and v, vi′ ∈ V . We call the vector v the leading term of Ψµ
λ. Note
that from (3.2), cl(λ) = wt(v) + cl(µ) = wt(vi′) + cl(wt(ui′)). We set
V µ
λ = {v ∈ V | wt(v) = cl(λ− µ)}.
Theorem 3.1 ([16, 17]). The map 〈 〉 : Ψµ
λ 7→ 〈id ⊗ u∗µ,Ψµ
λuλ〉 gives a C(q1/2)-linear isomor-
phism
HomU ′q(g)(Mλ, V ⊗ M̂µ) ∼−→ V µ
λ
This theorem tells that Ψµ
λ is determined by its leading term. Namely, for given v0 ∈ V µ
λ ,
there exists a unique VO satisfying
〈Ψµ
λ〉 = v0.
We denote such VO by Ψµ,v0
λ and corresponding Uq-intertwiner by Ψµ,v0
λ (z).
Proposition 3.2 ([17]). Let {vj} be a basis of V µ
λ . The set of VOs {Ψµ,vj
λ } forms a basis of
HomU ′q(g)(Mλ, V ⊗ M̂µ).
3.2 The q-KZ equation and connection matrices
Let λ, µ, ν ∈ h∗ be level-k elements. Let {vi} and {wj} be weight bases of V µ
λ and W ν
µ , respec-
tively. Consider the VOs Ψµ,vi
λ and Ψν,wj
µ given by
Ψµ,wi
λ (z1) : Mλ → Wz1⊗̂ Mµ,
Ψν,vj
µ (z2) : Mµ → Vz2⊗̂ Mν ,
and their composition(
id⊗Ψν,vi
µ (z2)
)
Ψµ,wj
λ (z1) : Mλ → Wz1⊗ Vz2⊗̂ Mν .
Setting
Ψ(ν,µ,λ)(z1, z2) =
〈
id⊗ id⊗ u∗ν ,
(
id⊗Ψν,vi
µ (z2)
)
Ψµ,wj
λ (z1)uλ
〉
,
we call Ψ(ν,µ,λ)(z1, z2) the two-point function.
Theorem 3.3 (q-KZ equation [16, 33]). The two-point function Ψ(ν,µ,λ)(z1, z2) satisfies the
q-KZ equation
Ψ(ν,µ,λ)(q2(k+h∨)z1, z2) = (q−πW (ν̄+λ̄+2ρ̄) ⊗ id)RWV (z)Ψ(ν,µ,λ)(z1, z2), (3.6)
where RWV (z) = (πW ⊗ πV )R(z).
Dynamical R Matrices of Elliptic Quantum Groups 9
Proof. See Appendix. �
Remark [16]. A solution of the q-KZ equation (3.6) is a function of z = z1/z2 and has
a form G(z)f(z1, z2). Here G(z) is a meromorphic function multiplied by a fractional power
of z determined from the normalization function of the R matrix RWV (z), while f(z1, z2) is
an analytic function in |z1| > |z2| 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 |z1| > |z2| 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.
Theorem 3.4 (Connection formula [16, 17]).
(PRWV (z1/z2)⊗ id)
(
id⊗Ψνvi
µ (z2)
)
Ψµwj
λ (z1)
=
∑
i′,j′,µ′
(
id⊗Ψ
νwj′
µ′ (z1)
)
Ψµ′vi′
λ (z2)CWV
λ wj µ
vi′ vi
µ′ wj′ ν
∣∣∣∣∣∣ z1
z2
,
where vi′ and wj′ are base vectors of V µ′
λ and W ν
µ′, respectively.
The matrix CWV is called the connection matrix. The following theorem states basic proper-
ties of the connection matrix.
Theorem 3.5 ([16, 17]). 1) The matrix elements of CWV are given by a ratio of elliptic
theta functions.
2) The matrix CWV satisfies i) the face type YBE and ii) the unitarity condition (the first
inversion relation):
i)
∑
vl,wi,uj ,µ′′
CV U
λ uj µ′′
wi′ wi
ω uj′ ν
∣∣∣∣∣∣ z2
z3
CWU
λ′ uj′′ µ′
vl′ vl
λ uj µ′′
∣∣∣∣∣∣ z1
z3
× CWV
µ′ wi′′ µ
vl vl′′
µ′′ wi ν
∣∣∣∣∣∣ z1
z2
=
∑
vl,wi,uj ,µ′′
CWV
λ′ wi µ′′
vl′ vl
λ wi′ ω
∣∣∣∣∣∣ z1
z2
CWU
µ′′ uj µ
vl vl′′
ω uj′ ν
∣∣∣∣∣∣ z1
z3
× CV U
λ′ uj′′ µ′
wi wi′′
µ′′ uj µ
∣∣∣∣∣∣ z2
z3
,
ii)
∑
vi′ ,wj′ ,µ
′
CWV
λ wj µ
vi′ vi
µ′ wj′ ν
∣∣∣∣∣∣ z
CV W
λ vi′ µ′
wj′′ wj′
µ′′ vi′′ ν
∣∣∣∣∣∣ z−1
= δwj ,wj′′ δvi,vi′′ δµ,µ′′ .
3) In the case V = W , CV V satisfies the second inversion relation
∑
vi′ ,wj′ ,λ
GλGν
GµGµ′
CV V
λ wj µ
vi′ vi
µ′ wj′ ν
∣∣∣∣∣∣ z−1
CV V
λ vi′ µ′
wj wj′′
µ vi′′ ν ′
∣∣∣∣∣∣ ξ−2z
10 H. Konno
= αV V (z)δvi,vi′′ δwj ,wj′′ δν,ν′ .
Here ξ = qth∨ (t = (long root)2/2), and Gλ and αV V (z) are given by (5.16) and (5.13)
in [17], respectively.
Furthermore, if Vz is self dual i.e. there exists an isomorphism of Uq-modules Q : Vξ−1z ' V ∗a
z ,
we have the following crossing symmetry.
Theorem 3.6 ([17]).
βV V (z−1)CV V
µ wj ν
vi wj′
λ vi′ µ′
∣∣∣∣∣∣ ξ−1z−1
=
∑
ĩ,j̃
γµĩi
λ gνj̃j′
µ′ CV V
λ vĩ µ
vi′ wj
µ′ wj̃ ν
∣∣∣∣∣∣ z
, (3.7)
where gνjj′
µ denotes a certain matrix element appearing in the inversion relation of the VO’s,
and γν
µ is its inverse matrix such that
∑
j̃ gνjj̃
µ γνj̃j′
µ = δjj′.
In Section 5, we will discuss the evaluation molude Vz with V being the vector representation
for g = A
(1)
n , B
(1)
n , C
(1)
n , D
(1)
n . There Vz is self dual except for A
(1)
n (n > 1), and is multiplicity
free. Therefore, dim V µ
λ = 1 etc. Hence the matrices gν
µ, γν
µ are scalars satisfying γν
µ = 1/gν
µ. In
this case, let us consider the gauge transformation
CV V
(
µ ν
λ µ′
∣∣∣∣ z) = f(z)
F (µ, ν)F (ν, µ′)
F (µ, λ)F (λ, µ′)
C̃V V
(
µ ν
λ µ′
∣∣∣∣ z)
with the choice
f(z)f(z−1) = 1, f(ξ−1z−1) = βV V (ξz)f(z),
F (ν, µ)F (µ, ν) = gν
µ
√
Gµ
Gν
.
Then we can change the crossing symmetry relation (3.7) to
C̃V V
(
µ ν
λ µ′
∣∣∣∣ ξ−1z−1
)
=
√
GνGλ
GµGµ′
C̃V V
(
λ µ
µ′ ν
∣∣∣∣ z) .
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 αV V (z) in the RHS
of 3), which is changed to 1.
3.3 Fusion matrices
We here follows the idea by Etingof and Varchenko [28], where the cases Uq(ḡ) with ḡ being
simple Lie algebras are discussed. We extend their results to the cases Uq(g) with g being affine
Lie algebras.
Definition 3.2 (Fusion matrix). Fix λ ∈ h∗. The fusion matrix is defined to be a h-linear
map JWV (λ) : W ⊗ V → W ⊗ V satisfying
JV W (λ) =
⊕
ν
JV W (λ)ν ,
JWV (λ)ν(wj ⊗ vi) =
〈
id⊗ id⊗ u∗ν ,
(
id⊗Ψν,vi
µ
)
Ψµ,wj
λ uλ
〉
∈ (W ⊗ V )cl(λ−ν),
for vi ∈ V ν
µ , wj ∈ Wµ
λ .
Dynamical R Matrices of Elliptic Quantum Groups 11
Note that from (3.2),
[h⊗ 1 + 1⊗ h, JWV (λ)] = 0 ∀h ∈ h. (3.8)
Noting (3.5) and the intertwining property of the vertex operators, we have
JWV (λ)ν(wj ⊗ vi) = wj ⊗ vi +
∑
l
Cl(λ)wl ⊗ vl, (3.9)
where wt(vl) < wt(vi), and Cl(λ) is a function of λ. Hence JV W (λ) is an upper triangular matrix
with all the diagonal components being 1. Therefore
Theorem 3.7. The fusion matrix JWV (λ) is invertible.
The definition of JWV (λ) indicates that the leading term of the intertwiner (id⊗Ψν,vi
µ ) Ψµ,wj
λ
is JWV (λ)(wj ⊗ vi). Therefore we write
Ψν,JWV (λ)(wj⊗vi)
λ =
(
id⊗Ψν,vi
µ
)
Ψµ,wj
λ .
Let us define JU,W⊗V (ω) and JU⊗W,V (ω) : U ⊗W ⊗ V → U ⊗W ⊗ V for ω ∈ h∗ by
JU,W⊗V (ω)(ul ⊗ wj ⊗ vi) =
〈
id⊗ id⊗ id⊗ u∗ν ,
(
id⊗Ψν,JWV (λ)(wj⊗vi)
λ
)
Ψλ,ul
ω uω
〉
,
JU⊗W,V (ω)(ul ⊗ wj ⊗ vi) =
〈
id⊗ id⊗ id⊗ u∗ν , (id⊗ id⊗Ψν,vi
µ )Ψµ,JWU (ω)(wj⊗ul)
ω uω
〉
,
where ul ∈ Uλ
ω .
Theorem 3.8. The fusion matrix satisfies the shifted cocycle condition
JU⊗W,V (ω)(JUW (ω)⊗ id) = JU,W⊗V (ω)(id⊗ JWV (ω − h(1))) on U ⊗W ⊗ V, (3.10)
where hvj = wt(vj)vj (vj ∈ V ) etc.
Proof. Consider the composition
(id⊗ id⊗Ψν,vi
µ )(id⊗Ψµ,wj
λ )Ψλ,ul
ω :
Mω
Ψ
λ,ul
ω−→ U ⊗ M̂λ
id⊗Ψ
µ,wj
λ−→ U ⊗W ⊗ M̂µ
id⊗id⊗Ψ
ν,vi
µ−→ U ⊗W ⊗ V ⊗ M̂ν
and express the highest to highest expectation value of it in two ways, and use cl(λ) = cl(ω)−
wt(ul). �
Remark. Regarding (∆ ⊗ id)J(ω) = JU⊗W,V (ω), (id ⊗ ∆)J(ω) = JU,W⊗V (ω), J (12)(ω) =
JUW (ω) ⊗ id and J (23)(ω) = id ⊗ JWV (ω), one obtains (2.5) from (3.10) by replacing J(ω) by
F−1(−ω).
Now let λ ∈ h∗ be a level-k element. By using the Uq-module VOs (3.1), we define a h-linear
map JWV (z1, z2;λ) : Wz1 ⊗ Vz2 → Wz1 ⊗ Vz2 by
JWV (z1, z2;λ) =
⊕
ν
JWV (z1, z2;λ)ν ,
JWV (z1, z2;λ)ν(wj ⊗ vi)
=
〈
id⊗ id⊗ u∗ν ,
(
id⊗ Ψ̃ν,vi
µ (z2)
)
Ψ̃µ,wj
λ (z1)uλ
〉
∈ (Wz1 ⊗ Vz2)cl(λ−ν),
for vj ∈ V ν
µ , wj ∈ Wµ
λ . Then from the q-KZ equation (3.6), one can derive the following
difference equation for JWV (z1, z2;λ).
12 H. Konno
Lemma 3.9.
JWV (q2(k+h∨)z1, z2;λ)(q−2πW (θ̄(−λ)) ⊗ id)
= (q−2πW (θ̄(−λ)) ⊗ id)qπW⊗V (T̄ )RWV (z1/z2)JWV (z1, z2, λ). (3.11)
Proof. See Appendix. �
From the remark below Theorem 3.3, JWV (z1, z2;λ) is a function of the ratio z = z1/z2. 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 FWV (z,−λ)−1 = (πW ⊗πV )F (z,−λ)−1 coincides with the q-KZ
equation (3.11) for JWV (z1, z2;λ) on wj ⊗ vi 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),〈
id⊗ id⊗ u∗ν ,
(
id⊗ Ψ̃ν,vi
µ (z2)
)
Ψ̃µ,wj
λ (z1)uλ
〉
= FWV (z1/z2,−λ)−1 (wj ⊗ vi).
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
ḡ , respectively, and the twistor is
identified with the two point function of the Uq(ḡ)-analogue of the type I VOs.
4 Dynamical R matrices and connection matrices
Let (πV , V ), (πW ,W ) be finite dimensional representations of U ′
q(g), and {vi}, {wj} be their
weight bases, respectively. Now we consider the dynamical R matrices given as the images of
the universal R matrix R(λ)
RWV (z, λ) = (πW ⊗ πV )R(z, λ)
= F
(21)
V W (z−1, λ)RWV (z)FWV (z, λ)−1. (4.1)
Note that F
(21)
V W (z−1, λ) = PFV W (z−1, λ)P .
By using Theorems 3.10 and 3.4, we show that the dynamical R matrix RWV (z, λ) coincides
with the corresponding connection matrix for the q-KZ equation of Uq(g) associated with the
representations (πV , V ) and (πW ,W ).
Theorem 4.1. For level-k λ ∈ h∗ in the form (2.12) with r = −(k + h∨), we have
RWV (z,−λ)(wj ⊗ vi) =
∑
i′,j′,µ′
CWV
λ wj µ
vi′ vi
µ′ wj′ ν
∣∣∣∣∣∣ z
(wj′ ⊗ vi′),
where vi ∈ V ν
µ , wj ∈ Wµ
λ , vi′ ∈ V µ′
λ and wj′ ∈ W ν
µ′.
Proof. From Theorem 3.10 and (4.1), we have
RWV (z1/z2,−λ)(wj ⊗ vi)
= F
(21)
V W (z−1,−λ)RWV (z)
〈
id⊗ id⊗ u∗ν ,
(
id⊗ Ψ̃ν,vi
µ (z2)
)
Ψ̃µ,wj
λ (z1)uλ
〉
,
where we set z = z1/z2. Then from Theorem 3.4, we obtain
RWV (z1/z2,−λ)(wj ⊗ vi) = F
(21)
V W (z−1,−λ)
Dynamical R Matrices of Elliptic Quantum Groups 13
×
∑
i′,j′,µ′
P
〈
id⊗ id⊗ u∗ν ,
(
id⊗ Ψ̃
ν,wj′
µ′ (z1)
)
Ψ̃µ,vi′
λ (z2)uλ
〉
CWV
λ wj µ
vi′ vi
µ′ wj′ ν
∣∣∣∣∣∣ z
= F
(21)
V W (z−1,−λ)
∑
i′,j′,µ′
PF (z2/z1,−λ)−1(vi′ ⊗ wj′)CWV
λ wj µ
vi′ vi
µ′ wj′ ν
∣∣∣∣∣∣ z
=
∑
i′,j′,µ′
CWV
λ wj µ
vi′ vi
µ′ wj′ ν
∣∣∣∣∣∣ z
(wj′ ⊗ vi′). �
Note that in view of Theorem 3.5, this theorem indicates that the dynamical R matrices of
Bq,λ(g) are elliptic.
5 Vector representations
In this section, we consider the vector representation of the universal R matrix R(λ) of Bq,λ(g)
for g = A
(1)
n , B
(1)
n , C
(1)
n , D
(1)
n , and show that they coincide with the corresponding face weights
obtained by Jimbo, Miwa and Okado [13].
5.1 Jimbo–Miwa–Okado’s solutions
Let us summarize Jimbo–Miwa–Okado’s elliptic solutions to the face type YBE.
Let (πV , V ) be the vector representation of U ′
q(g). We set dim V = N . Then N = n+1, 2n+1,
2n, 2n for g = A
(1)
n , B
(1)
n , C
(1)
n , D
(1)
n , respectively. Let us define an index set J by
J = {1, 2, . . . , n + 1} for A(1)
n
= {0,±1, . . . ,±n} for B(1)
n
= {±1, . . . ,±n} for C(1)
n , D(1)
n
and introduce a linear order ≺ in J by
1 ≺ 2 ≺ · · · ≺ n (≺ 0) ≺ −n ≺ · · · ≺ −2 ≺ −1.
We also use the usual order < in J .
Let Λ̄j (1 ≤ j ≤ n) be the fundamental weights of ḡ. Following Bourbaki [34] we introduce
orthonormal vectors {ε1, . . . , εn} with the bilinear form (εi|εj) = δi,j . Then one has the following
expression of Λ̄j as well as the set A of weights belonging to the vector representation of ḡ.
An : A = {ε1 − ε, . . . , εn+1 − ε},
Λ̄j = ε1 + · · ·+ εj − jε (1 ≤ j ≤ n), ε =
1
n + 1
n+1∑
j=1
εj ,
Bn : A = {±ε1, . . . ,±εn, 0},
Λ̄j = ε1 + · · ·+ εj (1 ≤ j ≤ n− 1),
=
1
2
(ε1 + · · ·+ εn) (j = n),
Cn : A = {±ε1, . . . ,±εn},
Λ̄j = ε1 + · · ·+ εj (1 ≤ j ≤ n),
Dn : A = {±ε1, . . . ,±εn},
14 H. Konno
Λ̄j = ε1 + · · ·+ εj (1 ≤ j ≤ n− 2),
=
1
2
(ε1 + · · ·+ εn−2 + εn−1 − εn) (j = n− 1),
=
1
2
(ε1 + · · ·+ εn−2 + εn−1 + εn) (j = n),
Now for µ ∈ J , let us define µ̂ ∈ A by
µ̂ = εµ − ε (1 ≤ µ ≤ n + 1) for An,
= ±εj or 0 (µ = ±j (1 ≤ j ≤ n), or µ = 0) for Bn,
= ±εj (µ = ±j (1 ≤ j ≤ n)) for Cn, Dn.
We then define a dynamical variable a ∈ h∗ of the face model of type g as follows.
aµ = (a + ρ|µ̂) (µ 6= 0),
= −1
2
(µ = 0).
We also set aµν = aµ − aν .
Proposition 5.1. If one parameterizes a ∈ h∗ such that a + ρ =
n∑
i=0
siΛi, one has
aµ =
1
n + 1
− µ−1∑
j=1
jsj +
n∑
j=µ
(n + 1− j)sj
(1 ≤ µ ≤ n + 1) for A(1)
n ,
= ±
n−1∑
j=i
sj +
sn
2
or − 1
2
(µ = ±i (1 ≤ i ≤ n), or 0) for B(1)
n ,
= ±
n∑
j=i
sj (µ = ±i (1 ≤ i ≤ n)) for C(1)
n ,
= ±
n−1∑
j=i
sj +
sn − sn−1
2
(µ = ±i (1 ≤ i ≤ n)) for D(1)
n .
Note that aµν =
ν−1∑
j=µ
sj for A
(1)
n and a−µ = −aµ for B
(1)
n , C
(1)
n , D
(1)
n .
Then Jimbo–Miwa–Okado’s solutions to the face type YBE are given as follows.
W
(
a b
c d
∣∣∣∣∣u
)
= κ(u)W
(
a b
c d
∣∣∣∣∣u
)
, (5.1)
(I) W
(
a a + µ̂
a + µ̂ a + 2µ̂
∣∣∣∣∣u
)
= 1 (µ 6= 0),
W
(
a a + µ̂
a + µ̂ a + µ̂ + ν̂
∣∣∣∣∣u
)
=
[1][aµν − u]
[1 + u][aµν ]
(µ 6= ±ν),
W
(
a a + ν̂
a + µ̂ a + µ̂ + ν̂
∣∣∣∣∣u
)
=
[u]
√
[aµν + 1][aµν − 1]
[1 + u][aµν ]
(µ 6= ±ν),
for A(1)
n , B(1)
n , C(1)
n , D(1)
n ,
Dynamical R Matrices of Elliptic Quantum Groups 15
(II) W
(
a a + ν̂
a + µ̂ a
∣∣∣∣∣u
)
=
[u][1][aµ,−ν + 1 + η − u]
[η − u][1 + u][aµ,−ν + 1]
√
GaµGaν (µ 6= ν),
W
(
a a + µ̂
a + µ̂ a
∣∣∣∣∣u
)
=
[η + u][1][aµ,−µ + 1 + 2η − u]
[η − u][1 + u][aµ,−µ + 1 + 2η]
− [u][1][aµ,−µ + 1 + η − u]
[η − u][1 + u][aµ,−µ + 1 + 2η]
∑
κ 6=µ
[aµ,−κ + 1 + 2η]
[aµ,−κ + 1]
Gaµ, for B(1)
n , C(1)
n , D(1)
n ,
where η = −th∨/2 (t = (long root)2/2) is the crossing parameter, and the symbol [u] denotes
the Jacobi elliptic theta function
[u] = qr/4eπi/4
(
− 2πi
log p
)−1/2
q
u2
r
−uΘp(q2u), p = q2r, (5.2)
Θp(z) = (z; p)∞(p/z; p)∞(p; p)∞, (z; p)∞ =
∞∏
n=0
(1− zpn).
The κ(u) denotes a function satisfying the following relations
κ(u)κ(−u) = 1 for A(1)
n , B(1)
n , C(1)
n , D(1)
n ,
κ(η − u)κ(η + u) =
[1 + η + u][1 + η − u]
[η + u][η − u]
for A(1)
n ,
κ(u)κ(η + u) =
[−u][1 + η + u]
[1− u][η + u]
for B(1)
n , C(1)
n , D(1)
n .
The Gaµ = Ga+µ̂
Ga
denotes a ratio of the principally specialized character Ga for the dual affine
Lie algebra g∨ [13]
Ga =
∏
1≤i<j≤n+1
[ai − aj ] for A(1)
n ,
= ε(a)
n∏
i=1
h(ai)
∏
1≤i<j≤n
[ai − aj ][ai + aj ] for B(1)
n , C(1)
n , D(1)
n .
Here ε(a) denotes a sign factor such that ε(a+ µ̂)/ε(a) = s. s and h(a) are listed in the following
Table
A
(1)
n B
(1)
n C
(1)
n D
(1)
n
h∨ n + 1 2n− 1 n + 1 2n− 2
t 1 1 2 1
s 1 1 −1 1
h(a) 1 [a] [2a] 1
Remark. Our normalization of the weights W and some notations are different from those
in [13]. Their relations are given as follows
W
(
a b
c d
∣∣∣∣∣u
)
=
[1]
[1 + u]
κ(u)WJMO
(
a b
c d
∣∣∣∣∣u
)
for A(1)
n ,
=
[1][η]
[1 + u][η − u]
κ(u)WJMO
(
a b
c d
∣∣∣∣∣u
)
for B(1)
n , C(1)
n , D(1)
n ,
r = LJMO, log p =
4π2
log pJMO
.
Here the symbols with subindex JMO denote the ones in [13]. Note [u] = [u]JMO.
The following theorem states basic properties of the face weights.
16 H. Konno
Theorem 5.2 ([13]). The face weight W satisfies i) the face type Yang–Baxter equation, ii) the
first and iii) the second inversion relations
i)
∑
g
W
(
f g
e d
∣∣∣∣∣u
)
W
(
a b
f g
∣∣∣∣∣u + v
)
W
(
b c
g d
∣∣∣∣∣v
)
=
∑
g
W
(
a b
g c
∣∣∣∣∣u
)
W
(
g c
e d
∣∣∣∣∣u + v
)
W
(
a g
f e
∣∣∣∣∣v
)
,
ii)
∑
g
W
(
a g
d c
∣∣∣∣∣u
)
W
(
a b
g c
∣∣∣∣∣−u
)
= δbd,
iii)
∑
g
GaGg
GbGd
W
(
a b
d g
∣∣∣∣∣−u
)
W
(
c d
b g
∣∣∣∣∣2η + u
)
= δac.
In addition, we have the crossing symmetry except for g = A
(1)
n (n > 1)
iv) W
(
a b
c d
∣∣∣∣∣u
)
=
√
GbGc
GaGd
W
(
c a
d b
∣∣∣∣∣η − u
)
.
The following theorem is communicated by Jimbo and Okado and is not written explicitly
in [13].
Theorem 5.3. For g = B
(1)
n , C
(1)
n , D
(1)
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). �
=a + µ
a a + µ
a + µ + ν
a + νa + µ + ν
a
u + v
u
v + a + µ
a a + µ
a + µ + ν
a + νa + µ + ν
a + µ + ν
u + v
u
v
a a + µ
a + νa + µ + ν
+
a + µ + ν
u + v
u
va + µ
a + µ
a a + µ
a + νa + µ + ν
a + µ + ν
u + v
u
va + µ
a + ν
=a + ν
a a + µ
a
a + νa + µ + ν
a
u + v
u
v + a + ν
a a + µ
a
a + νa + µ + ν
a + µ + ν
u + v
u
v
a a + µ
a + νa + µ + ν
+
a
u + v
u
va + ν
a + µ
a a + µ
a + νa + µ + ν
a
u + v
u
va + ν
a + ν
Figure 1. Two relevant equations (µ 6= ±ν).
5.2 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 RV V (z, λ) as the connection matrix in the vector representation (πV , V ),
and argue that it coincides with Jimbo–Miwa–Okado’s solution up to a gauge transformation.
Dynamical R Matrices of Elliptic Quantum Groups 17
Let us consider the difference equation (2.17) in the vector representation.
F (pz, λ) = (q2πV (θ̄(λ)) ⊗ id)F (z, λ)(q−2πV (θ̄(λ)) ⊗ id)qπV⊗V (T )R(pz), (5.3)
where λ is parameterized as (2.12), θ̄(λ) is given by (2.13), T = c ⊗ Λ0 + Λ0 ⊗ c +
n∑
i=1
h̄i ⊗ h̄i,
F (z, λ) = (πV ⊗ πV )F (z, λ), and R(z) = (πV ⊗ πV )R(z).
Let {vj |j ∈ J} be a basis of V and Ei,j be the matrix unit defined by Ei,jvk = δj,kvi. 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])
πV (e0) = En+1,1 for A(1)
n ,
= (−)n(E−1,2 − E−2,1) for B(1)
n ,
= (−)n−1(E−1,2 − E−2,1) for D(1)
n ,
= E−1,1 for C(1)
n ,
πV (ei) = Ei,i+1 (1 ≤ i ≤ n) for A(1)
n ,
= Ei,i+1 − E−i−1,−i (1 ≤ i ≤ n− 1) for B(1)
n , D(1)
n , C(1)
n ,
πV (en) =
√
[2]qn(En,0 − E0,−n) for B(1)
n ,
= En,−n for C(1)
n ,
= En−1,−n − En,−n+1 for D(1)
n ,
πV (t0) =
∑
j∈J
q−δj,1+δj,n+1Ej,j for A(1)
n ,
=
∑
j∈J
q−δj,1−δj,2+δj,−1+δj,−2Ej,j for B(1)
n , D(1)
n
=
∑
j∈J
q−2δj,1+2δj,−1Ej,j for C(1)
n ,
πV (ti) =
∑
j∈J
qδj,i−δj,i+1Ej,j (1 ≤ i ≤ n) for A(1)
n
=
∑
j∈J
qδj,i−δj,i+1+δj,−i−1−δj,−iEj,j (1 ≤ i ≤ n− 1) for B(1)
n , C(1)
n , D(1)
n ,
πV (tn) =
∑
j∈J
qδj,n−δj,−nEj,j for B(1)
n ,
=
∑
j∈J
q2δj,n−2δj,−nEj,j for C(1)
n ,
=
∑
j∈J
qδj,n−1+δj,n−δj,−n−δj,−n+1Ej,j for D(1)
n ,
and πV (fi) = πV (ei)t.
A basis {h̄i} of h̄ and its dual basis {h̄i} w.r.t (·|·) are given as follows
An : πV (h̄i) = Ei,i − Ei+1,i+1 (1 ≤ i ≤ n),
πV (h̄i) =
1
n + 1
(n− i + 1)
i∑
j=1
Ej,j − i
n+1∑
j=i+1
Ej,j
(1 ≤ i ≤ n),
Bn : πV (h̄i) = Ei,i − Ei+1,i+1 + E−i−1,−i−1 − E−i,−i (1 ≤ i ≤ n− 1),
18 H. Konno
πV (h̄n) = 2(En,n − E−n,−n),
πV (h̄i) =
i∑
j=1
(Ej,j − E−j,−j) (1 ≤ i ≤ n− 1),
πV (h̄n) =
1
2
n∑
j=1
(Ej,j − E−j,−j),
Cn : πV (h̄i) = Ei,i − Ei+1,i+1 + E−i−1,−i−1 − E−i,−i (1 ≤ i ≤ n− 1),
πV (h̄n) = En,n − E−n,−n,
πV (h̄i) =
i∑
j=1
(Ej,j − E−j,−j) (1 ≤ i ≤ n),
Dn : πV (h̄i) = Ei,i − Ei+1,i+1 + E−i−1,−i−1 − E−i,−i (1 ≤ i ≤ n− 1),
πV (h̄n) = En−1,n−1 + En,n − E−n,−n − E−n+1,−n+1,
πV (h̄i) =
i∑
j=1
(Ej,j − E−j,−j) (1 ≤ i ≤ n− 2),
πV (h̄n−1) =
1
2
n−1∑
j=1
(Ej,j − E−j,−j)−
1
2
(En,n − E−n,−n),
πV (h̄n) =
1
2
n−1∑
j=1
(Ej,j − E−j,−j) +
1
2
(En,n − E−n,−n).
Then one can easily verify the following.
Proposition 5.4.
qπV⊗V (T ) = q−
1
n+1
∑
i,j∈J
qδi,jEi,i ⊗ Ej,j for A(1)
n ,
=
∑
i,j∈J
qδi,j−δi,−jEi,i ⊗ Ej,j for B(1)
n , C(1)
n , D(1)
n .
Proposition 5.5. If we parameterize λ̄ such that λ̄ =
n∑
i=1
(si + 1)h̄i, we have
q−2πV (θ̄(λ)) = q
n
n+1
∑
j∈J
q2ajEj,j for A(1)
n ,
=
∑
j∈J
q2aj+1Ej,j for B(1)
n , C(1)
n , D(1)
n ,
where aj (j ∈ J) is given by Proposition 5.1.
The R matrix R(z) of Uq(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)
R(z) = ρ(z)
∑
i∈J
i6=0
Ei,i ⊗ Ei,i + b(z)
∑
i,j
i6=±j
Ei,i ⊗ Ej,j
+
∑
i≺j
i6=−j
(
c(z)Ei,j ⊗ Ej,i + zc(z)Ej,i ⊗ Ei,j
)
Dynamical R Matrices of Elliptic Quantum Groups 19
+
1
(1− q2z)(1− ξz)
∑
i,j
aij(z)Ei,j ⊗ E−i,−j
, (5.4)
b(z) =
q(1− z)
1− q2z
, c(z) =
1− q2
1− q2z
,
ρ(z) = q−
n
n+1
(q2z; ξ2)∞(q−2ξ2z; ξ2)∞
(z; ξ2)∞(ξ2z; ξ2)∞
for A(1)
n ,
= q−1 (q2z; ξ2)∞(ξz; ξ2)2∞(q−2ξ2z; ξ2)∞
(z; ξ2)∞(q−2ξz; ξ2)∞(q2ξz; ξ2)∞(ξ2z; ξ2)∞
for B(1)
n , C(1)
n , D(1)
n ,
aij(z) = 0 for A(1)
n ,
=
(q2 − ξz)(1− z) + δi,0(1− q)(q + z)(1− ξz) (i = j),
(1− q2)[εiεjq
j̄−ī(z − 1) + δi,−j(1− ξz)] (i ≺ j),
(1− q2)z[ξεiεjq
j̄−ī(z − 1) + δi,−j(1− ξz)] (i � j),
for B(1)
n , C(1)
n , D(1)
n .
Here ξ = qth∨ , and εj = 1 (j > 0), −1 (j < 0) for g = C
(1)
n and εj = 1 (j ∈ J) for the other
cases. The symbol j̄ is defined by
j̄ =
j − εj (j = 1, . . . , n),
n− εj (j = 0),
j + N − εj (j = −n, . . . ,−1).
Then due to the formula (2.4), we make the following ansatz for the twistor F (z, λ) in the vector
representation.
F (z, λ) = f(z)
∑
i∈J
i6=0
Ei,i ⊗ Ei,i +
∑
i,j
i6=±j
Xij
ij (z)Ei,i ⊗ Ej,j (5.5)
+
∑
i≺j
i6=−j
(
Xji
ij (z)Ei,j ⊗ Ej,i + Xij
ji (z)Ej,i ⊗ Ei,j
)
+
∑
i,j
Xj,−j
i,−i (z)Ei,j ⊗ E−i,−j
,
where Xkl
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 equa-
tion (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
1× 1 2× 2 N ×N
A
(1)
n n + 1 n(n+1)
2 0
B
(1)
n 2n 2n2 1
C
(1)
n 2n 2n(n− 1) 1
D
(1)
n 2n 2n(n− 1) 1
By using Propositions 5.4, 5.5 and (5.4), (5.5), we obtain the following equations.
1× 1 blocks:
f(pz) = q
n
n+1 ρ(pz)f(z) for An,
= qρ(pz)f(z) for Bn, Cn, Dn.
2× 2 blocks:(
Xij
ij (pz) Xji
ij (pz)
Xij
ji (pz) Xji
ji (pz)
)
= q−1
(
Xij
ij (z) w−1
ij Xji
ij (z)
wijX
ij
ji (z) Xji
ji (z)
)(
b(pz) c(pz)
pzc(pz) b(pz)
)
,
20 H. Konno
(i, j ∈ J, i ≺ j, i 6= −j)
where we set wij = q2(ai−aj).
N ×N block:
Xj,−j
i,−i (pz) =
q−2
(1− pq2z)(1− pξz)
∑
k∈J
q−2(ai−ak)akj(pz)Xk,−k
i,−i (z) (i, j ∈ J).
Here we dropped a scalar factor in the 2 × 2 and N × N blocks by using the equation in the
1× 1 block.
Note that the difference equations in the 2× 2 blocks have the same structure as the one in
the case g = ŝl2, which was analyzed completely in [23]. Let us summarize the essence of it. The
2× 2 block equation consists of two 2nd order q-difference equations of the type
(qc − qa+b+1z)u(q2z)− {(q + qc)− (qa + qb)qz}u(qz) + q(1− z)u(z) = 0.
This equation has two independent solutions of the form zα
∞∑
n=0
anzn around z = 0, which are
given by the basic hypergeometric series
2φ1
(
qa qb
qc ; q, z
)
=
∞∑
n=0
(qa; q)n(qb; q)n
(qc; q)n(q; q)n
zn,
and
z1−c
2φ1
(
qa−c+1 qb−c+1
q2−c ; q, z
)
,
where (x; q)n =
n−1∏
j=0
(1 − xqj), (x; q)0 = 1. The connection formula for these solutions is well
known:
2φ1
(
qa qb
qc ; q, 1/z
)
=
Γq(c)Γq(b− a)Θq(q1−az)
Γq(b)Γq(c− a)Θq(qz) 2φ1
(
qa qa−c+1
qa−b+1 ; q, qc−a−b+1z
)
+
Γq(c)Γq(a− b)Θq(q1−bz)
Γq(a)Γq(c− b)Θq(qz) 2φ1
(
qb qb−c+1
qb−a+1 ; q, qc−a−b+1z
)
, (5.6)
where
Γq(z) =
(q; q)∞
(qz; q)∞
(1− q)1−z.
By using this, one can derive the connection matrices for the 2× 2 block parts.
In our case, the initial condition (2.19) leads to
f(0) = 1,(
Xij
ij (0) Xji
ij (0)
Xij
ji (0) Xji
ji (0)
)
=
(
1 (q−q−1)wij
1−wij
0 1
)
.
The solutions to the 1× 1 and 2× 2 blocks are given as follows.
1× 1 block:
f(z) =
{pz}{pξ2z}
{pq2z}{pq−2ξ2z}
for An,
Dynamical R Matrices of Elliptic Quantum Groups 21
=
{pz}{pq−2ξz}{pq2ξz}{pξ2z}
{pq2z}{pξz}2{pq−2ξ2z}
for Bn, Cn, Dn,
where
{z} =
∞∏
n,m=0
(1− zξ2npm).
2× 2 block:
Xij
ij (z) = 2φ1
(
wijq
2 q2
wij
; p, pq−2z
)
,
Xji
ij (z) =
(q − q−1)wij
1− wij
2φ1
(
wijq
2 pq2
pwij
; p, pq−2z
)
,
Xij
ji (z) =
(q − q−1)pw−1
ij
1− pw−1
ij z
2φ1
(
pw−1
ij q2 pq2
p2w−1
ij
; p, pq−2z
)
,
Xji
ji (z) = 2φ1
(
pw−1
ij q2 q2
pw−1
ij
; p, pq−2z
)
.
Then due to the formulae (5.6) and (4.1) or Theorem 4.1, we determine the 1 × 1 and 2 × 2
blocks of the dynamical R matrix
(πV ⊗ πV )R(z, λ)
= ρell(z)
∑
i∈J
i6=0
Ei,i ⊗ Ei,i +
∑
i≺j
i6=±j
(
Rij
ij(z, wij)Ei,i ⊗ Ej,j + Rji
ji(z, wij)Ej,j ⊗ Ei,i
)
+
∑
i≺j
i6=−j
(
Rji
ij(z, wij)Ei,j ⊗ Ej,i + Rij
ji(z, wij)Ej,i ⊗ Ei,j
)
+
∑
i,j
Rj−j
i−i (z, wij)Ei,j ⊗ E−i,−j
as follows.
1× 1 block:
ρell(z) = f(z−1)ρ(z)f(z)−1
= q−
n
n+1
{q2z}{q−2ξ2z}{p/z}{pξ2/z}
{z}{ξ2z}{pq2/z}{pq−2ξ2/z}
for A(1)
n ,
= q−1 {q2z}{ξz}2{q−2ξ2z}{p/z}{pq−2ξ/z}{pq2ξ/z}{pξ2/z}
{z}{q−2ξz}{q2ξz}{ξ2z}{pq2/z}{pξ/z}2{pq−2ξ2/z}
for B(1)
n , C(1)
n , D(1)
n .
2× 2 blocks: for i ≺ j, i 6= −j,
Rij
ij(z, wij) = q
(pw−1
ij q2; p)∞(pw−1
ij q−2; p)∞
(pw−1
ij ; p)2∞
Θp(z)
Θp(q2z)
,
Rji
ji(z, wij) = q
(wijq
2; p)∞(wijq
−2; p)∞
(wij ; p)2∞
Θp(z)
Θp(q2z)
,
22 H. Konno
Rji
ij(z, wij) =
Θp(q2)
Θp(wij)
Θp(wijz)
Θp(q2z)
,
Rij
ji(z, wij) = z
Θp(q2)
Θp(pw−1
ij )
Θp(pw−1
ij z)
Θp(q2z)
.
By setting z = q2u 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.
Rkl
ij (z, wij) ⇔ W
(
a a + k̂
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.
Theorem 5.6. For g = A
(1)
n , B
(1)
n , C
(1)
n , D
(1)
n , the vector representation of the universal dyna-
mical 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 matri-
ces of the twisted Uq(g) to those of the B
(1)
n , C
(1)
n , D
(1)
n types. In fact, the resultant face weights
have the common 2×2 block part, as a function of aµ, to the cases g = A
(1)
n , B
(1)
n , C
(1)
n , D
(1)
n . The
simplest A
(2)
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.
Conjecture 5.7. Similar statement to Theorem 5.6 is true for Kuniba’s solution of A
(2)
2n , A
(2)
2n−1
types and for Kuniba–Suzuki’s solution of G
(1)
2 type.
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 Uq(g) and write
R =
∑
j
aj ⊗ bj , (A.1)
and set U =
∑
j S(bj)aj =
∑
j bjS
−1(aj) and Z = q2ρU .
Lemma A.1 ([38]).
(1) UxU−1 = S2(x) ∀x ∈ Uq,
(2) ZxZ−1 = x,
(3) Z|V (λ) = q(λ|λ+2ρ)idV (λ).
Dynamical R Matrices of Elliptic Quantum Groups 23
Lemma A.2. For (A.1),∑
j
aj ⊗∆(bj) =
∑
i,j
aiaj ⊗ bj ⊗ bi.
Proof. The statement follows (id⊗∆)R = R(13)R(12). �
Lemma A.3. Let Ψ(z) denote a vertex operator. Then we have
(id⊗ a)Ψ(z) =
∑
(S(a(1))⊗ 1)Ψ(z)a(2),
where we write ∆(a) =
∑
a(1) ⊗ a(2).
Proof.
RHS =
∑
(S(a(1))⊗ 1)∆(a(2))Ψ(z)
=
∑
(S(a(1))a
′
(2) ⊗ a′′(2))Ψ(z)
=
∑
(S(a′(1))a
′′
(1) ⊗ a(2))Ψ(z)
=
∑
(1⊗ ε(a(1))a(2))Ψ(z)
= LHS.
Here we wrote ∆(a(2)) =
∑
a′(2) ⊗ a′′(2) etc. and used (∆ ⊗ id)∆(a) = (id ⊗∆)∆(a) in the 3rd
line and m(S ⊗ id)∆(a) = ε(a) in the 4th line. �
Proof of Lemma 3.9. Let λ, µ, ν ∈ h∗ be level-k elements. Let us set p̃ = q2(k+h∨) and consider
Ψ̃(z1, z2) =
〈
id⊗ id⊗ u∗ν , (id⊗ Ψ̃ν
µ(z2)U)Ψ̃µ
λ(p̃z1)uλ
〉
,
where we abbreviate Ψ̃ν,vi
µ (z2) and Ψ̃µ,wj
λ (z1) as Ψ̃ν
µ(z2) and Ψ̃µ
λ(z1), respectively. We regard U
and its expression in terms of aj , bj as certain images of appropriate representations of Uq(g) in
the following processes. We evaluate Ψ̃(z1, z2) in the following two ways.
1) Substituting U = q−2ρZ and using the intertwining property (3.3) and Lemma A.1 (3),
we have
Ψ̃(z1, z2) = (id⊗ q−2ρ)q−(ν|2ρ)+(µ|µ+2ρ)JWV (p̃z1, z2;λ)(wj ⊗ vi)
= (id⊗ q−2ρ)q−(ν|2ρ)+(µ|µ+2ρ)+2(wt(wj)|ρ+λ)−(wt(wj)|wt(wj))
× JWV (p̃z1, z2;λ)(q−2πW (θ̄(λ)) ⊗ id)(wj ⊗ vi).
In the last line, we used (q−2πW (θ̄(λ)) ⊗ id)|wj⊗vi = q−2(wt(wj)|ρ+λ)+(wt(wj)|wt(wj)).
2) Using U =
∑
j bjS
−1(aj) and (A.2), we have
(id⊗ Ψ̃ν
µ(z2)U)Ψ̃µ
λ(z1) =
∑
j
(id⊗∆(bj)Ψ̃ν
µ(z2)S−1(aj))Ψ̃
µ
λ(z1)
id⊗ Ψ̃ν
µ(z2)U)Ψ̃µ
λ(z1) =
∑
i,j
(id⊗ (bi ⊗ bj)Ψ̃ν
µ(z2)S−1(ajai))Ψ̃
µ
λ(z1)
id⊗ Ψ̃ν
µ(z2)U)Ψ̃µ
λ(z1) =
∑
i,j
(id⊗ (S(bi)⊗ S(bj))Ψ̃ν
µ(z2)aiaj)Ψ̃
µ
λ(z1)
id⊗ Ψ̃ν
µ(z2)U)Ψ̃µ
λ(z1) =
∑
i,j
(id⊗ (S(bi)⊗ S(bj))Ψ̃ν
µ(z2))(1⊗ ai)(1⊗ aj)Ψ̃
µ
λ(z1)
24 H. Konno
In the 3rd line we used (id⊗S)R = (S−1⊗ id)R. Then apply Lemma A.3 and Lemma A.2 twice
each, we have
(id⊗ Ψ̃ν
µ(z2)U)Ψ̃µ
λ(z1) =
∑
i,j,k,l
(S(ai)S(aj)⊗ (S(bibk)⊗ S(bjbl))Ψ̃ν
µ(z2))Ψ̃
µ
λ(z1)akal.
Take the expectation value 〈id⊗ id⊗ u∗ν , uλ〉, and use〈
u∗ν ,
∑
l
S(bl)aluλ
〉
= q(λ|ν),∑
k
S(bk)⊗ akuλ = qkΛ0+λ̄ ⊗ uλ,∑
j
S(aj)⊗ u∗νS(bj) = q−kΛ0−ν̄ ⊗ u∗ν .
Noting further that (3.2) implies Ψ̃µ
λ(p̃z) = (p̃Λ0 ⊗ id)Ψ̃µ
λ(z), we obtain
Ψ̃(z1, z2) = (p̃Λ0 ⊗ id)q(λ|ν)(1⊗ qkΛ0+λ̄)(q−kΛ0−ν̄ ⊗ 1)
×
∑
i
(S(ai)⊗ S(bi))JWV (z1, z2)(wj ⊗ vi)
= q(λ|ν)+(λ+2ρ|wt(wj)+wt(vi))(q−2θ̄(λ) ⊗ 1)qπW⊗V (T̄ )
×RWV (z1/z2)JWV (z1, z2)(wj ⊗ vi).
Combining 1) and 2), we obtain (3.11). �
Acknowledgments
The author would like to thank Michio Jimbo and Masato Okado for stimulating discussions and
valuable suggestions. He also thanks Atsuo Kuniba and Atsushi Nakayashiki for discussions. He
is also grateful to the organizers of O’Raifeartaigh Symposium, Janos Balog, Laszlo Feher and
Zalan Horvath, for their kind invitation and hospitality during his stay in Budapest.
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1 Introduction
2 Affine quantum groups Uq(g) and elliptic quantum groups Bq,(g)
2.1 Affine quantum groups Uq(g)
2.2 Elliptic quantum groups Bq,(g)
3 Vertex operators and fusion matrices
3.1 The vertex operators of Uq(g)
3.2 The q-KZ equation and connection matrices
3.3 Fusion matrices
4 Dynamical R matrices and connection matrices
5 Vector representations
5.1 Jimbo-Miwa-Okado's solutions
5.2 The difference equations for the twistor
A Proof of Lemma 3.9
|