On different approaches to IRF lattice models. Part II

Abstract

A bstract
This paper represents a continuation of our previous work, where the Boltzmann weights (BWs) for several Interaction-Round-the Face (IRF) lattice models were computed using their relation to rational conformal field theories. Here, we focus on deriving solutions for the Boltzmann weights of the Interaction-Round the Face lattice model, specifically, the unrestricted face model, based on the su(3)k \mathfrak{su}{(3)}_k su 3 k affine Lie algebra. The admissibility conditions are defined by the adjoint representation. We find the BWs by determining the quantum R matrix of the Uq(sl(3)) {U}_q\left(\mathfrak{sl}(3)\right) U q sl 3 quantum algebra in the adjoint representation and then applying the so-called Vertex-IRF correspondence. The Vertex-IRF correspondence defines the BWs of IRF models in terms of R matrix elements.

Full text

JHEP02(2025)200
Published for SISSA by Springer
Received: September 15, 2024
Revised: January 6, 2025
Accepted: February 8, 2025
Published: February 27, 2025
On different approaches to IRF lattice models. Part II
Vladimir Belavin ,aDoron Gepner ,bJ. Ramos Cabezas aand Boris Runov a
aPhysics Department, Ariel University,
Ariel 40700, Israel
bDepartment of Particle Physics and Astrophysics, Weizmann Institute,
Rehovot 76100, Israel
E-mail: vladimirbe@ariel.ac.il,doron.gepner@weizmann.ac.il,
juanjose.ramoscab@msmail.ariel.ac.il,borisru@ariel.ac.il
Abstract: This paper represents a continuation of our previous work, where the Boltzmann
weights (BWs) for several Interaction-Round-the Face (IRF) lattice models were computed
using their relation to rational conformal field theories. Here, we focus on deriving solutions
for the Boltzmann weights of the Interaction-Round the Face lattice model, specifically, the
unrestricted face model, based on the
su
(3)
k
affine Lie algebra. The admissibility conditions
are defined by the adjoint representation. We find the BWs by determining the quantum
R
matrix of the
Uq
(
sl
(3)) quantum algebra in the adjoint representation and then applying
the so-called Vertex-IRF correspondence. The Vertex-IRF correspondence defines the BWs
of IRF models in terms of
R
matrix elements.
Keywords: Conformal and W Symmetry, Integrable Field Theories, Lattice Integrable
Models, Quantum Groups
ArXiv ePrint: 2409.05637
Open Access,©The Authors.
Article funded by SCOAP3.https://doi.org/10.1007/JHEP02(2025)200

JHEP02(2025)200
Contents
1 Introduction 1
2 Description of the unrestricted IRF model 2
3 Vertex-IRF correspondence approach 3
3.1 Jimbo’s method for the quantum R-matrix 4
4 Structure of the solution and connection to RCFTs 10
5 Conclusions 15
A Generators of the Uq(sl(3)) algebra 16
1 Introduction
This paper is a continuation of the previous work [
1
]. Here, we present the Boltzmann
weights (BWs) of the unrestricted Interaction-Round the Face (IRF) lattice model based
on the affine Lie algebra
su
(3)
k
and the
Uq
(
sl
(3)) quantum algebra, where the adjoint
representation defines the admissibility conditions of the face configurations. The BWs are
expressed in terms of the quantum
R
matrix.
It is well-established that certain lattice models at criticality are closely connected to
two-dimensional conformal field theories
1
(CFTs). In our previous paper [
1
], we utilized
this connection between CFT and restricted IRF models to determine the BWs for IRF
models based on the affine Lie algebras
su
(2)
k
and
su
(3)
k
, considering various levels
k
and
representations defining the admissibility conditions. We referred to that approach as the
“CFT approach”. In this paper, we adopt a different approach, namely, the Vertex-IRF
correspondence approach, to solve the aforementioned unrestricted IRF model. Ultimately,
we aim to develop a general procedure to solve generic IRF models, which may allow us to
explore some intriguing properties
2
of these models and further connections with CFTs.
The model under study is defined on a two-dimensional square lattice, where the fluc-
tuating variables residing on the lattice vertices (see figure 1) belong to the set of integral
weights of the algebra
su
(3)
k
. The adjoint representation of the corresponding finite Lie
algebra
su
(3) will be used to define the admissibility conditions for face configurations.
A comprehensive definition of the model will be provided in section 2. The BWs of the
model satisfy the Yang-Baxter equation (YBE). In section 3, we discuss the Vertex-IRF
correspondence and determine solutions for the BWs by finding the quantum
R
matrix and
applying the mentioned correspondence. All the non-zero
R
matrix elements are listed in
1
Some examples include the Ising model [
2
], Yang-Lee edge singularity [
3
], the tricritical Ising model [
4
,
5
],
the three-state Potts model [
6
,
7
], the eight-vertex SOS model [
8
,
9
], and IRF models based on affine Lie
algebras [10–12].
2For instance, those properties conjectured in [13–15] regarding the fixed point theory of IRF models.
– 1 –

JHEP02(2025)200
section 3.1. The derivation of the BWs for the model constitutes the main result of this
paper. In section 4, we explore the structure of our solution from the point of view of
representation theory and compare our results to the naive CFT approach. Finally, we
present our conclusions in section 5, and in appendix A, we provide a description of the
generators of
Uq
(
sl
(3)) in the adjoint representation.
2 Description of the unrestricted IRF model
Here, we present the unrestricted IRF model based on
su
(3)
k
. The model is defined on a
two-dimensional square lattice, as illustrated in figure 1, with fluctuating variables residing
on the lattice vertices. A face is formed by four nearest neighboring vertices (in figure 1,
a face with vertices
a, b, c
, and
d
is shown). The fluctuating variables take values from the
set of integral weights
P
of
su
(3)
k
P={a=a0Λ0+a1Λ1+a2Λ2= (a0, a1, a2), ai∈Z}.(2.1)
Let us introduce the weights of the adjoint representation of the corresponding finite algebra
su
(3) as follows
e1= (1,1), e2= (−1,2), e3= (2,−1), e4= (0,0)1,
e5= (0,0)2, e6= (1,−2), e7= (−2,1), e8= (−1,−1).(2.2)
Their corresponding affine extensions (at level zero) are
ˆe1= (−2,1,1),ˆe2= (−1,−1,2),ˆe3= (−1,2,−1),ˆe4= (0,0,0)1,
ˆe5= (0,0,0)2,ˆe6= (1,1,−2,),ˆe7= (1,−2,1),ˆe8= (2,−1,−1).(2.3)
For the unrestricted IRF model, we define the admissibility conditions as follows: A pair
(
a, b
)
∈P
is termed admissible if
b=a+ ˆei,for some i= 1,2,...,8.(2.4)
Now, let (
a, b, c, d
)be the values of the North-West (NW), NE, SE, and SW corners of a
face (e.g., the face shown in figure 1). The face configuration (
a, b, c, d
)is termed admissible
if the pairs (
a, b
),(
a, d
),(
b, c
), and (
d, c
)are all admissible. This definition of admissibility
conditions is based on [
16
–
18
]. For each face configuration, we assign a Boltzmann weight
ω a b
d c
u!.(2.5)
Thus, the BWs depend on the configuration (
a, b, c, d
), as well as on the spectral parameter
u
.
For non-admissible face configurations, we set the Boltzmann weight to zero. For admissible
face configurations, we require that the BWs satisfy the Yang-Baxter equation
X
g
ω a b
f g
u+v!ω f g
e d
u!ω b c
g d
v!=X
g
ω a g
f e
v!ω a b
g c
u!ω g c
e d
u+v!.(2.6)
We have successfully found solutions to the YBE for the BWs in terms of the quantum
R
matrix (which will be described in the next section). We have observed that for admissible face
– 2 –

JHEP02(2025)200
ab
c
d
u
Figure 1. Two-dimensional lattice.
configurations (
a, b, c, d
) that do not involve the null weights
ˆe4
= (0
,
0
,
0)
1
and
ˆe5
= (0
,
0
,
0)
2
,
all the BWs are non-zero. However, for some admissible face configurations involving
ˆe4
or
ˆe5
, certain BWs are zero. We interpret this result as an effect of the multiplicity of
these two null vectors.
Before concluding this section, let us highlight an important difference between the
definition (2.4) and the admissibility conditions of the restricted face model addressed in
paper [
1
]. For the restricted IRF model, we defined an ordered pair (
a, b
) as admissible
if
b
appears in the tensor product of
a
with the adjoint representation (
k−
2
,
1
,
1), that
is, if
b∈a⊗
(
k−
2
,
1
,
1). It is evident that the condition (2.4) is less restrictive than the
admissibility condition for restricted IRF models. Thus, the BWs of restricted IRF models
can be obtained within the BWs of unrestricted models, and it is significantly important
to note that, according to the authors of [
16
,
18
], the subset of “restricted” BWs satisfy
the YBE among themselves.
In the forthcoming section, we will explain the Vertex-IRF correspondence through which
we find solutions to the YBE for the BWs of the studied model.
3 Vertex-IRF correspondence approach
The Vertex-IRF correspondence is a method that can be used to determine the BWs of
unrestricted IRF models in terms of the quantum
ˆ
R
-matrix (for more details, see [
19
–
21
]). In
this section, we employ this method to find the BWs for the model described in the previous
section. To begin, let us define the quantum
ˆ
R
-matrix. By its definition, the quantum
ˆ
R
-matrix satisfies the Yang-Baxter equation in the form
(I⊗ˆ
R(u))( ˆ
R(u+v)⊗I)(I⊗ˆ
R(v)) = ( ˆ
R(v)⊗I)(I⊗ˆ
R(u+v))( ˆ
R(u)⊗I).(3.1)
Here, both sides of the equation act on the tensor product of three vector spaces:
V1⊗V2⊗V3
.
I
represents the identity operator, and
ˆ
R
is given by
ˆ
R=P·R, (3.2)
where
P
is the transposition operator
P a ⊗b
=
b⊗a
, and the operator
R
(also called the
quantum
R
-matrix) acts on the tensor product of two vector spaces. We denote its matrix
elements as
Rm,n
i,j
(
u
), namely,
R(u)(ei⊗ej) = X
m,n
Rm,n
i,j (u) (em⊗en).(3.3)
– 3 –

JHEP02(2025)200
For our purpose,
V1, V2, V3
will correspond to the vector space of the adjoint representation
of
su
(3), and hence
ei
(for
i
= 1
,
2
,...,
8) denotes the states (2.2). It is common to write
the YBE (3.1) in the form
R12(u)R13 (u+v)R23(v) = R23(v)R13(u+v)R12(u).(3.4)
In this form,
Rij
is an operator acting on the
i
th and
j
th components as
R
, and on the other
component as the identity operator. We can directly write the YBE (3.1), (3.4) in terms
of
R
. For a given sextuplet (
i1, i2, i3, f1, f2, f3
), we have
X
s1,s2,s3
Rf1,f2
s1,s2(u)Rs1,f3
i1,s3(u+v)Rs2,s3
i2,i3(v) = X
s1,s2,s3
Rf2,f3
s2,s3(v)Rf1,s3
s1,i3(u+v)Rs1,s2
i1,i2(u).(3.5)
As the adjoint representation contains 8 states, the matrix
R
becomes a 64
×
64 matrix.
However, many of its matrix elements are directly zero due to the property:
Rk,l
i,j = 0 if ei+ej=ek+el. (3.6)
This property arises because the
R
-matrix can be expressed as a projector (see e.g., [
17
,
20
]),
the matrix elements of which are the Clebsch-Gordan coefficients (up to some factor), and it
is known that the Clebsch-Gordan coefficients satisfy the property (3.6).
The Vertex-IRF correspondence (see e.g., [
19
,
20
]) states that the BWs of unrestricted
IRF models can be determined in terms of the matrix elements
3
of
ˆ
R
as follows
ω a b
d c
u!=


ˆ
Rk,l
i,j (u)if b−a= ˆek,c−b= ˆel,d−a= ˆei,c−d= ˆej
0otherwise. (3.7)
Indeed, if one defines the BWs according to this relation, one can see that equation (3.5)
becomes the YBE (2.6) for face models. In the following subsection, we will find the quantum
R
-matrix.
3.1 Jimbo’s method for the quantum R-matrix
In [
22
], Jimbo provided a scheme for computing the quantum
R
-matrix, and in this subsection,
we use this scheme. This scheme relies on certain elements. First, we need to introduce the
Uq
(
sl
(3)) quantum algebra, along with its generators in the adjoint representation. In this
algebra, for each simple root
αa
(for
a
= 1
,
2), there are three generators, namely, the Cartan,
raising and lowering generators denoted respectively as follows
Ha, Ea, Fa.(3.8)
3It is clear from the definition of ˆ
Rthat ˆ
Rk,l
i,j =Rl,k
i,j .
– 4 –

JHEP02(2025)200
The quantum algebra
slq
(3) is defined in terms of generators (3.8) as follows
ka=qHa/2,
kaEa=qEaka, kaEb=q−1/2Ebka,
kaFa=q−1Faka, kaFb=q1/2Fbka,
[k1, k2]=[E1, F2]=[E2, F1]=0,
[Ea, Fa] = k2
a−k−2
a
q−q−1,
E2
aEb−(q+q−1)EaEbEb+EbE2
a= 0,
F2
aFb−(q+q−1)FaFbFa+FbF2
a= 0,
(3.9)
where
a, b
= 1
,
2. In [
1
], we constructed the generators (3.8) in the adjoint representation,
satisfying the algebra (3.9). In appendix A, we provide their explicit expressions. We will need
the weights of the adjoint representation (2.2) in the Euclidean basis (these are eigenvectors
of
H1, H2
given by (A.1)). They are given by
e1= (1,0,0,0,0,0,0,0)T, e2= (0,1,0,0,0,0,0,0)T, e3= (0,0,0,0,0,1,0,0)T,
e4= (0,0,1,0,0,0,0,0)T, e5= (0,0,0,0,0,0,1,0)T, e6= (0,0,0,1,0,0,0,0)T,
e7= (0,0,0,0,0,0,0,1)T, e8= (0,0,0,0,1,0,0,0)T.
(3.10)
Additionally, in this scheme, one will need the following two important elements (see equa-
tion (2.2) of [
22
])
k0=q(H1+H2)/2, E0=q(H1−H2)/3F2F1−q−1F1F2.(3.11)
Jimbo showed that the quantum
R
-matrix (3.3), (3.5) satisfies the following system of linear
equations
4
(for which, up to an overall factor, the solution is unique)
R(u)Ea⊗k−1
a+ka⊗Ea=Ea⊗ka+k−1
a⊗EaR(u),
R(u)Fa⊗k−1
a+ka⊗Fa=Fa⊗ka+k−1
a⊗FaR(u),
[R(u), Ha⊗I+I⊗Ha]=0,
R(u)euE0⊗k0+k−1
0⊗E0=euE0⊗k−1
0+k0⊗E0R(u).
(3.12)
It is clear that, besides the spectral parameter
u
, the
R
-matrix also depends on the parameter
q
. To make identification with an IRF based on affine Lie algebra
su
(
n
)
k
, we must set it
to a specific root of unity:
q=eπi
k+g,(3.13)
where
g
is the dual Coxeter number of the Lie algebra
su
(
n
). Generally speaking, the
representation theory of
Uq
(
sl
(3)) mirrors that of undeformed
sl
(3) unless
q
is a root of
unity: for each finite-dimensional irreducible highest weight representation of
sl
(3) there is an
irreducible highest weight representation of
Uq
(
sl
(3)) of similar dimension. This is no longer
4We have noticed that work [23] partially overlaps with our discussion.
– 5 –

JHEP02(2025)200
true if
q
is a root of unity. However, any particular finite-dimensional representation irreducible
for generic
q
exists and remains irreducible at roots of unity except for a finite number of
values of
q
. For example, the adjoint representation remains irreducible unless
q6
= 1.
By using the generators (3.8) provided in appendix Aand substituting them into (3.12),
we have found solutions to this system of linear equations for the matrix elements
Rk,l
i,j
(
u
)
(for brevity, we denote
Rk,l
i,j
(
u
) =
Rk,l
i,j
). By using the relation (3.7), the matrix elements
Rk,l
i,j
provide us with solutions for the BWs of the model studied in this paper. Here, we
list the matrix elements we found, which constitute our main result. The solutions are
parameterized by the factor
s1
=
R4,5
2,6
, and we have verified that the following matrix
elements satisfy YBE (3.5).
R1,1
1,1=R2,2
2,2=R3,3
3,3=R6,6
6,6=R7,7
7,7=R8,8
8,8=s1e−uq2eu−12q6eu−1
(q2−1)2(q2+ 1) (eu−1) ,
R1,2
1,2=R1,3
1,3=R2,1
2,1=R2,7
2,7=R3,1
3,1=R3,6
3,6=R6,3
6,3=R6,8
6,8=R7,2
7,2=
R7,8
7,8=R8,6
8,6=R8,7
8,7=qs1e−uq2eu−1q6eu−1
(q2−1)2(q2+ 1) ,
R2,1
1,2=R3,1
1,3=R4,1
1,4=R7,2
2,7=R6,3
3,6=R8,4
4,8=R8,6
6,8=
R8,7
7,8=s1q2eu−1q6eu−1
(q4−1) (eu−1) ,
R1,4
1,4=R3,4
3,4=R4,5
4,5=R4,7
4,7=R4,8
4,8=R5,1
5,1=R5,2
5,2=R5,4
5,4=R6,5
6,5=
R8,5
8,5=q2s1e−u(eu−1) q6eu−1
(q2−1)2(q2+ 1) ,
R1,5
1,4=R3,5
1,6=R5,7
2,8=R1,5
3,2=R4,1
3,2=R3,5
3,4=R4,5
3,7=R5,4
3,7=R5,7
4,7=
R5,8
4,8=R4,1
5,1=R4,2
5,2=R4,5
6,2=R5,4
6,2=R6,4
6,5=R5,8
6,7=R8,4
6,7=R4,2
7,1=
R6,4
8,3=R8,4
8,5= 0,
R2,3
1,4=R3,2
1,5=R4,7
2,8=R7,4
2,8=R5,1
3,2=R6,7
4,8=R7,1
5,2=R8,3
6,4=√qs1q6eu−1
q4−1,
R3,2
1,4=R3,4
1,6=R4,3
1,6=R8,3
6,5=R8,5
6,7=s1q6eu−1
√q(q2−1) ,
R1,8
4,4=q2s1e−uq4−q2(eu−1) −1
(q2−1) (q2+ 1)2,
R7,3
4,4=s1q8eu−q2+ 1q2+ 1
q(q2−1) (q2+ 1)2,
R1,4
1,5=R3,4
3,5=R4,3
3,5=R5,1
4,1=R2,5
4,2=R5,2
4,2=R4,5
4,4=R5,4
4,4=R4,5
5,5=
R5,4
5,5=R4,7
5,7=R7,4
5,7=R4,8
5,8=R5,6
6,4=R6,5
6,4=R8,5
8,4=−q3s1
q2+ 1,
R1,5
1,5=R3,5
3,5=R4,1
4,1=R4,2
4,2=R5,7
5,7=R5,8
5,8=R6,4
6,4=
R8,4
8,4=q2s1e−uq2eu−1q4eu−1
(q2−1)2(q2+ 1) ,
– 6 –

JHEP02(2025)200
R2,3
1,5=R8,2
7,4=q3/2s1q4q2+ 1eu−1−1
q4−1,
R5,1
1,5=R8,5
5,8=s1q2+euq8eu−q4+ 2q2+ 1
(q4−1) (eu−1) ,
R1,6
1,6=R2,8
2,8=R3,2
3,2=R6,7
6,7=R7,1
7,1=R8,3
8,3=qs1e−ueu−q2q6eu−1
(q2−1)2(q2+ 1) ,
R6,1
1,6=R7,1
1,7=R8,2
2,8=R8,3
3,8=s1euq6eu−1
(q2+ 1) (eu−1) ,
R1,7
1,7=R2,3
2,3=R3,8
3,8=R6,1
6,1=R7,6
7,6=R8,2
8,2=q3s1e−u(eu−1) q4eu−1
(q2−1)2(q2+ 1) ,
R1,8
1,8=R2,6
2,6=R3,7
3,7=R6,2
6,2=R7,3
7,3=R8,1
8,1=q2s1e−ueu−q2q4eu−1
(q2−1)2(q2+ 1) ,
R6,2
1,8=R7,3
1,8=R8,1
2,6=R8,1
3,7=−q3s1eu
q2+ 1 ,
R8,1
1,8=s1euq2q4+ 1eu−1−1
(q2+ 1) (eu−1) ,
R1,2
2,1=R1,3
3,1=R1,5
5,1=R3,6
6,3=R2,7
7,2=R5,8
8,5=R6,8
8,6=R7,8
8,7=s1e−uq2eu−1q6eu−1
(q4−1) (eu−1) ,
R1,4
2,3=q9/2s1
q2+ 1, R1,5
2,3=q3/2s1e−uq2eu−1
q4−1, R4,1
2,3=q9/2s1q2eu−1
q4−1,
R3,2
2,3=R2,3
3,2=R4,3
3,4=R5,4
4,5=R7,4
4,7=R2,5
5,2=R4,5
5,4=R5,6
6,5=R7,6
6,7=
R6,7
7,6=s1q6eu−1
(q2+ 1) (eu−1) ,
R2,4
2,4=R4,3
4,3=R4,6
4,6=R7,4
7,4=q2s1e−ueuq6eu−2q4+q2−1+ 1
(q2−1)2(q2+ 1) ,
R2,5
2,4=R5,2
2,4=R5,3
4,3=R5,5
4,5=R5,6
4,6=R6,5
4,6=R5,5
5,4=R7,5
7,4=qs1
q2+ 1,
R4,2
2,4=R6,4
4,6=s1q2eu−1q4q2+ 1eu−1−1
(q4−1) (eu−1) ,
R1,7
2,5=R7,6
8,4=q3/2s1e−uq2q4+ 1eu−1−1
(q2−1) (q2+ 1)2,
R2,4
2,5=R4,4
4,5=R3,4
5,3=R4,3
5,3=R4,4
5,4=R4,6
5,6=R4,7
7,5=R7,4
7,5=q5s1
q2+ 1,
R2,5
2,5=R5,3
5,3=R5,6
5,6=R7,5
7,5=q2s1e−uq2euq4(eu−1) + q2−2+ 1
(q2−1)2(q2+ 1) ,
R1,4
3,2=R1,6
3,4=R3,2
4,1=R2,3
5,1=R2,8
5,7=R5,6
8,3=R6,5
8,3=R6,7
8,5=q3/2s1e−uq6eu−1
q4−1,
R1,6
3,5=R6,7
8,4=R7,6
8,5=q5/2s1e−uq6eu−1
(q2−1) (q2+ 1)2,
R5,3
3,5=R7,5
5,7=s1q2euq4q2+ 1eu−2−2+ 1+ 1
(q4−1) (eu−1) ,
– 7 –

JHEP02(2025)200
R1,4
4,1=R4,8
8,4=s1e−uq2euq6+q4(eu−2) −1+ 1
(q4−1) (eu−1) ,
R2,3
4,1=R3,8
5,6=√qs1e−uq2q4+ 1eu−1−1
q4−1,
R2,4
4,2=R4,6
6,4=s1e−uq2euq4q2+ 1eu−2−2+ 1+ 1
(q4−1) (eu−1) ,
R2,6
4,4=R3,7
5,5=qs1e−uq2q6+ 2q4+ 1eu−q2+ 12
(q2−1) (q2+ 1)2,
R4,4
4,4=s1e−u−q2+3q2−1q4+ 1eu+q6e2uq4+ 2q2sinh(u)−2q2−2
(q2−1)2(q2+ 1) (eu−1) ,
R6,2
4,4=R7,3
5,5=qs1q2+ 12q4eu−q4+ 2q2−1
(q2−1) (q2+ 1)2,
R3,7
4,5=R3,7
5,4=q2s1e−uq4−q2+ 1eu−1
q4−1,
R2,8
4,7=R3,2
5,1=R4,8
6,7=R2,5
7,1=R5,2
7,1=√qs1e−uq6eu−1
q2−1,
R7,1
4,2=R7,6
4,8=R6,7
5,8=q3/2s1q6eu−1
(q2−1) (q2+ 1)2,
R3,4
4,3=R4,7
7,4=s1q4q2+ 1eu−1−1
(q2+ 1) (eu−1) ,
R6,1
4,3=R7,6
5,8=q5/2s1q4q2+ 1eu−1−1
(q2−1) (q2+ 1)2,
R3,7
4,4=−qs1e−uq2+q4−q2−1eu
(q2−1) (q2+ 1)2,
R3,5
5,3=R5,7
7,5=s1e−uq2eu−1q2q4+ 1eu−1−1
(q4−1) (eu−1) ,
R5,5
5,5=s1e−ueuq22q2+ 1+euq8+q6(eu−3) + q4−3q2+ 1−1−q2
(q2−1)2(q2+ 1) (eu−1) ,
R1,8
5,5=−q2s1e−uq8eu−q2+ 1q2+ 1
(q2−1) (q2+ 1)2,
R2,6
5,5=q3s1e−u−q8+q6+q4eu−1
(q2−1) (q2+ 1)2,
R1,6
6,1=R1,7
7,1=R2,8
8,2=R3,8
8,3=s1e−uq6eu−1
(q2+ 1) (eu−1) ,
R2,6
6,2=R3,7
7,3=s1 q6−1
(q2+ 1) (eu−1) +e−u!,
R5,1
1,4=R8,5
4,8=s11−q6eu
q(q4−1) , R4,1
1,5=R8,4
5,8=−q3s1q2eu−1
q4−1,
R5,3
1,6=s11−q6eu
q3/2(q2−1) , R2,4
1,7=−q5/2s1,
– 8 –

JHEP02(2025)200
R2,5
1,7=R5,2
1,7=q3/2s1q4eu−1
q2−1, R4,2
1,7=−q5/2s1q2eu−1
q2−1,
R2,6
1,8=R8,1
7,3=q5s1eu−q2
q4−1, R3,7
1,8=R8,1
6,2=qs1q4eu−1
q4−1,
R4,4
1,8=−q2s1q2eu−1
q2−1, R4,5
1,8=R5,4
1,8=R5,5
3,7=qs1q4eu−1
q2−1,
R5,5
1,8=s11−q4eu
q2−1, R5,1
2,3=q3/2s1
q2+1 ,
R1,7
2,4=q5/2s1e−u(eu−1)
(q2−1)(q2+1)2, R7,1
2,4=s11−q6eu
√q(q2−1)(q2+1)2,
R4,2
2,5=R6,4
5,6=q5s1q2eu−1
q4−1, R7,1
2,5=R6,1
3,4=q5/2s1q6eu−1
(q2−1)(q2+1)2,
R5,2
2,5=R6,5
5,6=s1q2q4+1eu−1−1
(q2+1) (eu−1) , R1,8
2,6=R7,3
8,1=qs1e−ueu−q2
q4−1,
R6,2
2,6=R7,3
3,7=s1euq4q2+1eu−1−1
(q2+1) (eu−1) , R4,5
2,6=R5,4
2,6=R5,5
4,4=s1,
R3,7
2,6=R7,3
6,2=s1
q2+1 , R4,4
2,6=q3s1q2eu−1
q2−1,
R5,5
2,6=−s1
q, R3,5
6,1=−q3/2s1e−uq2eu−1
q2−1, R7,3
2,6=s11−q4eu
q4−1,
R7,5
2,8=s11−q6eu
√q(q4−1) , R5,3
3,4=R7,5
4,7=qs1q6eu−1
q4−1,
R6,1
3,5=−q11/2s1(eu−1)
(q2−1)(q2+1)2, R1,8
3,7=R6,2
8,1=q3s1e−uq4eu−1
q4−1,
R2,6
3,7=R6,2
7,3=q6s1
q2+1 , R4,4
3,7=−q3s1,
R6,2
3,7=q4s1q2−eu
q4−1, R4,6
3,8=−q7/2s1
q2+1 ,
R5,6
3,8=R6,5
3,8=q5/2s1q4eu−1
q4−1, R6,4
3,8=−q7/2s1q2eu−1
q4−1,
R1,5
4,1=R5,8
8,4=−q3s1e−uq2eu−1
q4−1, R1,7
4,2=−q9/2s1e−u(eu−1)
(q2−1)(q2+1)2,
R1,6
4,3=R1,7
5,2=q3/2s1e−uq6eu−1
(q2−1)(q2+1)2, R3,5
4,3=R5,7
7,4=qs1e−uq2eu−1
q4−1,
R8,1
4,4=s1q4−q2−1q4eu+1
(q2−1)(q2+1)2, R1,8
4,5=R1,8
5,4=qs1e−uq6eu+q4−q2−1
(q2−1)(q2+1)2,
R2,6
4,5=R2,6
5,4=q2s1e−u2q6eu−q2−1
(q2−1)(q2+1)2, R6,2
4,5=R6,2
5,4=q2s1q4(eu−1)+q2−1
q4−1,
R7,3
4,5=R7,3
5,4=q2s1q6+q4eu−2
(q2−1)(q2+1)2, R8,1
4,5=R8,1
5,4=q3s1q2q4+q2−1eu−1
(q2−1)(q2+1)2,
– 9 –

JHEP02(2025)200
R3,8
4,6=q3/2s1e−u(eu−1)
q4−1, R8,3
4,6=s11−q6eu
q3/2(q4−1) ,
R8,2
4,7=R8,3
5,6=q3/2s1q6eu−1
q4−1, R1,4
5,1=R4,8
8,5=−q3s1e−uq6eu−1
q4−1,
R2,4
5,2=R4,6
6,5=qs1e−uq6eu−1
q4−1, R1,6
5,3=−q9/2s1e−uq6eu−1
(q2−1)(q2+1)2,
R6,1
5,3=q15/2s1(eu−1)
(q2−1)(q2+1)2, R4,4
5,5=R4,5
7,3=R5,4
7,3=q4s1,
R6,2
5,5=−q5s1q4−q2(eu−1)−1
(q2−1)(q2+1)2, R8,1
5,5=q4s1q2+q4−q2−1eu
(q2−1)(q2+1)2,
R8,2
5,7=−q9/2s1(eu−1)
q4−1, R3,4
6,1=R4,3
6,1=√qs1e−uq4eu−1
q2−1,
R1,8
6,2=R1,8
7,3=R2,6
8,1=R3,7
8,1=−q3s1e−u
q2+1 , R3,7
6,2=q2s1e−uq2−eu
q4−1,
R4,4
6,2=R4,5
8,1=R5,4
8,1=qs1e−uq4eu−1
q2−1, R5,3
6,1=−q3/2s1, R5,5
6,2=−qs1,
R3,8
6,4=−q7/2s1e−u(eu−1)
q4−1, R3,8
6,5=R2,8
7,4=√qs1e−uq6eu−1
q4−1,
R2,4
7,1=−q3/2s1e−uq6eu−1
q2−1, R2,6
7,3=−q4s1e−uq4eu−1
q4−1,
R4,4
7,3=−q5s1, R8,4
7,6=q7/2s1q2eu−1
q2−1, R5,5
7,3=qs1e−uq2eu−1
q2−1,
R2,8
7,5=−q7/2s1e−uq6eu−1
q4−1, R8,2
7,5=q13/2s1(eu−1)
q4−1,
R4,8
7,6=q7/2s1, R8,5
7,6=√qs1, R5,8
7,6=√qs1e−uq2eu−1
q2−1,
R1,8
8,1=s1e−uq4q2+1eu−1−1
(q2+1) (eu−1) , R4,4
8,1=−q2s1e−uq4eu−1
q2−1,
R5,5
8,1=−q2s1e−uq2eu−1
q2−1, R4,7
8,2=R7,4
8,2=q3/2s1e−uq4eu−1
q4−1,
R5,7
8,2=−q5/2s1e−uq2eu−1
q4−1, R7,5
8,2=−q5/2s1
q2+1 , R4,6
8,3=−q5/2s1e−uq6eu−1
q4−1.
(3.14)
4 Structure of the solution and connection to RCFTs
Earlier, we discussed the approach to construction of integrable IRF models based on the
fusion rules of a rational conformal field theory. As mentioned in the previous paper [
1
], this
approach cannot be used for Uq(sl(3)) in the adjoint representation for generic qdue to the
presence of the multiplicities. On the other hand, the direct approach used in section 3.1
can be in principle applied for any representation of
Uq
(
sl
(
n
)) and for any
n
, but the
computational power requirements grow quickly with the dimension of representation spaces
– 10 –

JHEP02(2025)200
involved, rendering it impractical for larger representations. In this section, we combine
the insights about the structure of the
R
-matrix that can be extracted from the explicit
solution (3.14) with representation theory considerations to partially resolve the issue of
multiplicities in the CFT approach.
Let
ρ(q)
λ
be an irreducible highest weight representation of the quantum group
Uq
(
sl
(
n
)),
Vλ
corresponding irreducible module, and
ˆρ(q)
λ,u
corresponding evaluation representation of
the affine quantum group
Uq
(
ˆ
sl
(
n
)). Let
R ∈ Uq(ˆ
g)⊗Uq(ˆ
g)(4.1)
be the universal
R
-matrix of the affine quantum group
Uq
(
ˆ
sl
(
n
)), and
Rλλ
its representation:
Rλλ(u) = ˆρ(q)
λ,u ⊗ˆρ(q)
λ,0(R).(4.2)
The matrix
Rλλ
is a solution to Yang-Baxter equation (3.5), and
ˆ
Rλλ =P·Rλλ (4.3)
solves (3.1) respectively. The tensor product of representation spaces
Vλ⊗Vλ
can be
decomposed into a direct sum of irreducible
Uq
(
sl
(
n
)) modules:
Vλ⊗Vλ=
N
M
j=1
Vλj.(4.4)
Since
R
is the universal
R
-matrix, the matrix
ˆ
R
(
u
)must satisfy
[ˆ
R(u), ρ(q)
λ⊗ρ(q)
λ(∆(X))] = 0 ,∀X∈Uq(sl(n)) ,(4.5)
where ∆(
X
)denotes the coproduct of
X
in
Uq
(
sl
(
n
)). Thus, the matrix
ˆ
R
commutes with
all the generators of the “horizontal” quantum subgroup
Uq
(
sl
(
n
)), in particular with the
generators of the Cartan subalgebra, which implies it preserves the weight of any vector it
acts upon. Consequently, any highest weight vector is mapped to another highest weight
vector of the same weight. Therefore, at any value of the spectral parameter, the matrix
ˆ
R
(
u
)
can be decomposed into a sum of projectors onto irreducible submodules of
Uq
(
sl
(
n
)):
ˆ
R(u) =
N
X
j=1
fλj(u)P(λj)(u).(4.6)
There could be two kinds of multiplicities in the decomposition (4.6). Firstly, an
irreducible representation can appear multiple times in the decomposition (4.4) of the tensor
product of the representation spaces. Secondly, the UV limits
fλk
(
∞
)of some of the
eigenvalues might coincide. Both kinds of multiplicities are present in the case of
Uq
(
ˆ
sl
(3))
at generic level
k
considered in this work.
In the CFT approach, we recover
ˆ
R
(
u
)from its UV limit. If
λj
occurs only once in (4.4),
then
P(λj)
does not depend on
u
. We can, therefore, find the corresponding projector in
the UV limit as
P(λj)=Y
i=j
ˆ
R(∞)−fλi(∞)
fλj(∞)−fλi(∞)(4.7)
– 11 –

JHEP02(2025)200
and substitute it into (4.6) to get a manageable ansatz for an
R
-matrix. On the other hand,
if there are multiple submodules of the same highest weight in (4.4), then the decomposi-
tion (4.4) is not unique as there is an arbitrary choice of basis on the subspace of this weight.
Consequently, the
R
-matrix might mix different copies of
Vλj
. So, we can write the
R
-matrix as
ˆ
R=
N
X
m=1
#λm
X
i,j=1
M(λm)
ij (u)P(λm)
ij ,(4.8)
where
P(λm)
ii
is a projector onto
i
-th copy of
Vλm
in some fixed basis independent of the
spectral parameter,
P(λm)
ij
is an operator which maps
V(i)
λm
to
V(j)
λm
and commutes with all the
generators, #
λm
is the multiplicity of weight
λm
in the decomposition (4.4), and functions
M(λm)
ij
(
u
)capture the dependence on the spectral parameter. The operators
P(λm)
ij
have
the following property
P(λm)
ij P(λm′)
kl =δjk δmm′P(λm)
il , P (λm)
ij P(λk)= 0 .(4.9)
In the basis of eigenvectors of
M(λm)
(
∞
)one can still find projectors
P(λm)
ii
using formula (4.7).
On the other hand, finding explicit expression for the off-diagonal operators
P(λm)
ij
in terms
of
ˆ
R
(
∞
)remains an open problem.
The second kind of multiplicity is easier to deal with. Even though projectors onto
the corresponding irreducible submodules cannot be obtained directly from the UV limit
of the R-matrix using (4.7), they are still defined unambiguously and do not depend on
u
. Moreover, (4.7) can be generalised as
Pf=X
fλi(∞)=f
P(λi)=Y
fλi(∞)=f
ˆ
R(∞)−fλi(∞)
f−fλi(∞)(4.10)
to give the projector
Pf
onto the eigenspace of the
R
-matrix with eigenvalue
f
, which is a
direct sum of irreducible modules with that eigenvalue. As will be demonstrated below, in
the example (3.14) of
Uq
(
sl
(3)) in the adjoint representation, the eigenvalues coinciding at
the UV limit also coincide for all values of the spectral parameter. Based on this example,
we can speculate that such coincidences happen due to the symmetries of the weight system,
and therefore, the eigenvalues coinciding in the UV limit stay equal for all values of the
spectral parameter, in which case the knowledge of the projectors
Pf
would be sufficient
for the purpose of constructing the
R
-matrix.
Indeed, for any choice of the representation
λ
of the quantum group
Uq
(
sl
(
n
)) and any
pair of representations
λ1
and
λ2
of multiplicity one occurring in the r.h.s. of (4.4) the
functional form of the ratio of the corresponding eigenvalues can be found without solving
the full Yang-Baxter equation. To this end, we rewrite the last equation of (3.12) as follows
ˆ
R(u)euE0⊗k0+k−1
0⊗E0=euk−1
0⊗E0+E0⊗k0ˆ
R(u).(4.11)
The operators
E0⊗k0
and
k−1
0⊗E0
are not separately coproducts of any element of the
quantum group and, therefore, can mix different irreducible representations. Let us denote
– 12 –

JHEP02(2025)200
their matrix elements by
A
and
B
respectively, as follows
E0⊗k0|λm;µ;i⟩=
N
X
j=1 X
µj∈Ωλj
#µj
X
ij=1
Aλj;µj;ij
λm;µ;i|λj;µj;ij⟩,(4.12)
k−1
0⊗E0|λ;µ;i⟩=−
N
X
j=1 X
µj∈Ωλj
#µj
X
ij=1
Bλj;µj;ij
λm;µ;i|λj;µj;ij⟩,(4.13)
where we have assumed that on each of the irreducible submodules
Vλj
we have picked a
basis of eigenvectors of the Cartan subalgebra of the “horizontal” quantum group
Uq
(
sl
(
n
)),
and denoted the
i
-th vector of weight
µ
in representation of highest weight
λj
as
|λj
;
µ
;
i⟩
.
The symbol #
µj
above refers to the multiplicity of weight
µj
in the representation of highest
weight
λj
. Suppose the highest weights
λ1
and
λ2
both occur in the decomposition (4.4)
once. Acting with both sides of eq. (4.11) on the vector
|λ1
;
µ1
;
i1⟩
(i.e. the
i
-th vector of
weight
µ1
belonging to the irreducible representations of highest weight
λ1
), and inspecting
the coefficient in front of the basis vector
|λ2
;
µ2
;
i2⟩
we find that
fλ1(u)(Aλ2;µ2;i2
λ1;µ1;i1eu−Bλ2;µ2;i2
λ1;µ1;i1) = fλ2(u)(Aλ2;µ2;i2
λ1;µ1;i1−Bλ2;µ2;i2
λ1;µ1;i1eu).(4.14)
Since the ratio of eigenvalues does not depend on
µ1
,
µ2
,
i1
,
i2
, the quantity
ξλ2
λ1
defined as
eiξλ2
λ1=v
u
u
u
t
Aλ2;µ2;i2
λ1;µ1;i1
Bλ2;µ2;i2
λ1;µ1;i1
(4.15)
can also depend only on representations, but not on the particular choice of basis vectors
within the representations (provided
Bλ2;µ2;i2
λ1;µ1;i1
is nonzero). The ratio of eigenvalues then reads
fλ1(u)
fλ2(u)=−sinh u
2−iξλ2
λ1
sinh u
2+iξλ2
λ1,(4.16)
and the number
ξλ2
λ1
can be obtained from the UV limit. These considerations suggest that
if the eigenvalues coincide at infinity, the most likely scenario is that they coincide for all
values of the spectral parameter, in which case, for the purpose of constructing the R matrix
it is sufficient to know the projector onto the corresponding eigenspace given by eq. (4.10).
Since the
R
-matrix must be invertible, it follows from eq. (4.14) that the coefficients
Aλ2,µ2,i2
λ1,µ1,i1
and
Bλ2,µ2,i2
λ1,µ1,i1
can only vanish simultaneously for a given pair of vectors. It may
happen that for a given pair of representations, the coefficients
A
and
B
are identically zero
for all pairs of vectors in these representations. Then the ratio of two eigenvalues could be
found as a product of ratios of eigenvalues of other representations occurring in (4.4) with
multiplicity one, which are connected to
λ1
and
λ2
by the action of the operators
E0⊗k0
and
k−1
0⊗E0
. The equation (4.16) in this case should be generalized to
fλ1(u)
fλ2(u)= (−1)n+1 sinh u
2−iξλ′
1
λ1
sinh u
2+iξλ′
1
λ1
n−1
Y
j=1
sinh u
2−iξλ′
j+1
λ′
j
sinh u
2+iξλ′
j+1
λ′
j
sinh u
2−iξλ2
λ′
n
sinh u
2+iξλ2
λ′
n,(4.17)
– 13 –

JHEP02(2025)200
where we assume that the action of
E0⊗k0
connects consequently representations
λ′
j
to
λ′
j+1
for each
j
,
λ1
to
λ′
1
and
λ′
n
to
λ2
. The equation (4.17) provides the most general form of
the ratio of eigenvalues for representations of multiplicity one.
The analysis above is not applicable to the representations with multiplicities greater than
one, as the equation (4.14) turns into an underdetermined system of linear equations for func-
tions
M(λm)
ij
(
u
). We can only state that the functions
M(λm)
ij
(
u
)are rational functions of
eu
.
In the “CFT” approach, the eigenvalues of the
R
matrix in the UV limit are related to
the spectrum of dimensions of RCFT primaries, which are labeled by weights of irreducible
representations. These conformal dimensions read
∆λ=(λ, λ + 2ρ)
2(k+g),(4.18)
and the eigenvalues are expressed as
fλ=ϵλeiπ∆λ,(4.19)
where
ϵλ
=
±
1. For the
Uq
(
sl
(3)) at the generic level
k
the decomposition (4.4) reads
(1,1) ⊗(1,1) = (2,2) ⊕(3,0) ⊕(0,3) ⊕(1,1)1⊕(1,1)2⊕(0,0) ,(4.20)
and the corresponding RCFT conformal dimensions are given by
∆22 =8
k+ 3 ,∆30 = ∆03 =6
k+ 3 ,∆00 = 0 ,∆11 =3
k+ 3 .(4.21)
Let us compare these predictions of the CFT approach with the exact solution found in the
previous section. The solution (3.14) to the Yang-Baxter equation is fixed up to the choice
of normalization function
s1
. Normalizing it so that
R1,1
1,1(u)≡1,(4.22)
we obtain the following eigenvalues of
ˆ
R
in the limit
u→ ±∞
:
f22 = 1 , f30 =f03 =−q∓2, f00 =q∓8,(4.23)
f(σ)
11 =σq∓5, σ ∈ {−1,1}.(4.24)
Upon identification (3.13) the expression (4.19) from the CFT approach correctly reproduces
the UV limit (4.23), (4.24) of the
R
-matrix (3.14).
In previous work by one of the authors [
24
], the functions
fk
(
u
)were computed for the
R-matrices arising from RCFTs without multiplicities in the fusion rules. Their ratios read
fa(u)
fb(u)=
a−1
Y
r=b
sinh iξr−u
2
sinh iξr+u
2, ξr=π
2(∆r+1 −∆r).(4.25)
On the other hand, the inspection of our explicit solution (3.14) yields the following expressions
for the eigenvalues of the matrix
ˆ
R
:
f22(u)=1, f30(u) = f03(u) = −sinh u
2−iπ
k+3 
sinh u
2+iπ
k+3 ,(4.26)
f00(u) = sinh u
2−iπ
k+3 
sinh u
2+iπ
k+3 
sinh u
2−3iπ
k+3 
sinh u
2+3iπ
k+3 .(4.27)
– 14 –

JHEP02(2025)200
Notice that the eigenvalues
f03
(
u
)and
f30
(
u
)coincide identically. Furthermore, for the
eigenvalues associated with the representations of multiplicity one, there is an agreement
between (4.25) and our results, despite the fact that the whole decomposition (4.20) contains
multiplicities, and there are coinciding eigenvalues. Since the numbers
ξλj
λi
in (4.16), (4.17)
are determined, per eq. (4.19), by the conformal dimensions of RCFT primaries, we can state
that, unless said conformal dimensions are such that the r.h.s. of (4.17) tends to one in large
u
limit, the problem of the multiplicities of the second kind can be resolved by using eq. (4.10).
For the sake of completeness, we list below the expressions for the elements of the
matrix
M(1,1)
obtained from our explicit solution (3.14) using the property (4.9) of the
operators
P(λm)
ij
:
M(1,1)
11 =eu(q6−1) q2−1
(q2eu−1) (q6eu−1),(4.28)
M(1,1)
22 =eu(q2−eu)(q6−1) q2−1
(q2eu−1)2(q6eu−1) ,(4.29)
M(1,1)
12 =−(q2−eu)q3(eu−1)
(q2eu−1) (q6eu−1),(4.30)
M(1,1)
21 =−(q12 +q8−2q6+q4+ 1) −euq6−e−uq6(eu−1) eu
q(q2eu−1)2(q6eu−1).(4.31)
There is a freedom to choose a basis in two-dimensional space of highest weight vectors of
weight (1
,
1). In the above, we have chosen it so that
M(1,1)(∞) = q−5 0 1
1 0 !.(4.32)
There is no basis independent of the spectral parameter in which the matrix
M(1,1)
would
be diagonal for all values of the spectral parameter, which justifies the decomposition (4.8).
Rederiving the expressions (4.28)–(4.31) from the UV limit of the
R
-matrix represents an
interesting problem that we hope to address in the future.
5 Conclusions
In this paper, we have investigated the unrestricted IRF model based on the affine Lie
algebra
su
(3)
k
, where the fluctuating variables residing on the lattice vertices belong to the
set of integral weights
P
(2.1) of the mentioned algebra. The admissibility conditions of
the face configurations are defined by the adjoint representation of the corresponding finite
Lie algebra
su
(3). By utilizing the Vertex-IRF correspondence, we have determined the
solutions for the trigonometric BWs of this model, which are given by the relation (3.7) and
the quantum
R
matrix elements (3.14).
Furthermore, we have considered the structure of the solution of the Yang-Baxter
equation for the arbitrary irreducible representation of the quantum group
Uq
(
sl
(
n
)). The
R
-matrix acts on the tensor product of two copies of some representation of the quantum
group. We demonstrated that the tensor structure of the solution depends on the contents of
decomposition of the aforementioned tensor product into a sum of irreducible representations,
– 15 –

JHEP02(2025)200
in particular on the multiplicities of the representations occurring in this decomposition, and
described this structure in the general case. For representations of multiplicity one, we have
derived the formulae for the ratios of corresponding eigenvalues of the
R
-matrix. Moreover,
for the specific example of the quantum group
Uq
(
sl
(3)) in the adjoint representation, we
have obtained full decomposition of the
R
-matrix, including the matrix of coefficients
M(1,1)
corresponding to the adjoint representation which has multiplicity two in this example.
One of the motivations for the present work was to generalize the construction of the
R
-matrix from its UV limit pioneered in [
24
]. The original construction had two limitations:
it did not allow finding the
R
matrix in cases where the eigenvalues coincided in the UV limit,
and it could not describe the cases with multiplicities greater than one. Our results solve the
first problem. To solve the second problem, one needs to find general expressions, similar
to (4.25), for the matrices
M(λm)
, which contain the coefficients of the decomposition of the
R
-matrix for representations with multiplicity greater than one. The explicit expression for
the matrix
M(1,1)
obtained in this work might be able to facilitate this task. We intend
to address this problem in future works.
In principle, the procedure applied in section 3can be extended to solve any unrestricted
IRF model based on a generic affine Lie algebra and a generic representation (defining the
admissibility conditions) of this algebra by finding the corresponding quantum
R
-matrix.
As mentioned in the introduction, one of our aims in studying IRF models is to develop a
method (or combinations of methods) through which one can find the BWs for more complex
IRF models (e.g., those based on the extended current algebras discussed in [
1
,
25
,
26
]),
and to explore further connections with CFTs. This research direction for IRF models is
both intriguing and interesting, particularly considering the relevance of CFTs in current
times. We plan to explore it further in future works.
Acknowledgments
J.R. thanks the organizers of the Workshop on Integrability 2023 held at the University of
Amsterdam, where the result of this work was announced. The work of V.B. and J.R. is
supported in part by “Program of Support of High Energy Physics”, Grant RA2300000222
by the Israeli Council for Higher Education.
A Generators of the Uq(sl(3)) algebra
The generators of the
Uq
(
sl
(3)) quantum algebra (3.9) in the adjoint representation can be
written in the following manner. The Cartan generators are given by
H1=
















1 0 0 0 0 0 0 0
0−1 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 −1 0 0 0
0 0 0 0 0 2 0 0
0 0 0 0 0 0 0 0
0000000−2
















, H2=
















1 0 0 0 0 0 0 0
0 2 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 −2 0 0 0 0
0 0 0 0 −1 0 0 0
0 0 0 0 0 −1 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
















,(A.1)
– 16 –

JHEP02(2025)200
the raising and lowering generators are
E1=
















0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0
001000q2+1
q0
000000 0 q2+1
q
0 0 0 0 0 0 0 0
















, E2=
















0 0 0 0 0 1 0 0
0 0 q2+1
q0 0 0 1 0
0 0 0 q2+1
q0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
















,(A.2)
F1=
















0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 q
q2+1 00010
















, F2=
















0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
001000 q
q2+1 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
















.(A.3)
Data Availability Statement. This article has no associated data or the data will not
be deposited.
Code Availability Statement. This article has no associated code or the code will not
be deposited.
Open Access. This article is distributed under the terms of the Creative Commons Attri-
bution License (CC-BY4.0), which permits any use, distribution and reproduction in any
medium, provided the original author(s) and source are credited.
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