Vertex Models and Spin Chains in Formulas and Pictures
We systematise and develop a graphical approach to the investigations of quantum integrable vertex statistical models and the corresponding quantum spin chains. The graphical forms of the unitarity and various crossing relations are introduced. Their explicit analytical forms for the case of integra...
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| Cite this: | Vertex Models and Spin Chains in Formulas and Pictures / K.S. Nirov, A.V. Razumov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 77 назв. — англ. |
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| citation_txt | Vertex Models and Spin Chains in Formulas and Pictures / K.S. Nirov, A.V. Razumov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 77 назв. — англ. |
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| description | We systematise and develop a graphical approach to the investigations of quantum integrable vertex statistical models and the corresponding quantum spin chains. The graphical forms of the unitarity and various crossing relations are introduced. Their explicit analytical forms for the case of integrable systems associated with the quantum loop algebra Uq(L(slₗ₊₁)) are given. The commutativity conditions for the transfer operators of lattices with a boundary are derived by the graphical method. Our consideration reveals useful advantages of the graphical approach for certain problems in the theory of quantum integrable systems.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 068, 67 pages
Vertex Models and Spin Chains
in Formulas and Pictures
Khazret S. NIROV †1†2†3 and Alexander V. RAZUMOV †4
†1 Institute for Nuclear Research of the Russian Academy of Sciences,
7a 60th October Ave., 117312 Moscow, Russia
E-mail: nirov@inr.ac.ru
†2 Faculty of Mathematics, National Research University “Higher School of Economics”,
119048 Moscow, Russia
†3 Mathematics and Natural Sciences, University of Wuppertal, 42097 Wuppertal, Germany
E-mail: nirov@uni-wuppertal.de
†4 NRC “Kurchatov Institute — IHEP”, 142281 Protvino, Moscow region, Russia
E-mail: Alexander.Razumov@ihep.ru
Received March 19, 2019, in final form August 30, 2019; Published online September 13, 2019
https://doi.org/10.3842/SIGMA.2019.068
Abstract. We systematise and develop a graphical approach to the investigations of quan-
tum integrable vertex statistical models and the corresponding quantum spin chains. The
graphical forms of the unitarity and various crossing relations are introduced. Their explicit
analytical forms for the case of integrable systems associated with the quantum loop algebra
Uq(L(sll+1)) are given. The commutativity conditions for the transfer operators of lattices
with a boundary are derived by the graphical method. Our consideration reveals useful ad-
vantages of the graphical approach for certain problems in the theory of quantum integrable
systems.
Key words: quantum loop algebras; integrable vertex models; integrable spin models; graphi-
cal methods; open chains
2010 Mathematics Subject Classification: 17B37; 17B80; 16T05; 16T25
Contents
1 Introduction 2
2 Quantum loop algebras and integrability objects 4
2.1 Quantum loop algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Some information on loop algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Definition of a quantum loop algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Poincaré–Birkhoff–Witt basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.4 Universal R-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.5 Modules and representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.6 Spectral parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Integrability objects and their graphical representations . . . . . . . . . . . . . . . . . . . 12
2.2.1 Introductory words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 R-operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Unitarity relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.4 Crossing relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.5 Double duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.6 Yang–Baxter equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.7 Monodromy operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.8 Transfer operators and Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . 32
malto:nirov@inr.ac.ru
mailto:nirov@uni-wuppertal.de
mailto:Alexander.Razumov@ihep.ru
https://doi.org/10.3842/SIGMA.2019.068
2 Kh.S. Nirov and A.V. Razumov
3 Integrability objects for the case of quantum loop algebra Uq(L(sll+1)) 36
3.1 Quantum group Uq(gll+1) and some its representations . . . . . . . . . . . . . . . . . . . . 36
3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.2 Representation π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.3 Representation π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Representations of Uq(L(sll+1)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Jimbo’s homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.2 Representation ϕζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.3 Representation ϕζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.4 Representations ϕ∗
ζ and ∗ϕζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Integrability objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.1 Poincaré–Birkhoff–Witt basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.2 Monodromy operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.3 Explicit form of R-operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.4 Crossing and unitarity relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.5 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.6 Case of Uq(L(sl2)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Graphical description of open chains 51
4.1 Transfer operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Commutativity of transfer operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5 Conclusions 59
A.1 Tensor products and symmetric group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
A.2 Partial transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A.3 Partial trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
References 65
Alice was beginning to get very tired of
sitting by her sister on the bank, and of
having nothing to do: once or twice she
had peeped into the book her sister was
reading, but it had no pictures or
conversations in it, “and what is the use
of a book”, thought Alice “without
pictures or conversations”?
Alice’s Adventures in Wonderland
Lewis Carroll
1 Introduction
Graphical methods have proven useful for many branches of theoretical and mathematical
physics. First of all, it is the method of Feynman diagrams which is the main working tool
of quantum field theory [32, 69]. Rather developed graphical methods are used in the quan-
tum theory of angular momentum [6, 74, 76], the general relativity [58, 59, 60], and physical
applications of the group theory [27]. The graphical methods used in the theory of quantum in-
tegrable models of statistical physics [5] were successfully applied to the problems of enumerative
combinatorics [3, 4, 14, 34, 35, 47, 48, 62, 63, 64].
In this paper, we systematise and develop the graphical approach to the investigation of in-
tegrable vertex statistical models and the corresponding quantum spin chains. Here the most
common vertex model is a two-dimensional quadratic lattice formed by vertices connected by
Vertex Models and Spin Chains in Formulas and Pictures 3
edges. The vertices have weights determined by the states of the adjacent edges. The consid-
eration of such systems begins with the definition of suitable integrability objects that possess
necessary properties. The initial objects here are R-operators and basic monodromy opera-
tors encoding the weights of the vertices. An R-operator acts in the tensor square of a vector
space called the auxiliary space, and a monodromy operator acts in the tensor product of the
auxiliary space and an additional one called the quantum space. To ensure integrability, the
R-operator must satisfy the Yang–Baxter equation, and the monodromy operator the so-called
RMM -equation which, in the case when the auxiliary space coincides with the quantum one,
reduces to the Yang–Baxter equation [5]. The necessary equations are satisfied automatically if
one obtains integrability objects using the quantum group approach formulated in the most clear
form by Bazhanov, Lukyanov and Zamolodchikov [9, 10, 11]. The method proved to be efficient
for the construction of R-operators [15, 16, 23, 24, 42, 49, 52, 72, 77], monodromy operators and
L-operators [9, 10, 11, 12, 15, 16, 17, 18, 19, 56, 61], and for the proof of functional relations
[8, 11, 12, 18, 19, 46, 54, 56].
A quantum group is a special kind of a Hopf algebra arising as a deformation of the universal
enveloping algebra of a Kac–Moody algebra. The concept of the quantum group was introduced
by Drinfeld [30] and Jimbo [37]. Any quantum group possesses the universal R-matrix connecting
its two comultiplications. The universal R-matrix is an element of the tensor square of two copies
of the quantum group. In the framework of the quantum group approach, the integrability
objects are obtained by choosing representations for the factors of that tensor product and
applying them to the universal R-matrix. Here one identifies the representation space of the
first factor with the auxiliary space, and the representation space of the second one with the
quantum space. However, the roles of the factors can be interchanged. The universal R-matrix
satisfies the universal Yang–Baxter equation. This leads to the fact that the received objects
have certain required properties. Besides, such integrability objects satisfy some additional
relations, such as unitarity and crossing relations, which follow from the general properties of
the universal R-matrix and used representations.
The structure of the paper is as follows. In Section 2 we give the definition of the class
of quantum groups, called quantum loop algebras, used in the quantum group approach to
the study of integrable vertex models of statistical physics. Then we discuss properties of
integrability objects and introduce their graphical representations.
Section 3 is devoted to the case of quantum loop algebras Uq(L(sll+1)). We describe some
finite-dimensional representations and derive an expression for the R-operator associated with
the first fundamental representation of Uq(L(sll+1)). Explicit forms of the unitarity and crossing
relations are discussed.
The graphical methods of Section 2 are used in Section 4 to derive the commutativity condi-
tions for the transfer matrices of lattices with boundary. Such conditions are relations connecting
the corresponding R-operator with left and right boundary operators. For the first time the com-
mutativity conditions for lattices with boundary were given by Sklyanin in paper [68] based on
a previous work by Cherednik [26]. In paper [68] rather restrictive conditions on the form of the
R-operators were imposed. In a number of subsequent works [29, 31, 53] these limitations were
weakened with the corresponding modification of the commutativity conditions. Finally, Vlaar
[75] gave the commutativity condition in the form which requires no essential limitations on the
R-operator. It is this form which is obtained by using the graphical method.
We use the standard notations for q-numbers
[ν]q =
qν − q−ν
q − q−1
, ν ∈ C, [n]q! =
n∏
k=1
[k]q, n ∈ Z≥0.
Depending on the context, the symbol 1 means the unit of an algebra or the unit matrix.
4 Kh.S. Nirov and A.V. Razumov
2 Quantum loop algebras and integrability objects
2.1 Quantum loop algebras
2.1.1 Some information on loop algebras
Let g be a complex finite-dimensional simple Lie algebra of rank l [36, 67], h a Cartan subalgebra
of g, and ∆ the root system of g relative to h. We fix a system of simple roots αi, i ∈ [1 . . l]. It
is known that the corresponding coroots hi form a basis of h, so that
h =
l⊕
i=1
Chi.
The Cartan matrix A = (aij)i,j∈[1 . . l] of g is defined by the equation
aij = 〈αj , hi〉. (2.1)
Note that any Cartan matrix is symmetrizable. It means that there exists a diagonal matrix
D = diag(d1, . . . , dl) such that the matrix DA is symmetric and di, i ∈ [1 . . l], are positive
integers. Such a matrix is defined up to a nonzero scalar factor. We fix the integers di assuming
that they are relatively prime.
Denote by (·|·) an invariant nondegenerate symmetric bilinear form on g. Any two such forms
are proportional one to another. We will fix the normalization of (·|·) below. The restriction of
(·|·) to h is nondegenerate. Therefore, one can define an invertible mapping ν : h → h∗ by the
equation
〈ν(x), y〉 = (x|y),
and the induced bilinear form (·|·) on h∗ by the equation
(λ|µ) =
(
ν−1(λ)|ν−1(µ)
)
.
We use one and the same notation for the bilinear form on g, for its restriction to h and for the
induced bilinear form on h∗.
Using the mapping ν, given any root α of g, one obtains the following expression for the
corresponding coroot
αˇ =
2
(α|α)
ν−1(α). (2.2)
Hence, we can write
aij =
2
(αi|αi)
(αj |αi) =
2
(αi|αi)
(αi|αj).
It is clear that the numbers (αi|αi)/2 are proportional to the integers di. We normalize the
bilinear form (·|·) assuming that
1
2
(αi|αi) = di. (2.3)
Denote by θ the highest root of g [36, 67]. Remind that the extended Cartan matrix A(1) =
(aij)i,j∈[0 . . l] is defined by relation (2.1) and by the equations
a00 = 〈θ, θ 〉̌, a0j = −〈αj , θ 〉̌, ai0 = −〈θ, hi〉, (2.4)
Vertex Models and Spin Chains in Formulas and Pictures 5
where i, j ∈ [1 . . l]. We have
θ =
l∑
i=1
aiαi, θˇ =
l∑
i=1
ǎihi
for some positive integers ai and ǎi with i ∈ [1 . . l]. These integers, together with
a0 = 1, a0̌ = 1,
are the Kac labels and the dual Kac labels of the Dynkin diagram associated with the extended
Cartan matrix A(1). Recall that the sums
h =
l∑
i=0
ai, hˇ =
l∑
i=0
ǎi
are called the Coxeter number and the dual Coxeter number of g. Using (2.2), one obtains
θˇ =
2
(θ|θ)ν
−1(θ) =
2
(θ|θ)
l∑
i=1
aiν
−1(αi) =
l∑
i=1
(αi|αi)
(θ|θ) aihi.
It follows that
ǎi =
(αi|αi)
(θ|θ) ai
for any i ∈ [1 . . l].
It is clear that
a00 = 2, a0j = −
l∑
i=1
ǎiaij , j ∈ [1 . . l], ai0 = −
l∑
j=1
aijaj , i ∈ [1 . . l]. (2.5)
We see that for the extended Cartan matrix A(1) one has
l∑
j=0
aijaj = 0, i ∈ [0 . . l],
l∑
i=0
ǎiaij = 0, j ∈ [0 . . l].
Since the Cartan matrix of g is symmetrizable, so is the extended Cartan matrix. Indeed,
using relation (2.2), one can rewrite equations (2.4) as
a00 = 2, a0j = −2
(αj |θ)
(θ|θ) , ai0 = −2
(θ|αi)
(αi|αi)
.
We see that the symmetricity condition
diaij = djaji, i, j ∈ [0 . . l],
for the extended Cartan matrix is equivalent to the equations
d0
(θ|θ) =
di
(αi|αi)
, diaij = djaji, i, j ∈ [1 . . l].
We take as di, i ∈ [1 . . l], the relatively prime positive integers symmetrizing the Cartan matrix A
of g, then, using (2.3), we see that
d0 =
1
2
(θ|θ). (2.6)
6 Kh.S. Nirov and A.V. Razumov
Note that, for our normalization of the quadratic form, (θ|θ) = 4 for the types Bl, Cl and F4,
(θ|θ) = 6 for the type G2, and (θ|θ) = 2 for all other cases. Therefore, we have relatively prime
positive integers di, i ∈ [0 . . l], which define the diagonal matrix symmetrizing the extended
Cartan matrix A(1).
Following Kac [41], we denote by L(g) the loop algebra of g, by L̃(g) its standard central
extension by a one-dimensional centre CK, and by L̂(g) the Lie algebra obtained from L̃(g) by
adding a natural derivation d. By definition
L̂(g) = L(g)⊕ CK ⊕ Cd,
and we use as a Cartan subalgebra of L̂(g) the space
ĥ = h⊕ CK ⊕ Cd.
Introducing an additional coroot
h0 = K −
l∑
i=1
ǎihi,
we obtain
ĥ =
l⊕
i=0
Chi ⊕ Cd.
It is worth to note that
K = h0 +
l∑
i=1
ǎihi =
l∑
i=0
ǎihi.
We identify the space h∗ with the subspace of ĥ∗ defined as
h∗ =
{
λ ∈ ĥ∗ | 〈λ,K〉 = 0, 〈λ, d〉 = 0
}
.
It is also convenient to denote
h̃ = h⊕ CK
and identify the space h∗ with the subspace of h̃∗ which consists of the elements λ̃ ∈ h̃∗ satisfying
the condition
〈λ̃,K〉 = 0. (2.7)
Here and everywhere below we mark such elements of h̃∗ by a tilde. Explicitly the identification
is performed as follows. The element λ̃ ∈ h̃∗ satisfying (2.7) is identified with the element λ ∈ h∗
defined by the equations
〈λ, hi〉 = 〈λ̃, hi〉, i ∈ [1 . . l].
In the opposite direction, given an element λ ∈ h∗, we identify it with the element λ̃ ∈ h̃∗
determined by the relations
〈λ̃, h0〉 = −
l∑
i=1
ǎi〈λ, hi〉, 〈λ̃, hi〉 = 〈λ, hi〉, i ∈ [1 . . l].
It is clear that λ̃ satisfies (2.7).
Vertex Models and Spin Chains in Formulas and Pictures 7
After all we denote by δ the element of ĥ∗ defined by the equations
〈δ, hi〉 = 0, i ∈ [0 . . l], 〈δ, d〉 = 1,
and define the root α0 ∈ ĥ∗ corresponding to the coroot h0 as
α0 = δ − θ,
so that for the entries of the extended Cartan matrix we have
aij = 〈αj , hi〉, i, j ∈ [0 . . l],
see equations (2.1) and (2.4). We stress that in the above relation 〈·, ·〉 means the pairing of the
spaces ĥ∗ and ĥ, while in equations (2.1) and (2.4) it means the pairing of the spaces h∗ and h.
Thus, the elements αi, i ∈ [0 . . l], are the simple roots and hi, i ∈ [0 . . l], are the corresponding
coroots forming a minimal realization of the generalized Cartan matrix A(1) [41]. Let ∆+ be the
full system of positive roots of g, then the full system ∆̂+ of positive roots of the Lie algebra L̂(g)
is
∆̂+ = {γ + nδ | γ ∈ ∆+, n ∈ Z≥0} ∪ {nδ |n ∈ Z>0} ∪ {(δ − γ) + nδ | γ ∈ ∆+, n ∈ Z≥0}.
The system of negative roots ∆̂− is ∆̂− = −∆̂+, and the full system of roots is
∆̂ = ∆̂+ t ∆̂− = {γ + nδ | γ ∈ ∆, n ∈ Z} ∪ {nδ |n ∈ Z \ {0}}.
Recall that the roots ±nδ are imaginary, all other roots are real [41]. It is worth to note here
that the set formed by the restriction of the simple roots αi to h̃ is linearly dependent. In fact,
we have
δ|
h̃
=
l∑
i=0
aiαi|h̃ = 0. (2.8)
This is the main reason to pass from L̃(g) to L̂(g).
We fix a non-degenerate symmetric bilinear form on ĥ by the equations
(hi|hj) = aijd
−1
j , (hi|d) = δi0d
−1
0 , (d|d) = 0,
where i, j ∈ [0 . . l]. Then, for the corresponding symmetric bilinear form on ĥ∗ one has
(αi|αj) = diaij .
It follows from this relation that
(δ|γ) = 0, (δ|δ) = 0
for any γ ∈ ∆.
2.1.2 Definition of a quantum loop algebra
Let ~ be a nonzero complex number such that q = exp ~ is not a root of unity. For each i ∈ [0 . . l]
we set
qi = qdi .
8 Kh.S. Nirov and A.V. Razumov
and assume that
qν = exp(~ν)
for any ν ∈ C.
The quantum loop algebra Uq(L(g)) is a unital associative C-algebra generated by the ele-
ments
ei, fi, i = 0, 1, . . . , l, qx, x ∈ h̃,
satisfying the relations
qνK = 1, ν ∈ C, qx1qx2 = qx1+x2 , (2.9)
qxeiq
−x = q〈αi,x〉ei, qxfiq
−x = q−〈αi,x〉fi, (2.10)
[ei, fj ] = δij
qhii − q−hii
qi − q−1
i
, (2.11)
1−aij∑
n=0
(−1)n
e
1−aij−n
i
[1− aij − n]qi !
ej
eni
[n]qi !
= 0,
1−aij∑
n=0
(−1)n
f
1−aij−n
i
[1− aij − n]qi !
fj
fni
[n]qi !
= 0. (2.12)
Here, relations (2.10) and (2.11) are valid for all i, j ∈ [0 . . l]. The last line of the relations is
valid for all distinct i, j ∈ [0 . . l].
The quantum loop algebra Uq(L(g)) is a Hopf algebra. Here the multiplication mapping
µ : Uq(L(g))⊗Uq(L(g))→ Uq(L(g)) is defined as
µ(a⊗ b) = ab,
and for the unit mapping ι : C→ Uq(L(g)) we have
ι(ν) = ν1.
The comultiplication ∆, the antipode S, and the counit ε are given by the relations
∆(qx) = qx ⊗ qx, ∆(ei) = ei ⊗ 1 + qhii ⊗ ei, ∆(fi) = fi ⊗ q−hii + 1⊗ fi, (2.13)
S(qx) = q−x, S(ei) = −q−hii ei, S(fi) = −fiqhii , (2.14)
ε(qh) = 1, ε(ei) = 0, ε(fi) = 0. (2.15)
For the inverse of the antipode one has
S−1(qx) = q−x, S−1(ei) = −eiq−hii , S−1(fi) = −qhii fi. (2.16)
2.1.3 Poincaré–Birkhoff–Witt basis
The abelian group
Q̂ =
l⊕
i=0
Zαi
is called the root lattice of L̂(g). The algebra Uq(L(g)) can be considered as Q̂-graded if we
assume that
ei ∈ Uq(L(g))αi , fi ∈ Uq(L(g))−αi , qx ∈ Uq(L(g))0
Vertex Models and Spin Chains in Formulas and Pictures 9
for any i ∈ [0 . . l] and x ∈ h̃. An element a of Uq(L(g)) is called a root vector corresponding
to a root γ of ĥ∗ if a ∈ Uq(L(g))γ . In particular, the generators ei and fi are root vectors
corresponding to the roots αi and −αi.
One can construct linearly independent root vectors corresponding to all roots from ∆̂, see, for
example, papers [42, 43, 44, 72], and papers [13, 28] for an alternative approach. If some ordering
of roots is chosen, then appropriately ordered monomials constructed from these vectors form
a Poincaré–Birkhoff–Witt basis of Uq(L(g)). In fact, in applications to the theory of quantum
integrable systems one uses the so-called normal orderings. The definition and an example for
the case of g = sll+1 is given in Section 3.3.1.
2.1.4 Universal R-matrix
Let Π be the automorphism of the algebra Uq(L(g))⊗Uq(L(g)) defined by the equation
Π(a⊗ b) = b⊗ a,
see Appendix A.1. One can show that the mapping
∆′ = Π ◦∆
is a comultiplication in Uq(L(g)) called the opposite comultiplication.
Let Uq(L(g)) be a quantum loop algebra. There exists an element R of Uq(L(g))⊗Uq(L(g))
connecting the two comultiplications in the sense that
∆′(a) = R∆(a)R−1 (2.17)
for any a ∈ Uq(L(g)), and satisfying in Uq(L(g))⊗Uq(L(g))⊗Uq(L(g)) the equations
(∆⊗ id)(R) = R(13)R(23), (id⊗∆)(R) = R(13)R(12). (2.18)
The meaning of the superscripts in the above relations is explained in Appendix A.1. The
element R is called the universal R-matrix. One can show that it satisfies the universal Yang–
Baxter equation
R(12)R(13)R(23) = R(23)R(13)R(12) (2.19)
in Uq(L(g))⊗Uq(L(g))⊗Uq(L(g)).
It should be noted that we define the quantum loop algebra as a C-algebra. It can be also
defined as a C[[~]]-algebra, where ~ is considered as an indeterminate. In this case one really has
a universal R-matrix. In our case, the universal R-matrix exists only in some restricted sense,
see, for example, paper [70], and the discussion in Section 3.3.2 for the case of g = sll+1.
There are two main approaches to the construction of the universal R-matrices for quantum
loop algebras. One of them was proposed by Khoroshkin and Tolstoy [42, 43, 44, 72], and
another one is related to the names Beck and Damiani [13, 28].
2.1.5 Modules and representations
Let ϕ be a representation of a quantum loop algebra Uq(L(g)), and V the corresponding
Uq(L(g))-module. The generators qx, x ∈ h̃, form an abelian group in Uq(L(g)). Let vector
v ∈ V be a common eigenvector for all operators ϕ(qx), then
qxv = q〈µ,x〉v
10 Kh.S. Nirov and A.V. Razumov
for some unique element µ ∈ h̃∗. Using the first relation of (2.9), we obtain
qνKv = qν〈µ,K〉v = v
for any ν ∈ C. Therefore, the element µ satisfies the equation
〈µ,K〉 = 0,
and there is a unique element λ ∈ h∗ such that µ = λ̃. For the definition of λ̃ see Section 2.1.1.
A Uq(L(g))-module V is said to be a weight module if
V =
⊕
λ∈h∗
Vλ,
where
Vλ =
{
v ∈ V | qxv = q〈λ̃,x〉v for any x ∈ h̃
}
.
This means that any vector of V has the form
v =
∑
λ∈h∗
vλ,
where vλ ∈ Vλ for any λ ∈ h∗, and vλ = 0 for all but finitely many of λ. The space Vλ is called
the weight space of weight λ, and a nonzero element of Vλ is called a weight vector of weight λ.
We say that λ ∈ h∗ is a weight of V if Vλ 6= {0}.
We say that a Uq(L(g))-module V is in the category O if
(i) V is a weight module all of whose weight spaces are finite-dimensional;
(ii) there exists a finite number of elements λ1, . . . , λs ∈ h∗ such that every weight of V belongs
to the set
s⋃
i=1
{λ ∈ h∗ |λ ≤ λi},
where ≤ is the usual partial order in h∗ [36].
In this paper we deal only with Uq(L(g))-modules in the category O and in its dual O?, see
Section 2.2.4.
Let V1, V2 be two Uq(L(g))-modules, and ϕ1, ϕ2 the corresponding representations. The
tensor product of the vector spaces V1 and V2 can be supplied with the structure of a Uq(L(g))-
module corresponding to the representation
ϕ1 ⊗∆ ϕ2 = (ϕ1 ⊗ ϕ2) ◦∆.
We denote the obtained Uq(L(g))-module as V1 ⊗∆ V2.
Using the opposite comultiplication, one can construct another representation
ϕ1 ⊗∆′ ϕ2 = (ϕ1 ⊗ ϕ2) ◦∆′
and define the corresponding Uq(L(g))-module V1 ⊗∆′ V2. However, one can show that there is
a natural isomorphism
ϕ1 ⊗∆ ϕ2
∼= ϕ2 ⊗∆′ ϕ1.
Vertex Models and Spin Chains in Formulas and Pictures 11
2.1.6 Spectral parameter
In applications to the theory of quantum integrable systems, one usually considers families of
representations of a quantum loop algebra parametrized by a complex parameter called a spectral
parameter. We introduce a spectral parameter in the following way. Assume that a quantum
loop algebra Uq(L(g)) is Z-graded,
Uq(L(g)) =
⊕
m∈Z
Uq(L(g))m, Uq(L(g))mUq(L(g))n ⊂ Uq(L(g))m+n,
so that any element of a ∈ Uq(L(g)) can be uniquely represented as
a =
∑
m∈Z
am, am ∈ Uq(L(g))m.
Given ζ ∈ C×, we define the grading automorphism Γζ by the equation
Γζ(a) =
∑
m∈Z
ζmam.
It is worth noting that
Γζ1ζ2 = Γζ1 ◦ Γζ2 (2.20)
for any ζ1, ζ2 ∈ C×. Now, for any representation ϕ of Uq(L(g)) we define the corresponding
family ϕζ of representations as
ϕζ = ϕ ◦ Γζ .
If V is the Uq(L(g))-module corresponding to the representation ϕ, we denote by Vζ the
Uq(L(g))-module corresponding to the representation ϕζ .
A common way to endow Uq(L(g)) by a Z-gradation is to assume that
qx ∈ Uq(L(g))0, ei ∈ Uq(L(g))si , fi ∈ Uq(L(g))−si ,
where si are arbitrary integers. We denote
s =
l∑
i=0
aisi, (2.21)
where ai are the Kac labels of the Dynkin diagram associated with the extended Cartan mat-
rix A(1) and assume that s is non-zero. It is clear that for such a Z-gradation one has
Γζ(q
x) = qx, Γζ(ei) = ζsiei, Γζ(fi) = ζ−sifi. (2.22)
Further, it follows from the explicit expression for the universal R-matrix [13, 28, 42, 43, 44, 72]
that
(Γζ ⊗ Γζ)(R) = R (2.23)
for any ζ ∈ C×. Besides, equations (2.14) and (2.16) give
S ◦ Γζ = Γζ ◦ S, S−1 ◦ Γζ = Γζ ◦ S−1. (2.24)
12 Kh.S. Nirov and A.V. Razumov
α1 α2
ζ
δα1
α2
Figure 2.1.
2.2 Integrability objects and their graphical representations
In this section we use the Einstein summation convention: if the same index appears in a single
term exactly twice, once as an upper index and once as a lower index, summation is implied.
Some additional information on integrability objects can be found in the remarkable paper by
Frenkel and Reshetikhin [33] and in papers [17, 19].
2.2.1 Introductory words
What we mean by integrability objects are certain linear mappings acting between representation
spaces of quantum groups, which are, in general, tensor products of some basic representation
spaces. Certainly, the simplest mapping is the unit operator on a basic representation space.
We use for its matrix elements the depiction given in Fig. 2.1. In fact, we associate with a basic
representation space an oriented line, which can be single, double, etc. The direction of a line is
represented as an arrow. The arrowhead corresponds to the input, and the tail to the output of
the operator. The spectral parameter associated with the representation is placed in the vicinity
of the line. The unit operator acting on a tensor product of representation spaces is depicted as
a bunch of oriented lines corresponding to the factors of the tensor product.
2.2.2 R-operators
A more complicated object is an R-operator. It depends on two spectral parameters and is
defined as follows. Let V1, V2 be two Uq(L(g))-modules, ϕ1, ϕ2 the corresponding representations
of Uq(L(g)), and ζ1, ζ2 the spectral parameters associated with the representations. We define
the R-operator RV1|V2(ζ1|ζ2)1 by the equation
ρV1|V2(ζ1|ζ2)RV1|V2(ζ1|ζ2) = (ϕ1ζ1 ⊗ ϕ2ζ2)(R), (2.25)
where ρV1|V2(ζ1|ζ2) is a scalar normalization factor. It follows from (2.20) and (2.23) that
(ϕ1ζ1ν ⊗ ϕ2ζ2ν)(R) = ((ϕ1 ⊗ ϕ2) ◦ (Γζ1 ⊗ Γζ2) ◦ (Γν ⊗ Γν))(R) = (ϕ1ζ1 ⊗ ϕ2ζ2)(R)
for any ν ∈ C×. We will assume that the normalization factor in equation (2.25) is chosen in
such a way that
ρV1|V2(ζ1ν|ζ2ν) = ρV1|V2(ζ1|ζ2) (2.26)
for any ν ∈ C×. In this case
RV1|V2(ζ1ν|ζ2ν) = RV1|V2(ζ1|ζ2),
and one has
RV1|V2(ζ1|ζ2) = RV1|V2
(
ζ1(ζ2)−1|1
)
= RV1|V2
(
ζ1(ζ2)−1
)
, (2.27)
1The notation Rϕ1|ϕ2
(ζ1|ζ2) is also used.
Vertex Models and Spin Chains in Formulas and Pictures 13
α1
α2
β1
β2
ζ1
ζ2
RV1|V2(ζ1|ζ2)α1β1
α2β2
Figure 2.2.
α1
α2
β1
β2
ζ1
ζ2
(RV1|V2(ζ1|ζ2)−1)α1β1
α2β2
Figure 2.3.
=
β1
β2
β3
α1
α2
α3
β1 β3
α1 α3
ζ2
ζ1
ζ2
ζ1
Figure 2.4.
=
α1
α2
α3
β1
β2
β3
α1 α3
β1 β3
ζ1
ζ2
ζ1
ζ2
Figure 2.5.
where
RV1|V2(ζ) = RV1|V2(ζ|1).
Below we sometimes use the notation
ζij = ζi(ζj)
−1.
Using this notation, we can, for example, write (2.27) as
RV1|V2(ζ1|ζ2) = RV1|V2(ζ12|1) = RV1|V2(ζ12).
It is clear that the operator RV1|V2(ζ1|ζ2) acts on V1 ⊗ V2. Fixing bases, say (eα) and (fβ),
of V1 and V2 we can write
RV1|V2(ζ1|ζ2)(eα2 ⊗ fβ2) = (eα1 ⊗ fβ1)RV1|V2(ζ1|ζ2)α1β1
α2β2 .
We use for the matrix elements of RV1|V2(ζ1|ζ2) the depiction which can be seen in Fig. 2.2. Here
we associate with V1 and V2 a single and a double line respectively. It is worth to note that the
indices in the graphical image go clockwise.
For the matrix elements of the inverse RV1|V2(ζ1|ζ2)−1 of the R-operator RV1|V2(ζ1|ζ2) we use
the depiction given in Fig. 2.3. Here we use a grayed circle for the operator and the counter-
clockwise order for the indices. This allows one to have a natural graphical form of the equations
(
RV1|V2(ζ1|ζ2)−1
)α1β1
α2β2RV1|V2(ζ1|ζ2)α2β2
α3β3 = δα1
α3
δβ1β3 ,
RV1|V2(ζ1|ζ2)α1β1
α2β2
(
RV1|V2(ζ1|ζ2)−1
)α2β2
α3β3 = δα1
α3
δβ1β3 ,
see Figs. 2.4 and 2.5. One can see that to represent a product of operators we connect outcoming
and incoming lines corresponding to the indices common for the operators. It is clear that the
notation used for the indices and spectral parameters are arbitrary. Therefore, when it does
not lead to a misunderstanding, we do not write them explicitly in pictures. In fact, in such
a case we obtain a depiction not for a matrix element, but for an operator itself. For example,
we associate Figs. 2.6 and 2.7 with the operator equations
RV1|V2(ζ1|ζ2)−1RV1|V2(ζ1|ζ2) = 1, RV1|V2(ζ1|ζ2)RV1|V2(ζ1|ζ2)−1 = 1.
14 Kh.S. Nirov and A.V. Razumov
=
Figure 2.6.
=
Figure 2.7.
∼
α1
α2
β1
β2
ζ1
ζ2
R̃V1|V2(ζ1|ζ2)α1β1
α2β2
Figure 2.8.
∼
α2
α1
β1
β2
ζ1
ζ2
(R̃V1|V2(ζ1|ζ2)−1)α1β1
α2β2
Figure 2.9.
∼ =
Figure 2.10.
∼ =
β3
β2
β1
α1
α2
α3
β3 β1
α1 α3
ζ2
ζ1
ζ2
ζ1
Figure 2.11.
It is worth to note that the modules V1 and V2 are arbitrary. Therefore the above equations re-
main valid if we interchange them. Respectively, the graphical equations represented by Figs. 2.6
and 2.7 also remain valid if we interchange the single and double lines. This remark is applicable
to all similar situations.
It is in order to formulate some general rules. To obtain a graphical representation of an
operator, we first specify the types of lines corresponding to the basic vector spaces and associate
with each basic vector space a spectral parameter. Then we choose some shape which will
represent the operator. This shape with the appropriate number of outcoming and incoming
lines depicts the matrix element, or the operator itself. To depict the matrix element of the
product of two operators we connect the lines corresponding to the common indices over which
the summation is carried out.
It turns out to be useful to introduce new R-operators, which, at first sight, drop out of the
general scheme described above.2 We denote these operators by R̃V1|V2(ζ1|ζ2) and their inverses
by R̃V1|V2(ζ1|ζ2)−1. As the usual R-operators, they act on the tensor product V1 ⊗ V2. The
depiction of the corresponding matrix elements can be seen in Figs. 2.8 and 2.9. We require the
operator R̃V1|V2(ζ1|ζ2)−1 to be the ‘skew inverse’ of the operator RV1|V2(ζ1|ζ2). By this we mean
the validity of the graphical equation given in Fig. 2.10. Marking out this figure with indices,
we come to Fig. 2.11. We see that in terms of matrix elements the equation given in Fig. 2.10
has the form
(
R̃V1|V2(ζ1|ζ2)−1
)α2β1
α3β2RV1|V2(ζ1|ζ2)α1β2
α2β3 = δα3
α1δβ1β3 .
One can rewrite this as
((
R̃V1|V2(ζ1|ζ2)−1
)t1)
α3
β1α2
β2
(
RV1|V2(ζ1|ζ2)t1
)
α2
β2α1
β3 = δα3
α1δβ1β3 .
2The relation to the usual R-operators can be understood from the results of Section 2.2.5.
Vertex Models and Spin Chains in Formulas and Pictures 15
∼ =
Figure 2.12.
≈
α1
α2
β1
β2
ζ1
ζ2
≈
RV1|V2(ζ1|ζ2)α1β1
α2β2
Figure 2.13.
≈
α2
α1
β1
β2
ζ1
ζ2
(
≈
RV1|V2(ζ1|ζ2)−1)α1β1
α2β2
Figure 2.14.
Here t1 denotes the partial transpose with respect to the space V1, see Appendix A.2. Note
that RV1|V2(ζ1|ζ2)t1 and
(
R̃V1|V2(ζ1|ζ2)−1
)t1 are linear operators on V ?
1 ⊗V2.3 Thus, we have the
following operator equation
(
R̃V1|V2(ζ1|ζ2)−1
)t1RV1|V2(ζ1|ζ2)t1 = 1 (2.28)
on V ?
1 ⊗ V2, and we come to the equation
R̃V1|V2(ζ1|ζ2) =
((
(RV1|V2(ζ1|ζ2)t1)−1
)t1)−1
.
Certainly, equation (2.28) can be also written as
RV1|V2(ζ1|ζ2)t1
(
R̃V1|V2(ζ1|ζ2)−1
)t1 = 1. (2.29)
The corresponding graphical image is given in Fig. 2.12. Transposing equations (2.28) and (2.29),
we obtain
RV1|V2(ζ1|ζ2)t2
(
R̃V1|V2(ζ1|ζ2)−1
)t2 = 1,
(
R̃V1|V2(ζ1|ζ2)−1
)t2RV1|V2(ζ1|ζ2)t2 = 1,
where t2 denotes the partial transpose with respect to the space V2, see again Appendix A.2.
One can get convinced that this does not lead to new pictures. However, using any of these
equations, we obtain
R̃V1|V2(ζ1|ζ2) =
(((
RV1|V2(ζ1|ζ2)t2
)−1)t2)−1
. (2.30)
For completeness we introduce the R-operators denoted by
≈
RV1|V2(ζ1|ζ2), with the inverses
≈
RV1|V2(ζ1|ζ2)−1, acting on V1 ⊗ V2 and depicted by Figs. 2.13 and 2.14. Now we require the
operator
≈
RV1|V2(ζ1|ζ2) to be the ‘skew inverse’ of the operator RV1|V2(ζ1|ζ2)−1. By this we mean
the validity of the graphical equation given in Fig. 2.15. Similarly as above, we determine that
it is equivalent to the following operator equation
≈
RV1|V2(ζ1|ζ2)t1
(
RV1|V2(ζ1|ζ2)−1
)t1 = 1, (2.31)
3We denote by V ? the restricted dual space of V , see Section 2.2.4. If V is finite-dimensional V ? coincides
with the usual dual space.
16 Kh.S. Nirov and A.V. Razumov
≈ =
Figure 2.15.
≈ =
Figure 2.16.
and, therefore,
≈
RV1|V2(ζ1|ζ2) =
(((
RV1|V2(ζ1|ζ2)−1
)t1)−1)t1 . (2.32)
Rewriting equation (2.31) as
(
RV1|V2(ζ1|ζ2)−1
)t1≈RV1|V2(ζ1|ζ2)t1 = 1, (2.33)
we come to the graphical equation given in Fig. 2.16. After all, transposing equations (2.31)
and (2.33), we obtain
(
RV1|V2(ζ1|ζ2)−1
)t2≈RV1|V2(ζ1|ζ2)t2 = 1,
≈
RV1|V2(ζ1|ζ2)t2
(
RV1|V2(ζ1|ζ2)−1
)t2 = 1.
Using any of these equations, we obtain
≈
RV1|V2(ζ1|ζ2) =
(((
RV1|V2(ζ1|ζ2)−1
)t2)−1)t2 .
2.2.3 Unitarity relations
Applying the mapping Π to both sides of the equation
Π(∆(a)) = R∆(a)R−1, a ∈ Uq(L(g)),
and using again the same equation, we obtain
∆(a) = Π(R)Π(∆(a))Π
(
R−1
)
= Π(R)R∆(a)R−1Π(R)−1.
Therefore,
∆(a)Π(R)R = Π(R)R∆(a). (2.34)
Let ϕ1 and ϕ2 be representations of Uq(L(g)) on the vector spaces V1 and V2 respectively. For
any v ∈ V1, w ∈ V2 and a, b ∈ Uq(L(g)) one has
(ϕ1ζ1 ⊗ ϕ2ζ2)(Π(a⊗ b))(v ⊗ w) = (ϕ1ζ1(b)⊗ ϕ2ζ2(a))(v ⊗ w)
= (ϕ1ζ1(b))(v)⊗ (ϕ2ζ2(a))(w)
= PV2|V1((ϕ2ζ2(a))(w)⊗ (ϕ1ζ1(b))(v))
= (PV2|V1((ϕ2ζ2 ⊗ ϕ1ζ1)(a⊗ b))PV1|V2)(v ⊗ w).
It follows that
(ϕ1ζ1 ⊗ ϕ2ζ2)(Π(R)) = PV2|V1((ϕ2ζ2 ⊗ ϕ1ζ1)(R))PV1|V2
= ρV2|V1(ζ2|ζ1)(PV2|V1RV2|V1(ζ2|ζ1)PV1|V2).
Vertex Models and Spin Chains in Formulas and Pictures 17
Now, applying to both sides of equation (2.34) the mapping ϕ1ζ1 ⊗ ϕ2ζ2 , we see that for any
a ∈ Uq(L(g)) one has
(ϕ1ζ1 ⊗∆ ϕ2ζ2)(a)(PV2|V1RV2|V1(ζ2|ζ1))(PV1|V2RV1|V2(ζ1|ζ2))
= (PV2|V1RV2|V1(ζ2|ζ1))(PV1|V2RV1|V2(ζ1|ζ2))(ϕ1ζ1 ⊗∆ ϕ2ζ2)(a).
Hence, if the representation ϕ1ζ1 ⊗∆ ϕ2ζ2 is irreducible for a general value of the spectral pa-
rameters,4 then
ŘV2|V1(ζ2|ζ1)ŘV1|V2(ζ1|ζ2) = CV1|V2(ζ1|ζ2) idV1⊗V2 , (2.35)
where CV1|V2(ζ1|ζ2) is a scalar factor, and we use the notation
ŘV2|V1(ζ2|ζ1) = PV2|V1RV2|V1(ζ2|ζ1), ŘV1|V2(ζ1|ζ2) = PV1|V2RV1|V2(ζ1|ζ2).
Equation (2.35) is called the unitarity relation. Since the representations and spectral parameters
in (2.35) are arbitrary, we also have
ŘV1|V2(ζ1|ζ2)ŘV2|V1(ζ2|ζ1) = CV2|V1(ζ2|ζ1) idV2⊗V1 . (2.36)
From the other hand, multiplying (2.35) from the left by ŘV2|V1(ζ2|ζ1)−1 and from the right by
ŘV2|V1(ζ2|ζ1) we obtain
ŘV1|V2(ζ1|ζ2)ŘV2|V1(ζ2|ζ1) = CV1|V2(ζ1|ζ2) idV2⊗V1 .
It follows from the last two equations that
CV1|V2(ζ1|ζ2) = CV2|V1(ζ2|ζ1). (2.37)
Again fixing bases (eα) and (fβ) of V1 and V2 we write
ŘV2|V1(ζ2|ζ1)(fβ2 ⊗ eα2) = (eα1 ⊗ fβ1)ŘV2|V1(ζ2|ζ1)α1β1
β2α2 ,
ŘV1|V2(ζ1|ζ2)(eα2 ⊗ fβ2) = (fβ1 ⊗ eα1)ŘV1|V2(ζ1|ζ2)β1α1
α2β2 .
It is easy to see that
ŘV2|V1(ζ2|ζ1)α1β1
β2α2 = RV2|V1(ζ2|ζ1)β1α1
β2α2 ,
ŘV1|V2(ζ1|ζ2)β1α1
α2β2 = RV1|V2(ζ1|ζ2)α1β1
α2β2 .
Hence, in terms of matrix elements equations (2.35) and (2.36) look as
RV2|V1(ζ2|ζ1)β1α1
β2α2RV1|V2(ζ1|ζ2)α2β2
α3β3 = CV1|V2(ζ1|ζ2)δα1
α3δ
β1
β3 ,
RV1|V2(ζ1|ζ2)α1β1
α2β2RV2|V1(ζ2|ζ1)β2α2
β3α3 = CV2|V1(ζ2|ζ1)δα1
α3δ
β1
β3 .
These two equations are depicted in Figs. 2.17 and 2.18.
Instead of (2.35) and (2.36) we can also write
ŘV1|V2(ζ1|ζ2) = CV1|V2(ζ1|ζ2)ŘV2|V1(ζ2|ζ1)−1,
ŘV2|V1(ζ2|ζ1) = CV2|V1(ζ2|ζ1)ŘV1|V2(ζ1|ζ2)−1.
These equations can be recognized in Figs. 2.19 and 2.20. For completeness, we also redraw
Figs. 2.19 and 2.20 in the form of Figs. 2.21 and 2.22.
4This is a common situation in the quantum theory of integrable spin chains.
18 Kh.S. Nirov and A.V. Razumov
= C
Figure 2.17.
= C
Figure 2.18.
= C
Figure 2.19.
= C
Figure 2.20.
= C−1
Figure 2.21.
= C−1
Figure 2.22.
2.2.4 Crossing relations
Let V be Uq(L(g))-module in the category O. Define two dual modules V ∗ and ∗V . As vector
spaces both V ∗ and ∗V coincide with the restricted dual space
V ? =
⊕
λ∈h∗
(Vλ)∗
of V . This means that any element µ ∈ V ? has the form
µ =
∑
λ∈h∗
µλ,
where µλ ∈ (Vλ)∗ for any λ ∈ h∗, and µλ = 0 for all but finitely many of λ. The action of an
element µ ∈ V ? on a vector v ∈ V is given by the equation
〈µ, v〉 =
∑
λ∈h∗
〈µλ, vλ〉,
where the sum in the right hand side is finite. If V is a finite-dimensional module, the restricted
dual space coincides with the usual dual space. The module operation for the module V ∗ is
defined by the equation
〈aµ, v〉 = 〈µ, S(a)v〉, µ ∈ V ?, v ∈ V
and for ∗V by the equation
〈aµ, v〉 =
〈
µ, S−1(a)v
〉
, µ ∈ V ?, v ∈ V.
For any two Uq(L(g))-modules V and W there are natural isomorphisms
(V ⊗∆ W )∗ ∼= V ∗ ⊗∆′ W
∗ ∼= W ∗ ⊗∆ V ∗, ∗(V ⊗∆ W ) ∼= ∗V ⊗∆′
∗W ∼= ∗W ⊗∆
∗V.
We now define the category O? containing the dual modules V ∗ and ∗V . We say that
a Uq(L(g))-module V is in the category O? if
Vertex Models and Spin Chains in Formulas and Pictures 19
(i) V is a weight module all of whose weight spaces are finite-dimensional;
(ii) there exists a finite number of elements λ1, . . . , λs ∈ h∗ such that every weight of V belongs
to the set
s⋃
i=1
{λ ∈ h∗ |λi ≤ λ},
where, as in the definition of the category O, ≤ is the usual partial order in h∗.
It is clear that for any module V in the category O the modules V ∗ and ∗V are objects of the
category O?.
Let V be in the category O, and ϕ the corresponding representation of Uq(L(g)). For any
M ∈ End(V ) one defines the transpose of M as an element M t ∈ End(V ?) defined by the
equation
〈M tµ, v〉 = 〈µ,Mv〉, µ ∈ V ?, v ∈ V.
Denote by ϕ∗ and ∗ϕ the representations of Uq(L(g)) corresponding to the modules V ∗ and ∗V
respectively. Now one has
ϕ∗(a) = ϕ(S(a))t, ∗ϕ(a) = ϕ
(
S−1(a)
)t
, a ∈ Uq(L(g)). (2.38)
Note that, using equation (2.24), one obtains
(ϕ∗)ζ = (ϕζ)
∗, (∗ϕ)ζ = ∗(ϕζ) (2.39)
for any ζ ∈ C×. Therefore, we write instead of (ϕ∗)ζ and (ϕζ)
∗ just ϕ∗ζ , and instead of (∗ϕ)ζ and
∗(ϕζ) just ∗ϕζ .
Let V be a Uq(L(g))-module in the category O. Define a mapping ηV : V → V ?? by the
equality
〈ηV (v), µ〉 = 〈µ, v〉
for all v ∈ V and µ ∈ V ?. It can be shown that ηV is an isomorphism of vector spaces. It is
easy to see that for any M ∈ End(V ) one has
η−1
V
(
M t
)t
ηV = M. (2.40)
In what follows we identify the spaces V and V ??, and whether an element belongs to the
space V or to the space V ?? will be determined by the context. Equation (2.40) becomes the
identification
(
M t
)t
= M. (2.41)
Consider now the modules ∗(V ∗) and (∗V )∗. These modules as vector spaces are identical to
the vector space V ?? = V . Equations (2.38) and (2.41) give
∗(ϕ∗) = ϕ, (∗ϕ)∗ = ϕ,
and we have the identification of the corresponding modules
∗(V ∗) = V, (∗V )∗ = V.
Similarly as above, we see that the notations ∗ϕ∗ζ and ∗V ∗ζ have a unique sense.
20 Kh.S. Nirov and A.V. Razumov
According to the definition of an R-operator (2.25), we write
ρV ∗1 |V2(ζ1|ζ2)RV ∗1 |V2(ζ1|ζ2) = (ϕ∗1ζ1 ⊗ ϕ2ζ2)(R).
Using the decomposition
R =
∑
i
ai ⊗ bi,
we determine that
(ϕ∗1ζ1 ⊗ ϕ2ζ2)(R) =
∑
i
ϕ∗1ζ1(ai)⊗ ϕ2ζ2(bi) =
∑
i
ϕ1ζ1(S(ai))
t ⊗ ϕ2ζ2(bi)
=
(∑
i
ϕ1ζ1(S(ai))⊗ ϕ2ζ2(bi)
)t1
= (ϕ1ζ1 ⊗ ϕ2ζ2)((S ⊗ id)(R))t1 .
Now, using the equation
(S ⊗ id)(R) = R−1,
see, for example, [25, p. 124], we come to the equation
(ϕ∗1ζ1 ⊗ ϕ2ζ2)(R) = (ϕ1ζ1 ⊗ ϕ2ζ2)
(
R−1
)t1 .
We have
1 = (ϕ1ζ1 ⊗ ϕ2ζ2)
(
RR−1
)
= (ϕ1ζ1 ⊗ ϕ2ζ2)(R)(ϕ1ζ1 ⊗ ϕ2ζ2)
(
R−1
)
,
therefore,
(ϕ1ζ1 ⊗ ϕ2ζ2)
(
R−1
)
= ((ϕ1ζ1 ⊗ ϕ2ζ2)(R))−1 = ρV1|V2(ζ1|ζ2)−1RV1|V2(ζ1|ζ2)−1.
Hence, we obtain
RV ∗1 |V2(ζ1|ζ2) = D(ζ1|ζ2)
(
RV1|V2(ζ1|ζ2)−1
)t1 , (2.42)
where
D(ζ1|ζ2) = ρV ∗1 |V2(ζ1|ζ2)−1ρV1|V2(ζ1|ζ2)−1.
We call relation (2.42), and any similar to it, a crossing relation.
Any crossing relation has the form of an equation whose left and right hand sides contain an
R-operators or the inverse of an R-operator. The right hand side contains also a scalar coeffi-
cient D whose concrete form is determined by the following rules. If the left hand side contains
an R-operator RV1|V2(ζ1|ζ2) or its inverse, the factor D contains the factor ρV1|V2(ζ1|ζ2)−1 or
ρV1|V2(ζ1|ζ2). Respectively, if the right hand side contains an R-operator RW1|W2
(η1|η2) or its
inverse, the factor D contains the factor ρW1|W2
(η1|η2) or ρW1|W2
(η1|η2)−1.
In the same way as above, using the identity
(
id⊗ S−1
)
(R) = R−1,
one comes to the equation
RV1|∗V2(ζ1|ζ2) = D(ζ1|ζ2)
(
RV1|V2(ζ1|ζ2)−1
)t2 . (2.43)
Vertex Models and Spin Chains in Formulas and Pictures 21
= D
Figure 2.23.
= D
Figure 2.24.
= D
Figure 2.25.
= D
Figure 2.26.
= D
Figure 2.27.
= D
Figure 2.28.
The graphical representation of the crossing relations (2.42) and (2.43) are given in Figs. 2.23
and 2.24. Here and below, for the representation ϕ∗ we use the dotted variant of the line used
for the representation ϕ, and for the representation ∗ϕ we use the dashed variant of that line.
Further, we have
(∗ϕ1ζ1 ⊗ ϕ2ζ2)
(
R−1
)
= (ϕ1ζ1 ⊗ ϕ2ζ2)
((
S−1 ⊗ id
)(
R−1
))t1 = (ϕ1ζ1 ⊗ ϕ2ζ2)(R)t1 .
It follows from this equation that
R∗V1|V2(ζ1|ζ2)−1 = D(ζ1|ζ2)RV1|V2(ζ1|ζ2)t1 . (2.44)
Similarly,
RV1|V ∗2 (ζ1|ζ2)−1 = D(ζ1|ζ2)RV1|V2(ζ1|ζ2)t2 . (2.45)
One can see that Figs. 2.25 and 2.26 are the depiction of the crossing relations (2.44) and (2.45).
Concluding this section, we give two crossing relations obtained as a result of combining the
crossing relations given above. They are
RV ∗1 |V ∗2 (ζ1|ζ2) = D(ζ1|ζ2)RV1|V2(ζ1|ζ2)t, (2.46)
and
R∗V1|∗V2(ζ1|ζ2) = D(ζ1|ζ2)RV1|V2(ζ1|ζ2)t. (2.47)
One can see the graphical representation of these relations in Figs. 2.27 and 2.28. For com-
pleteness we give in Figs. 2.29 and 2.30 the graphical images of the crossing relations obtained
from (2.46) and (2.47) by inversion.
22 Kh.S. Nirov and A.V. Razumov
= D
Figure 2.29.
= D
Figure 2.30.
2.2.5 Double duals
Let us proceed to the module V ∗∗. Certainly, as a vector space it is again the vector space V ??.
Now for any a ∈ Uq(L(g)) we have
〈aηV (v), µ〉 = 〈ηV (v), S(a)µ〉 = 〈S(a)µ, v〉 =
〈
µ, S2(a)v
〉
=
〈
ηV
(
S2(a)v
)
, µ
〉
.
It means that ηV intertwines the representations ϕ∗∗ and ϕ◦S2, and since ηV is an isomorphism
of vector spaces we have the isomorphism of representations
ϕ∗∗ ∼= ϕ ◦ S2.
In the same way we prove the isomorphism
∗∗ϕ ∼= ϕ ◦ S−2.
It follows from (2.39) that
(ϕ∗∗)ζ = (ϕζ)
∗∗, (∗∗ϕ)ζ = ∗∗(ϕζ)
for any ζ ∈ C×, hence, the notations ϕ∗∗ζ and ∗∗ϕζ are unambiguous.
Using (2.14), we obtain
S2(qx) = qx, S2(ei) = q−2diei, S2(fi) = q2difi.
For the image of S2(ei) in the representation ϕζ we have
ϕζ
(
S2(ei)
)
= ϕζ
(
q−2diei
)
.
Thus, in this representation the action of S2 on ei is realized as a rescaling. Looking at (2.10),
one can try to perform such rescaling by conjugation with an appropriate element qx. One has
qxeiq
−x = q〈αi,x〉ei = qµiei,
where
µi = 〈αi, x〉, (2.48)
and, using relation (2.8), we obtain
l∑
i=0
aiµi = 0. (2.49)
It is clear that it impossible to find an element x ∈ h̃ such that the similarity transformation
determined by qx gives the desired result. However, one can simultaneously with such transfor-
mation modify the spectral parameter. Let ζ̃ be a new spectral parameter. We have to satisfy
the equation
ζsiq−2di = ζ̃siqµi . (2.50)
Vertex Models and Spin Chains in Formulas and Pictures 23
It follows from equation (2.49) that
ζsq
−2
l∑
i=0
aidi
= ζ̃s,
where s is defined by equation (2.21). Using equations (2.6) and (2.3), we find
l∑
i=0
aidi =
1
2
[
(θ|θ) +
l∑
i=1
ai(αi|αi)
]
=
(θ|θ)
2
l∑
i=0
ǎi =
(θ|θ)
2
h .̌
Hence, we come to the following expression for the new spectral parameter
ζ̃ = q−(θ|θ)hˇ/sζ. (2.51)
Finally, we find that equation (2.50) is satisfied if
µi = −2di + (θ|θ)hˇsi/s. (2.52)
Note that in the case where si = di we have µi = 0, and, therefore, x = 0.
The element x can be written as
x =
l∑
i=0
λihi (2.53)
for some numbers λi ∈ C. Using (2.48) and (2.1), we obtain the following system of equations
for λi:
l∑
j=0
λjaji = µi. (2.54)
The solution of this equation is not unique. We fix the ambiguity by the condition
λ0 = 0,
and, using (2.49), rewrite the system (2.54) as
l∑
j=1
λjaj0 = −
l∑
i=1
ajµj , (2.55)
l∑
j=1
λjaji = µi, i ∈ [1 . . l]. (2.56)
The system (2.56) has the unique solution
λi =
l∑
j=1
µjbji, i ∈ [1 . . l], (2.57)
where bij are the matrix elements of the matrix B inverse to the Cartan matrix A = (aij)i,j∈[1 . . l]
of the Lie algebra g. Substituting this solution into (2.55) and taking into account the last
equation of (2.5), we see that equation (2.55) is satisfied identically.
Let us obtain another expression for the element x, cf. paper [33]. To this end recall that
the elements ωi ∈ h, i ∈ [1 . . l], defined by the equation
〈ωi, hj〉 = δij
24 Kh.S. Nirov and A.V. Razumov
are called the fundamental weights. Their sum
ρ =
l∑
i=1
ωi
satisfies the equation
〈ρ, hi〉 = 1
for any i ∈ [1 . . l]. We obtain
〈
αi, ν
−1(ρ)
〉
=
(
ν−1(αi)|ν−1(ρ)
)
=
(αi|αi)
2
(
hi|ν−1(ρ)
)
=
(αi|αi)
2
〈ρ, hi〉 = di
and
〈
α0, ν
−1(ρ)
〉
= −
l∑
i=1
ai
〈
αi, ν
−1(ρ)
〉
= −
l∑
i=1
aidi = −(θ|θ)
2
(hˇ− 1).
This gives
x = −2ν−1(ρ) + y, (2.58)
where for the components
νi = 〈αi, y〉
we have the expressions
ν0 = (θ|θ)h (̌s0 − s)/s, νi = (θ|θ)hˇsi/s.
In the case
s0 = 1, si = 0, i ∈ [1 . . l],
we see that y = 0.
Thus, for any i ∈ [0 . . l] we have
ϕ∗∗ζ (ei) = ϕ
ζ̃
(
qxeiq
−x),
where the new spectral parameter ζ̃ is given by (2.51) and element x is determined either by
equation equations (2.53), (2.57) and (2.52), or by equation (2.58). In a similar way we obtain
ϕ∗∗ζ (fi) = ϕ
ζ̃
(
qxfiq
−x)
for any i ∈ [0 . . l]. Summarizing, we see that
ϕ∗∗ζ (a) = ϕ
(
qx
)
ϕ
ζ̃
(a)ϕ
(
q−x
)
. (2.59)
for any a ∈ Uq(L(g)). This means that we have the isomorphism
V ∗∗ζ ∼= V
ζ̃
.
In a similar way we obtain the equation
∗∗ϕζ(a) = ϕ
(
q−x
)
ϕ
ζ̃
(a)ϕ
(
qx
)
,
Vertex Models and Spin Chains in Formulas and Pictures 25
where x is determined again either by equations (2.53), (2.57) and (2.52), or by equation (2.58),
while ζ̃ is now defined as
ζ̃ = q(θ|θ)hˇ/sζ.
Using equation (2.42), we obtain
RV ∗∗1 |V2(ζ1|ζ2) = ρV ∗∗1 |V2(ζ1|ζ2)−1ρV ∗1 |V2(ζ1|ζ2)−1
(
RV ∗1 |V2(ζ1|ζ2)−1
)t1 . (2.60)
Using (2.42) again, we come to the equation
RV ∗∗1 |V2(ζ1|ζ2) = ρV ∗∗1 |V2(ζ1|ζ2)−1ρV1|V2(ζ1|ζ2)
(((
RV1|V2(ζ1|ζ2)−1
)t1)−1)t1 . (2.61)
Comparing it with (2.32), we see that
≈
RV1|V2(ζ1|ζ2) is proportional to RV ∗∗1 |V2(ζ1|ζ2). It follows
from (2.59) and (2.51) that
RV ∗∗1 |V2(ζ1|ζ2) = ρV ∗∗1 |V2(ζ1|ζ2)−1ρV1|V2
(
q−(θ|θ)hˇ/sζ1|ζ2
)
× (XV1 ⊗ idV2)RV1|V2
(
q−(θ|θ)hˇ/sζ1|ζ2
)(
X−1
V1
⊗ idV2
)
. (2.62)
Here and below for a Uq(L(g))-module V and the corresponding representation ϕ we denote
XV = ϕ
(
qx
)
for x given either by equations (2.53), (2.57) and (2.52), or by equation (2.58). Comparing
equations (2.61) and (2.62), we come to the equation
(((
RV1|V2(ζ1|ζ2)−1
)t1)−1)t1 = ρV1|V2(ζ1|ζ2)−1ρV1|V2
(
q−(θ|θ)hˇ/sζ1|ζ2
)
× (XV1 ⊗ idV2)RV1|V2
(
q−(θ|θ)hˇ/sζ1|ζ2
)(
X−1
V1
⊗ idV2
)
.
Further, comparing equations (2.62) and (2.60), we come to the crossing relation
RV ∗1 |V2(ζ1|ζ2)−1 = D(ζ1|ζ2)
((
X−1
V1
)t ⊗ idV2
)
RV1|V2
(
q−(θ|θ)hˇ/sζ1|ζ2
)t1(XtV1 ⊗ idV2
)
, (2.63)
where
D(ζ1|ζ2) = ρV ∗1 |V2(ζ1|ζ2)ρV1|V2
(
q−(θ|θ)hˇ/sζ1|ζ2
)
= ρV ∗1 |V2(ζ1|ζ2)ρV1|V2
(
ζ1|q(θ|θ)hˇ/sζ2
)
.
Here equation (2.26) is used.
Starting with the R-matrix RV1|∗∗V2(ζ1|ζ2), we come to the equation
(((
RV1|V2(ζ1|ζ2)−1
)t2)−1)t2 = ρV1|V2(ζ1|ζ2)−1ρV1|V2
(
ζ1|q(θ|θ)hˇ/sζ2
)
×
(
idV1 ⊗ X−1
V2
)
RV1|V2
(
ζ1|q(θ|θ)hˇ/sζ2
)t2(idV1 ⊗ XV2)
and to the crossing relation
RV1|∗V2(ζ1|ζ2)−1 = D(ζ1|ζ2)
(
idV1 ⊗ XtV2
)
RV1|V2
(
ζ1|q(θ|θ)hˇ/sζ2
)t2(idV1 ⊗
(
X−1
V2
)t)
, (2.64)
where
D(ζ1|ζ2) = ρV1|∗V2(ζ1|ζ2)ρV1|V2
(
ζ1|q(θ|θ)hˇ/sζ2
)
= ρV1|∗V2(ζ1|ζ2)ρV1|V2
(
q−(θ|θ)hˇ/sζ1|ζ2
)
.
Let us give a graphical representation of the crossing relations (2.63) and (2.64). For the
matrix elements of the operator XV and its inverse we use the depiction given in Figs. 2.31
and 2.32.
26 Kh.S. Nirov and A.V. Razumov
ζ q−(θ|θ)hˇ/sζ
α1 α2
(XV )α1
α2
Figure 2.31.
q−(θ|θ)hˇ/sζ ζ
α1 α2
(X−1
V )α1
α2
Figure 2.32.
=
Figure 2.33.
=
Figure 2.34.
=
Figure 2.35.
=
Figure 2.36.
= D
Figure 2.37.
= D
Figure 2.38.
Note that the equation
ϕ∗
(
qx
)
= ∗ϕ
(
qx
)
=
(
ϕ
(
qx
)−1)t
results in four graphical equations given in Figs. 2.33–2.36. It can be demonstrated now that
Figs. 2.37 and 2.38 represent the crossing relations (2.63) and (2.64).
Finally, starting with the R-operators R∗∗V1|V2(ζ1|ζ2) and RV1|V ∗∗2
(ζ1|ζ2), we obtain two more
equations
(((
RV1|V2(ζ1|ζ2)t1
)−1)t1)−1
= ρV1|V2(ζ1|ζ2)−1ρV1|V2
(
q(θ|θ)hˇ/sζ1|ζ2
)
×
(
X−1
V1
⊗ idV2
)
RV1|V2
(
q(θ|θ)hˇ/sζ1|ζ2
)
(XV1 ⊗ idV2),
(((
RV1|V2(ζ1|ζ2)t2
)−1)t2)−1
= ρV1|V2(ζ1|ζ2)−1ρV1|V2
(
ζ1|q−(θ|θ)hˇ/sζ2
)
× (idV1 ⊗ XV2)RV1|V2
(
ζ1|q−(θ|θ)hˇ/sζ2
)(
idV1 ⊗ X−1
V2
)
,
and two more crossing relations
R∗V1|V2(ζ1|ζ2)t1 = D(ζ1|ζ2)
(
X−1
V1
⊗ idV2
)
RV1|V2
(
q(θ|θ)hˇ/sζ1|ζ2
)−1(XV1 ⊗ idV2
)
(2.65)
and
RV1|V ∗2 (ζ1|ζ2)t2 = D(ζ1|ζ2)(idV1 ⊗ XV2)RV1|V2
(
ζ1|q−(θ|θ)hˇ/sζ2
)−1(
idV1 ⊗ X−1
V2
)
, (2.66)
where
D(ζ1|ζ2) = ρ∗V1|V2(ζ1|ζ2)−1ρV1|V2
(
q(θ|θ)hˇ/sζ1|ζ2
)−1
= ρ∗V1|V2(ζ1|ζ2)−1ρV1|V2
(
ζ1|q−(θ|θ)hˇ/sζ2
)−1
Vertex Models and Spin Chains in Formulas and Pictures 27
= D
Figure 2.39.
= D
Figure 2.40.
=
Figure 2.41.
=
Figure 2.42.
and
D(ζ1|ζ2) = ρV1|V ∗2 (ζ1|ζ2)−1ρV1|V2
(
ζ1|q−(θ|θ)hˇ/sζ2
)−1
= ρV1|V ∗2 (ζ1|ζ2)−1ρV1|V2
(
q(θ|θ)hˇ/sζ1|ζ2
)−1
respectively. The crossing relations (2.65) and (2.66) are depicted in Figs. 2.39 and 2.40. Now,
similarly as for the case of
≈
RV1|V2(ζ1|ζ2), one can demonstrate that R̃V1|V2(ζ1|ζ2) is proportional
to R∗∗V1|V2(ζ1|ζ2).
2.2.6 Yang–Baxter equation
Now, let V1, V2, V3 be Uq(L(g))-modules, ϕ1, ϕ2, ϕ3 the corresponding representations of
Uq(L(g)), and ζ1, ζ2, ζ3 the spectral parameters associated with the representations. We asso-
ciate with V1, V2 and V3 a single, double and triple lines, respectively. Applying to both sides of
equation (2.19) the mapping ϕ1ζ1 ⊗ϕ2ζ2 ⊗ϕ3ζ3 and using the definition of an R-operator (2.25),
we obtain the Yang–Baxter equation
R
(12)
V1|V2(ζ1|ζ2)R
(13)
V1|V3(ζ1|ζ3)R
(23)
V2|V3(ζ2|ζ3) = R
(23)
V2|V3(ζ2|ζ3)R
(13)
V1|V3(ζ1|ζ3)R
(12)
V1|V2(ζ1|ζ2).
It is natural, slightly abusing notation, to denote RVi|Vj (ζi|ζj)(ij) simply by RVi|Vj (ζi|ζj). Now
the above equation takes the form
RV1|V2(ζ1|ζ2)RV1|V3(ζ1|ζ3)RV2|V3(ζ2|ζ3) = RV2|V3(ζ2|ζ3)RV1|V3(ζ1|ζ3)RV1|V2(ζ1|ζ2). (2.67)
One can recognize the graphical image of this equation in Fig. 2.41. As one can see, we have
three external arrowheads and three external arrowtails there. It is worth to stress that the heads
and the tails are grouped together, and the graphical equation given in Fig. 2.42, where there is
no such grouping, is not a true Yang–Baxter equation. However, as it is shown below, in the case
when the corresponding R-operators satisfy the unitarity relations this equation is also true.
Multiplying both sides of equation (2.67) on the left and right by RV1|V2(ζ1|ζ2)−1, we obtain
RV1|V3(ζ1|ζ3)RV2|V3(ζ2|ζ3)RV1|V2(ζ1|ζ2)−1
= RV1|V2(ζ1|ζ2)−1RV2|V3(ζ2|ζ3)RV1|V3(ζ1|ζ3). (2.68)
28 Kh.S. Nirov and A.V. Razumov
=
Figure 2.43.
=
Figure 2.44.
∼
=
∼
Figure 2.45.
It is instructive to obtain this equation by the graphical method. It is clear that the multiplica-
tion of (2.67) by RV1|V2(ζ1|ζ2)−1 is equivalent to transition from the equation given in Fig. 2.41
to the equation given in Fig. 2.43. Now, using the graphical equations given in Figs. 2.6 and 2.7,
we come to the graphical image of equation (2.68) given in Fig. 2.44.
In a similar way one can obtain a lot of graphical versions of the Yang–Baxter equation.
We give here only one additional example, shown in Fig. 2.45. One can get convinced that the
analytical form of that graphical equation is
RV1|V2(ζ1|ζ2)t1RV2|V3(ζ2|ζ3)
(
R̃V1|V3(ζ1|ζ3)−1
)t1
=
(
R̃V1|V3(ζ1|ζ3)−1
)t1RV2|V3(ζ2|ζ3)RV1|V2(ζ1|ζ2)t1 , (2.69)
or, equivalently,
(
R̃V1|V3(ζ1|ζ3)−1
)t2RV1|V2(ζ1|ζ2)RV2|V3(ζ2|ζ3)t2
= RV2|V3(ζ2|ζ3)t2RV1|V2(ζ1|ζ2)
(
R̃V1|V3(ζ1|ζ3)−1
)t2 . (2.70)
Let us demonstrate how equations (2.69) and (2.70) can be obtained analytically. For sim-
plicity, we denote RVi|Vj (ζi|ζj) just by Rij . Transposing the Yang–Baxter equation (2.67) with
respect to V1 and using equations (A.5) and (A.4), we come to the equation
(R12R13)t1R23 = R23(R13R12)t1 .
Taking into account equation (A.3), we see that it is equivalent to
(R13)t1(R12)t1R23 = R23(R12)t1(R13)t1 .
Multiplying both sides of this equation on the left and right by
(
(R13)t1
)−1
we obtain
(R12)t1R23
(
(R13)t1
)−1
=
(
(R13)t1
)−1
R23(R12)t1 .
It follows from (2.30) that this equation is equivalent to (2.69). Equation (2.70) can be obtained
in a similar way.
More examples of graphical Yang–Baxter equations can be found in Section 4.2. One should
keep in mind that initially we have only one Yang–Baxter equation with many analytical and
graphical reincarnations.
Vertex Models and Spin Chains in Formulas and Pictures 29
=
Figure 2.46.
=
Figure 2.47.
α
β
i
j
ζ
η
MV |W (ζ|η)αiβj
Figure 2.48.
α
β
i
j
ζ
η
(MV |W (ζ|η)−1)αiβj
Figure 2.49.
Let us show now that if the corresponding R-operators satisfy the unitarity relations we can
obtain the Yang–Baxter equation given in Fig. 2.42. We start with the Yang–Baxter equation
depicted in Fig. 2.46. Using the crossing relation in Fig. 2.24, we come to the Yang–Baxter
equation in Fig. 2.47. Now if the corresponding R-operators satisfy the unitarity relations
described in Section 2.2.3 we obtain the Yang–Baxter equation given in Fig. 2.42.
It is not difficult to demonstrate that if the corresponding unitarity relations are satisfied,
all possible Yang–Baxter equations with all possible types of the vertices and directions of the
arrows are correct. However, if the unitarity relations are not true, one has to check whether
the used Yang–Baxter equations can be obtained without them.
2.2.7 Monodromy operators
The representation spaces of the basic modules used to construct integrability objects are of
two types: auxiliary and quantum spaces. Although the boundary between these two types is
rather conventional, such a division proves useful. When both spaces used to define a basic
integrability object are auxiliary, we call it an R-operator. When one of the spaces is auxiliary
and another one is quantum, we say about a monodromy operator. In this paper we use for
auxiliary spaces as before a single line, double lines, etc., and indices α, β, γ, etc. For quantum
spaces we use waved lines, and indices i, j, k, etc.
The definition of a basic monodromy operator is in fact the same as the definition of an
R-operator, except the notation. Let V and W be two Uq(L(g))-modules, and ϕ and ψ the
associated representations, corresponding to auxiliary and quantum spaces respectively. We
define a monodromy operator MV |W (ζ|η) as
ρϕ|ψ(ζ|η)MV |W (ζ|η) = (ϕζ ⊗ ψη)(R).
The graphical representation of the matrix elements of MV |W (ζ|η) and its inverse is practically
the same as for R-matrices. However, for completeness, we present it in Figs. 2.48 and 2.49.
From the point of view of spin chains, the monodromy operator MV |W (ζ|η) corresponds to
a chain of one site. For a general spin chain we use instead of the module Wη the tensor product
Wη1 ⊗∆ Wη2 ⊗∆ · · · ⊗∆ WηN ,
30 Kh.S. Nirov and A.V. Razumov
ζ
ηN η2 η1
iN i2 i1
jN j2 j1
α
β
MV |W (ζ|η1, η2, . . . , ηN )αi1i2...iN βj1j2...jN
Figure 2.50.
and define the monodromy operator MV |W (ζ|η1, η2, . . . , ηN ) as
ρV |W (ζ|η1)ρV |W (ζ|η2) · · · ρV |W (ζ|ηN )MV |W (ζ|η1, η2, . . . , ηN )
= (ϕζ ⊗ (ψη1 ⊗∆ ψη2 ⊗∆ · · · ⊗∆ ψηN ))(R).
Let us establish the connection of the general monodromy operator with the basic one. Con-
sider the case of N = 2. By definition, we have
ρV |W (ζ|η1)ρV |W (ζ|η2)MV |W (ζ|η1, η2) = (ϕζ ⊗ (ψη1 ⊗∆ ψη2))(R)
= (ϕζ ⊗ ψη1 ⊗ ψη2)((id⊗∆)(R)). (2.71)
Using the second equation of (2.18) in (2.71), we obtain
MV |W (ζ|η1, η2) = R
(13)
V |W (ζ|η2)R
(12)
V |W (ζ|η1).
It is not difficult to see that for a general N one has
MV |W (ζ|η1, η2, . . . , ηN ) = R
(1,N+1)
V |W (ζ|ηN ) · · ·R(13)
V |W (ζ|η2)R
(12)
V |W (ζ|η1), (2.72)
or in terms of matrix elements
MV |W (ζ|η1, η2, . . . , ηN )αi1i2...iN βj1j2...jN
= RV |W (ζ|ηN )αiN γ1jN · · ·RV |W (ζ|η2)γN−2i2
γN−1j2RV |W (ζ|η1)γN−1i1
βj1 .
Now it is clear that these matrix elements can be depicted as in Fig. 2.50. The inverse mono-
dromy operator has the form
MV |W (ζ|η1, η2, . . . , ηN )−1 = R
(12)
V |W (ζ|η1)−1R
(13)
V |W (ζ|η2)−1 · · ·R(1,N+1)
V |W (ζ|ηN )−1,
and the graphical representation of its matrix elements can be found in Fig. 2.51. A graphical
exposition of the fact that the operators depicted in Figs. 2.50 and 2.51 are mutually inverse is
given in Figs. 2.52–2.55.
Below we numerate auxiliary spaces by primed numbers and quantum spaces by usual num-
bers. For example, we assume that the monodromy operator (2.72) acts on the space
U1′ ⊗ U1 ⊗ U2 ⊗ · · · ⊗ UN = V ⊗W ⊗W ⊗ · · · ⊗W,
and write
MV |W (ζ|η1, η2, . . . , ηN ) = R
(1′N)
V |W (ζ|ηN ) · · ·R(1′2)
V |W (ζ|η2)R
(1′1)
V |W (ζ|η1).
The most important property of monodromy operators is the relation
RV1|V2(ζ1|ζ2)(1′2′)MV1|W (ζ1|η1, η2, . . . , ηN )(1′1...N)MV2|W (ζ2|η1, η2, . . . , ηN )(2′1...N)
= MV2|W (ζ2|η1, η2, . . . , ηN )(2′1...N)MV1|W (ζ1|η1, η2, . . . , ηN )(1′1...N)RV1|V2(ζ1|ζ2)(1′2′).
Here we have two auxiliary spaces labeled by 1′ and 2′. The well known graphical proof of the
above relation is presented in Figs. 2.56–2.59.
Vertex Models and Spin Chains in Formulas and Pictures 31
ζ
ηN η2 η1
iN i2 i1
jN j2 j1
β
α
(MV |W (ζ|η1, η2, . . . , ηN )−1)αi1i2...iN βj1j2...jN
Figure 2.51.
Figure 2.52. Figure 2.53.
Figure 2.54. Figure 2.55.
Figure 2.56. Figure 2.57.
Figure 2.58. Figure 2.59.
32 Kh.S. Nirov and A.V. Razumov
2.2.8 Transfer operators and Hamiltonians
By definition, a nonzero element a ∈ Uq(L(g)) is group-like if
∆(a) = a⊗ a.
The counit axiom says
(ε⊗ id)(∆(a)) = (id⊗ ε)(∆(a)) = a,
where in the last equality the canonical identification C⊗Uq(L(g)) ' Uq(L(g)) is used. Hence,
for a group-like element a we have
ε(a)a = aε(a) = a,
therefore,
ε(a) = 1.
Now, using the antipode axiom,
µ((id⊗ S)(∆(a))) = µ((S ⊗ id)(∆(a))) = ι(ε(a))
we obtain
aS(a) = S(a)a = 1.
Thus, any group-like element is invertible, and, since 1 is group-like, the set of all group-like
elements form a group.
It follows from (2.22) and (2.13) that
∆ ◦ Γζ = (Γζ ⊗ Γζ) ◦∆.
This equation implies that
Γζ(a) = a
for any group-like element a. Therefore, for any representation ϕ of Uq(L(g)) we have
ϕζ(a) = ϕ(a).
By definition, for any group-like element we have
∆(a) = ∆′(a).
It follows that
R(a⊗ a) = (a⊗ a)R.
Hence, for any two representations ϕ1, ϕ2 of Uq(L(g)) and the corresponding Uq(L(g))-modu-
les V1, V2 one can obtain the equation
RV1|V2(ζ1|ζ2)(AV1 ⊗ AV2) = (AV1 ⊗ AV2)RV1|V2(ζ1|ζ2). (2.73)
Here and below for any representation ϕ of Uq(L(g)) and the corresponding Uq(L(g))-module V
we denote
AV = ϕζ(a) = ϕ(a).
Vertex Models and Spin Chains in Formulas and Pictures 33
α1 α2
(AV )α1
α2
Figure 2.60.
α1 α2
(A−1
V )α1
α2
Figure 2.61.
=
Figure 2.62.
=
Figure 2.63.
=
Figure 2.64.
=
Figure 2.65.
=
Figure 2.66.
Figure 2.67. Figure 2.68.
For the matrix elements of the operator AV and its inverse we use the depiction given in Figs. 2.60
and 2.61. One can easily recognize the graphical image of equation (2.73) in Fig. 2.62. Note
that the equation
ϕ∗(a) = ∗ϕ(a) =
(
ϕ(a)−1
)t
results in four graphical equations given in Figs. 2.63–2.66.
We call a group-like element a the twisting element, and the corresponding operators AV the
twisting operators. The twisted transfer operator is defined by the equation
TV |W (ζ|η1, η2, . . . , ηN ) = trV (MV |W (ζ|η1, η2, . . . , ηN )(AV ⊗ idW ))
with the depiction given in Fig. 2.67. Here trV means the partial trace with respect to the
space V , see Appendix A.3, and hooks at the ends of the line mean that it is closed in an
evident way.
The most important property of transfer operators is their commutativity
[TV1|W (ζ1|η1, η2, . . . , ηN ), TV2|W (ζ2|η1, η2, . . . , ηN )] = 0.
The graphical proof of this property starts with a picture representing the product of two transfer
operators, see Fig. 2.68. Then one makes the four steps described by Figs. 2.69–2.72.
34 Kh.S. Nirov and A.V. Razumov
Figure 2.69. Figure 2.70.
Figure 2.71. Figure 2.72.
ζ
η
ζ, η=1
=
Figure 2.73.
The commutativity property is the source of commuting quantities of quantum integrable
systems. The most interesting here are local quantities. An example of such a quantity is
a Hamiltonian, which is usually constructed in the following way. Assume that the quantum
space W coincides with the auxiliary space V . Further, assume that the R-operator RV |V (1|1) is
proportional to the permutation operator P12. In fact, as it follows from the definition below, the
Hamiltonian does not change after multiplying of the R-operator by a constant factor, and we
assume that RV |V (1|1) coincides with the permutation operator. The equation given in Fig. 2.73
is the graphical representation of this fact. The Hamiltonian HN for the chain of length N is
constructed from the homogeneous transfer operator
TV (ζ) = TV |V (ζ|1, 1, . . . , 1)
with the help of the equation
HN = ζ
d
dζ
log TV (ζ)
∣∣∣∣
ζ=1
= ζ
dTV (ζ)
dζ
∣∣∣∣
ζ=1
TV (1)−1 = T ′V (1)TV (1)−1.
It is evident that Figs. 2.74 and 2.75 represent the depiction of the operators TV (1) and TV (1)−1,
respectively. In fact, they are shifts operators multiplied by the twisting operator or its inverse.
We use for the derivative of the R-operators the depiction given in Fig. 2.76. It is clear that HN
is a sum of N terms arising from differentiation of the R-operators entering the transfer mat-
rix TV (ζ). As is shown above AV does not depend on ζ and therefore there are no corresponding
derivative terms. We meet three different situations depicted in Figs. 2.77, 2.78 and 2.79. Note
that for clarity the pictures in Figs. 2.77 and 2.79 are rotated. It is clear that the twisting ope-
Vertex Models and Spin Chains in Formulas and Pictures 35
Figure 2.74. Figure 2.75.
=
d
dζ
ζ
η
ζ, η=1
Figure 2.76.
Figure 2.77.
Figure 2.78.
Figure 2.79.
rators are retained only in the first situation, and we come to the following analytical expression
for the Hamiltonian
HN =
∑
i∈[1 . . N−1]
H(i,i+1) + A(1)
V H(N,1)
(
A−1
V
)(1)
, (2.74)
where we use the notation
H(k,l) = Ř′V |V (1|1)(k,l). (2.75)
36 Kh.S. Nirov and A.V. Razumov
3 Integrability objects for the case
of quantum loop algebra Uq(L(sll+1))
To construct integrability objects for the quantum loop algebra Uq(L(sll+1)) we need its repre-
sentations. The simplest way to construct such representations is to use Jimbo’s homomorphism
from Uq(L(sll+1)) to the quantum group Uq(gll+1). Therefore, we start with a short reminder
of some basics facts on finite-dimensional representations of the quantum group Uq(gll+1).
3.1 Quantum group Uq(gll+1) and some its representations
3.1.1 Definition
The standard basis of the standard Cartan subalgebra kl+1 of gll+1 is formed by the matrices Ki,
i ∈ [1 . . l + 1], with the matrix entries
(Ki)jm = δijδim.
There are l simple roots αi ∈ k∗l+1, which are defined by the equation
〈αi,Kj〉 = cji,
where
cij = δij − δi,j+1.
The full system of positive roots of gll+1 is
∆+ = {αij | 1 ≤ i < j ≤ l + 1},
where
αij =
j−1∑
k=i
αk, 1 ≤ i < j ≤ l + 1.
It is clear that αi = αi,i+1. Certainly, the negative roots are −αij .
The special linear Lie algebra sll+1 is a subalgebra of gll+1. The standard Cartan subalge-
bra hl+1 of sll+1 is formed by the elements
Hi = Ki −Ki+1, i ∈ [1 . . l].
The positive and negative roots of sll+1 are the restrictions of αij and −αij to hl+1 respectively.
Here we have
〈αj , Hi〉 = aij ,
where
aij = cij − ci+1,j = − δi−1,j + 2δij − δi+1,j (3.1)
are the entries of the Cartan matrix of the Lie algebra sll+1. The highest root of sll+1 is
θ = α1,l+1 =
l∑
i=1
αi.
Vertex Models and Spin Chains in Formulas and Pictures 37
One can easily see that
(θ|θ) = 2. (3.2)
The Kac labels and the dual Kac labels are given by the equations
ai = 1, ǎi = 1, i ∈ [1 . . l],
and, therefore, for the Coxeter number and the dual Coxeter number of sll+1 one has
h = l + 1, hˇ = l + 1. (3.3)
Let q be the exponential of a complex number ~, such that q is not a root of unity. We define
the quantum group Uq(gll+1) as a unital associative C-algebra generated by the elements5
Ei, Fi, i = 1, . . . , l, qX , X ∈ kl+1,
satisfying the following defining relations
q0 = 1, qX1qX2 = qX1+X2 ,
qXEiq
−X = q〈αi,X〉Ei, qXFiq
−X = q−〈αi,X〉Fi,
[Ei, Fj ] = δij
qKi−Ki+1 − q−Ki+Ki+1
q − q−1
.
Besides, we have the Serre relations
EiEj = EjEi, FiFj = FjFi, |i− j| ≥ 2,
E2
i Ei±1 − [2]qEiEi±1Ei + Ei±1E
2
i = 0, F 2
i Fi±1 − [2]qFiFi±1Fi + Fi±1F
2
i = 0.
From the point of view of quantum integrable systems, it is important that Uq(gll+1) is a Hopf
algebra with respect to appropriately defined comultiplication, antipode and counit. However,
we do not use the explicit form of the Hopf algebra structure in the present paper. The quantum
group Uq(sll+1) can be considered as a Hopf subalgebra of Uq(gll+1) generated by the elements
Ei, Fi, i = 1, . . . , l, qX , X ∈ hl+1.
Following Jimbo [38], introduce the elements Eij and Fij , 1 ≤ i < j ≤ l+ 1, with the help of
the relations
Ei,i+1 = Ei, i ∈ [1 . . l],
Eij = Ei,j−1Ej−1,j − qEj−1,jEi,j−1, j − i > 1,
and
Fi,i+1 = Fi, i = 1, . . . , l,
Fij = Fj−1,jFi,j−1 − q−1Fi,j−1Fj−1,j , j − i > 1.
The appropriately ordered monomials constructed from Eij , Fij , 1 ≤ i < j ≤ l + 1, and qX ,
X ∈ kl+1, form a Poincaré–Birkhoff–Witt basis of Uq(gll+1).
A Uq(gll+1)-module V is said to be a weight module if
V =
⊕
λ∈k∗l+1
Vλ,
5We use capital letters to distinguish between generators of Uq(gll+1) and Uq(L(sll+1)).
38 Kh.S. Nirov and A.V. Razumov
where
Vλ =
{
v ∈ V | qXv = q〈λ,X〉v for any X ∈ kl+1
}
.
The space Vλ is called the weight space of weight λ, and a nonzero element of Vλ is called
a weight vector of weight λ. We say that λ ∈ k∗l+1 is a weight of V if Vλ 6= {0}.
The Uq(gll+1)-module V is called a highest weight Uq(gll+1)-module with highest weight λ if
there exists a weight vector vλ ∈ V satisfying the relations
Eiv
λ = 0, i = 1, . . . , l, qXvλ = q〈λ,X〉vλ, X ∈ kl+1, λ ∈ k∗l+1,
Uq(gll+1)vλ = V.
Given λ ∈ k∗l+1, denote by Ṽ λ the corresponding Verma Uq(gll+1)-module, see, for example [45].
This is a highest weight module with the highest weight λ. Below we sometimes identify any
weight λ with the set of its components (λ1, . . . , λl+1), where
λi = 〈λ,Ki〉.
We denote by π̃λ the representation of Uq(gll+1) corresponding to Ṽ λ. The structure and
properties of Ṽ λ and π̃λ for l = 1 and l = 2 are considered in much detail in papers [17, 18, 19,
20, 56] and for a general l in papers [21, 55, 57]. It is clear that Ṽ λ and π̃λ are infinite-dimensional
for a general weight λ ∈ k∗l+1. However, if all the differences λi − λi+1, i = 1, . . . , l, are non-
negative integers, there is a maximal submodule, such that the respective quotient module is
finite-dimensional. We denote this quotient by V λ and the corresponding representation by πλ.
For any i ∈ [1 . . l] the finite-dimensional representation πωi with
ωi = (1, . . . , 1
i
, 0, . . . , 0
l+1−i
)
is called the i-th fundamental representation of Uq(gll+1). It is clear that ωi can be also defined
as
ωi(Kj) =
{
1, 1 ≤ j ≤ i,
0, i < j ≤ l + 1.
Hence, it is evident that
ωi(Hj) = δij .
The weights ωi, i ∈ [1 . . l], are called the fundamental weights of Uq(gll+1). The dimensions of
the corresponding fundamental representations πωi are
(
l+1
i
)
, i ∈ [1 . . l].
3.1.2 Representation π
We denote by π the first fundamental representation πω1 of the quantum group Uq(gll+1).
This representation is (l + 1)-dimensional and can be realised as follows. Assume that the
representation space is the free vector space generated by the set {vk}k∈[1 . . l+1] and denote
by Eij , i, j ∈ [1 . . l + 1], the endomorphisms of this space defined by the equation
Eijvk = viδjk. (3.4)
Vertex Models and Spin Chains in Formulas and Pictures 39
One can verify that the equations
π(qνKi) = qνEii +
l+1∑
k=1
k 6=i
Ekk, i ∈ [1 . . l + 1],
π(Ei) = Ei,i+1, π(Fi) = Ei+1,i, i ∈ [1 . . l],
describe an (l + 1)-dimensional representation of Uq(gll+1) with the highest weight ω1, as is
required. It is useful to have in mind the equations
π(Eij) = Eij , π(Fij) = Eji, 1 ≤ i < j ≤ i+ 1.
One can see that
v2 = F1v1, v3 = F2F1v1, . . . , vl+1 = Fl · · ·F2F1v1.
Therefore, vk is a weight vector of weight
λk = ω1 −
k−1∑
i=1
αi. (3.5)
3.1.3 Representation π
The last fundamental representation π = πωl of Uq(gll+1) is also (l + 1)-dimensional. We again
assume that the representation space is the free vector space generated by the set {vk}k∈[1 . . l+1]
and denote by Eij , i, j ∈ [1 . . l + 1], the endomorphisms of this space defined by equation (3.4).
Here we have
π(qνKi) = qν
l+1∑
k=1
k 6=l−i+2
Ekk + El−i+2, l−i+2, i ∈ [1 . . l + 1],
π(Ei) = El−i+1,l−i+2, π(Fi) = El−i+2,l−i+1, i ∈ [1 . . l].
It is not difficult to determine that
π(Eij) = (−1)−i+j−1q−i+j−1El−j+2,l−i+2, 1 ≤ i < j ≤ l + 1,
π(Fij) = (−1)−i+j−1qi−j+1El−i+2,l−j+2, 1 ≤ i < j ≤ l + 1.
We obtain successively
v2 = Flv1, v3 = Fl−1Flv1, . . . , vl+1 = F1 · · ·Fl−1Flv1,
and see that vk is a weight vector of weight
λk = ωl −
l∑
i=l−k+2
αi. (3.6)
3.2 Representations of Uq(L(sll+1))
All representations of Uq(L(sll+1)) considered in this section are (l + 1)-dimensional, and we
always assume that their representation space is the free vector space generated by the set
{vk}k∈[1 . . l+1] and denote by Eij , i, j ∈ [1 . . l + 1], the endomorphisms of this space defined by
equation (3.4).
40 Kh.S. Nirov and A.V. Razumov
3.2.1 Jimbo’s homomorphism
To construct representations of Uq(L(sll+1)), it is common to use Jimbo’s homomorphism ε
from the quantum loop algebra Uq(L(sll+1)) to the quantum group Uq(gll+1) defined by the
equations [38]
ε(qνh0) = qν(Kl+1−K1), ε(qνhi) = qν(Ki−Ki+1), (3.7)
ε(e0) = F1,l+1q
K1+Kl+1 , ε(ei) = Ei,i+1, (3.8)
ε(f0) = E1,l+1q
−K1−Kl+1 , ε(fi) = Fi,i+1, (3.9)
where i runs from 1 to l.
3.2.2 Representation ϕζ
We denote by ϕζ the representation π ◦ ε ◦ Γζ of Uq(L(sll+1)), where π is the representation
of Uq(gll+1) considered in Section 3.1.2, and Γζ is defined by equation (2.22). Using Jimbo’s
homomorphism, we obtain
ϕζ
(
qνh0
)
= q−νE11 + qνEl+1,l+1 +
l∑
k=2
Ekk, (3.10)
ϕζ
(
qνhi
)
= qνEii + q−νEi+1,i+1 +
l+1∑
k=1
k 6=i,i+1
Ekk, i ∈ [1 . . l], (3.11)
and, further,
ϕζ(e0) = ζs0qEl+1,1, ϕζ(ei) = ζsiEi,i+1, i ∈ [1 . . l],
ϕζ(f0) = ζ−s0q−1E1,l+1, ϕζ(fi) = ζ−siEi+1,i, i ∈ [1 . . l].
It is clear that the vector vk is a weight vector of weight defined as the restriction of the weight λk,
given by equation (3.5), to the Cartan subalgebra hl+1 of sll+1.
3.2.3 Representation ϕζ
From the representation π of Uq(gll+1), described in Section 3.1.3 we obtain the representation
ϕ = π ◦ ε ◦ Γζ of Uq(L(sll+1)). Here we have
ϕζ
(
qνh0
)
= q−νE11 + qνEl+1,l+1 +
l∑
k=2
Ekk,
ϕζ
(
qνhi
)
= qνEl−i+1,l−i+1 + q−νEl−i+2,l−i+2 +
l+1∑
k=1
k 6=l−i+1
k 6=l−i+2
Ekk, i ∈ [1 . . l],
and, further,
ϕζ(e0) = ζs0(−1)l−1q−l+2El+1,1, ϕζ(ei) = ζsiEl−i+1,l−i+2, i ∈ [1 . . l],
ϕζ(f0) = ζ−s0(−1)l−1ql−2E1,l+1, ϕζ(fi) = ζ−siEl−i+2,l−i+1, i ∈ [1 . . l].
Here again vk is a weight vector of weight which is obtained by the restriction of the weight λk,
given by equation (3.6), to the Cartan subalgebra hl+1 of sll+1.
Vertex Models and Spin Chains in Formulas and Pictures 41
3.2.4 Representations ϕ∗
ζ and ∗ϕζ
The dual representations are defined and discussed in Section 2.2.4. Consider first the represen-
tation ϕ∗ζ . For the generators qνhi we obtain
ϕ∗ζ
(
qνh0
)
= qνE11 + q−νEl+1,l+1 +
l∑
k=2
Ekk,
ϕ∗ζ
(
qνhi
)
= q−νEii + qνEi+1,i+1 +
l+1∑
k=1
k 6=i,i+1
Ekk, i ∈ [1 . . l],
and some simple calculations lead to the relations
ϕ∗ζ(e0) = −ζs0E1,l+1, ϕ∗ζ(ei) = −ζsiq−1Ei+1,i, i ∈ [1 . . l],
ϕ∗ζ(f0) = −ζ−s0El+1,1, ϕ∗ζ(fi) = −ζ−siqEi,i+1, i ∈ [1 . . l].
Now vk is a weight vector of weight
λk = ωl −
l∑
i=k
αi, (3.12)
where ωl and αi are treated as elements of h∗l+1. The representations ϕ∗ζ and ϕζ are equivalent
up to a rescaling of the spectral parameter. In fact, for any a ∈ Uq(L(sll+1)) one has
Pϕ∗
q2/sζ
(a)P−1 = ϕζ(a),
where the operator P is given by the equation
P =
l+1∑
i=1
(−1)i−1q
−2
i−1∑
k=1
sk/s+i−1
El−i+2,i.
The representation ∗ϕζ is very similar to the representation ϕ∗ζ . Here for the generators qνhi
of Uq(L(sll+1)) we obtain
∗ϕζ
(
qνh0
)
= qνE11 + q−νEl+1,l+1 +
l∑
k=2
Ekk,
∗ϕζ
(
qνhi
)
= q−νEii + qνEi+1,i+1 +
l+1∑
k=1
k 6=i,i+1
Ekk, i ∈ [1 . . l].
Then it is not difficult to come to the relations
∗ϕζ(e0) = −ζs0q2E1,l+1,
∗ϕζ(ei) = −ζsiqEi+1,i, i ∈ [1 . . l],
∗ϕζ(f0) = −ζ−s0q−2El+1,1,
∗ϕζ(fi) = −ζ−siq−1Ei,i+1, i ∈ [1 . . l].
The vector vk is a weight vector of the weight given again by equation (3.12). The representa-
tions ∗ϕζ and ϕζ are again equivalent up to a rescaling of the spectral parameter. It can be
verified that
P∗ϕq−2l/sζ(a)P−1 = ϕζ(a),
where now the operator P is given by the equation
P =
l+1∑
i=1
(−1)i−1q2l
∑i−1
k=1 sk/s−i+1El−i+2,i.
42 Kh.S. Nirov and A.V. Razumov
3.3 Integrability objects
3.3.1 Poincaré–Birkhoff–Witt basis
Recall that to construct a Poincaré–Birkhoff–Witt basis one has to define root vectors corre-
sponding to all roots of L̂(sll+1). To construct root vectors we follow the procedure proposed
by Khoroshkin and Tolstoy based on a normal ordering of positive roots [42, 43, 44, 72].
For the case of a finite-dimensional simple Lie algebra an order relation ≺ is called a normal
order [1, 50, 71] when if a positive root γ is a sum of two positive roots α ≺ β, then α ≺ γ ≺ β.
In our case we assume additionally that
α+ kδ ≺ mδ ≺ (δ − β) + nδ (3.13)
for any α, β ∈ ∆+, k, n ∈ Z≥0 and m ∈ Z>0.
Assume that some normal ordering of positive roots is chosen. We say that a pair (α, β) of
positive roots generates a root γ if γ = α+β and α ≺ β. A pair of positive roots (α, β) generating
a root γ is said to be minimal if there is no other pair of positive roots (α′, β′) generating γ such
that α ≺ α′ ≺ β′ ≺ β.
It is convenient to denote a root vector corresponding to a positive root γ by eγ , and a root
vector corresponding to a negative root −γ by fγ . Following [43, 72], we define root vectors by
the following inductive procedure. Given a root γ ∈ ∆+, let (α, β) be a minimal pair of positive
roots generating γ. Now, if the root vectors eα, eβ and fα, fβ are already constructed, we define
the root vectors eγ and fγ as
eγ = [eα, eβ]q, fγ = [fβ, fα]q.
Here and below we use the q-commutator of root vectors [·, ·]q defined as
[eα, eβ]q = eαeβ − q−(α|β)eβeα, [fα, fβ]q = fαfβ − q(α|β)fβfα,
where (·|·) denotes the symmetric bilinear form on ĥ∗ described in Section 2.1.1.
We use the normal order of the positive roots of L̂(sll+1) defined as follows. First order
the positive roots of sll+1 assuming that αij ≺ αkl if i < k, or if i = k and j < l. Then,
α + mδ ≺ β + nδ, with α, β ∈ ∆+ and m,n ∈ Z≥0, if α ≺ β, or α = β and m < n. Further,
(δ−α) +mδ ≺ (δ−β) +nδ, with α, β ∈ ∆+, if α ≺ β, or α = β and m > n. Finally, we assume
that relation (3.13) is valid.
The root vectors are defined inductively. We start with the root vectors corresponding to the
roots ±αi, i ∈ [1 . . l], which we identify with the generators ei and fi, i ∈ [1 . . l], of Uq(L(sll+1)),
eαi = eαi,i+1 = ei, fαi = fαi,i+1 = fi.
The next step is to construct root vectors eαij and fαij for all roots αij ∈ ∆+. We assume that
eαij = eαi,j−1eαj−1,j − qeαj−1,jeαi,j−1 , fαij = fαj−1,jfαi,j−1 − q−1fαi,j−1fαj−1,j
for j − i > 1. Further, taking into account that
α0 = δ − α1,l+1,
we put
eδ−α1,l+1
= e0, fδ−α1,l+1
= f0,
and define
eδ−αi,l+1
= eαi−1,ieδ−αi−1,l+1
− qeδ−αi−1,l+1
eαi−1,i ,
Vertex Models and Spin Chains in Formulas and Pictures 43
fδ−αi,l+1
= fδ−αi−1,l+1
fαi−1,i − q−1fαi−1,ifδ−αi−1,l+1
for i > 1, and
eδ−αij = eαj,j+1eδ−αi,j+1
− qeδ−αi,j+1
eαj,j+1 ,
fδ−αij = fδ−αi,j+1
fαj,j+1 − q−1fαj,j+1fδ−αi,j+1
for j < l+ 1. The root vectors corresponding to the roots δ and −δ are indexed by the elements
of ∆+ and defined by the relations
e′δ,αij = eαijeδ−αij − q2eδ−αijeαij , f ′δ,αij = fδ−αijfαij − q−2fαijfδ−αij .
The remaining definitions are
eαij+nδ = [2]−1
q
(
eαij+(n−1)δe
′
δ,αij
− e′δ,αijeαij+(n−1)δ
)
, (3.14)
fαij+nδ = [2]−1
q
(
f ′δ,αijfαij+(n−1)δ − fαij+(n−1)δf
′
δ,αij
)
, (3.15)
e(δ−αij)+nδ = [2]−1
q
(
e′δ,αije(δ−αij)+(n−1)δ − e(δ−αij)+(n−1)δe
′
δ,αij
)
, (3.16)
f(δ−αij)+nδ = [2]−1
q
(
f(δ−αij)+(n−1)δf
′
δ,αij
− f ′δ,αijf(δ−αij)+(n−1)δ
)
, (3.17)
e′nδ,αij = eαij+(n−1)δeδ−αij − q2eδ−αijeαij+(n−1)δ, (3.18)
f ′nδ,αij = fδ−αijfαij+(n−1)δ − q−2fαij+(n−1)δfδ−αij . (3.19)
Note that, among all imaginary root vectors e′nδ,αij and f ′nδ,αij only the root vectors e′nδ,αi,i+1
and f ′nδ,αi,i+1
, i ∈ [1 . . l], are independent and required for the construction of the Poincaré–
Birkhoff–Witt basis. However, the vectors e′δ,γ and f ′δ,γ with arbitrary γ ∈ ∆+ are needed for
the iterations (3.14)–(3.19).
The prime in the notation for the root vectors corresponding to the imaginary roots nδ and
−nδ, n ∈ Z>0 is explained by the fact that to construct the expression for the universal R-matrix
one uses another set of root vectors corresponding to these roots. They are introduced by the
functional equations
−κqeδ,γ(u) = log(1− κqe′δ,γ(u)),
κqfδ,γ(u−1) = log(1 + κqf
′
δ,γ(u−1)),
where the generating functions
e′δ,γ(u) =
∞∑
n=1
e′nδ,γu
n, eδ,γ(u) =
∞∑
n=1
enδ,γu
n,
f ′δ,γ(u−1) =
∞∑
n=1
f ′nδ,γu
−n, fδ,γ(u−1) =
∞∑
n=1
fnδ,γu
−n
are defined as formal power series, and κq is defined by the equation
κq = q − q−1.
3.3.2 Monodromy operators
The expression for the universal R-matrix of Uq(L(sll+1)) considered as a C[[~]]-algebra can be
constructed using the procedure proposed by Khoroshkin and Tolstoy [42, 43, 44, 72]. Here we
treat the quantum group as an associative C-algebra. In fact, one can use the expression for
the universal R-matrix from papers [42, 43, 44, 72] in this case as well, having in mind that
44 Kh.S. Nirov and A.V. Razumov
the quantum group is quasitriangular only in some restricted sense. Namely, all the relations
involving the universal R-matrix should be considered as valid only for the weight representations
of Uq(L(sll+1)), see in this respect paper [70] and the discussion below.
Let V , W be two weight Uq(L(sll+1))-modules in the category O, and ϕ, ψ the corresponding
representations. Define the monodromy operator MV |W (ζ|η) as
MV |W (ζ|η) = ρV |W (ζ|η)(ϕζ ⊗ ψη)(R≺δR∼δR�δ)KV |W . (3.20)
Here R≺δ, R∼δ and R�δ are elements of Uq(L(sll+1))⊗Uq(L(sll+1)), while KV |W is an element
of End(V ⊗W ).
Explicitly, the element R≺δ is the product over the set of roots αij +nδ of the q-exponentials
Rαij+nδ = expq2
(
−κqeαij+nδ ⊗ fαij+nδ
)
.
The order of the factors in R≺δ coincides with the chosen normal order of the roots αij + nδ.
The element R∼δ is defined as
R∼δ = exp
−κq
∑
n∈Z>0
l∑
i,j=1
un,ijenδ,αi ⊗ fnδ,αj
,
where for each n ∈ Z>0 the quantities un,ij are the matrix elements of the matrix Un inverse to
the matrix Tn with the matrix elements
tn,ij = (−1)n(i+j) 1
n
[naij ]q = (−1)n(i+j) [n]q
n
[aij ]qn ,
where aij are the matrix elements of the Cartan matrix A of the Lie algebra sll+1. The matrix Tn
is tridiagonal. Using the results of paper [73], one can see that
un,ij = (−1)n(i+j) n
[n]q
[i]qn [l − j + 1]qn
[l + 1]qn
, i ≤ j,
un,ij = (−1)n(i+j) n
[n]q
[l − i+ 1]qn [j]qn
[l + 1]qn
, i > j.
The definition of the element R�δ is similar to the definition of the element R≺δ. It is the
product over the set of roots (δ − αij) + nδ of the q-exponentials
R(δ−αij)+nδ = expq2
(
−κqe(δ−αij)+nδ ⊗ f(δ−αij)+nδ
)
.
The order of the factors inR�δ coincides with the chosen normal order of the roots (δ − αij)+nδ.
The endomorphism KV |W is defined as follow. Let v ∈ V and w ∈ W be weight vectors of
weights λ and µ respectively. Then we assume that
KV |W v ⊗ w = q
−
l∑
i,j=1
bij〈λ,hi〉〈µ,hj〉
v ⊗ w, (3.21)
where bij are the matrix elements of the matrix B inverse to the Cartan matrix A = (aij)i,j∈[1 . . l]
of the Lie algebra sll+1, see equation (3.1). Using again the results of paper [73], we obtain
bij =
i(l − j + 1)
l + 1
, i ≤ j, bij =
(l − i+ 1)j
l + 1
, i > j. (3.22)
One can show that it is possible to work with MV |W (ζ|η) defined by (3.20) as if it is defined
by the universal R-matrix satisfying equations (2.17) and (2.18).
Vertex Models and Spin Chains in Formulas and Pictures 45
3.3.3 Explicit form of R-operator
In this section we obtain explicit expressions for the R-operators arising in the case V = W =
V ω1 , see also paper [52]. The explicit formulas for the action of the generators of Uq(L(sll+1))
on the basis of the representation space are given in Section 3.2.2. We have
RV |V (ζ1|ζ2) = ρV |V (ζ1|ζ2)(ϕζ1 ⊗ ϕζ2)(R≺δ)(ϕζ1 ⊗ ϕζ2)(R∼δ)(ϕζ1 ⊗ ϕζ2)(R�δ)KV |V .
First construct the expression for the operator KV |V . Using equations (3.21) and (3.5), we
see that
KV |V vm ⊗ vn = q
−
l∑
i,j=1
〈
ω1−
m−1∑
q=1
αq ,hi
〉〈
ω1−
n−1∑
p=1
αp,hj
〉
bij
vm ⊗ vn
= q
−
(
b11−
m−1∑
q=1
δ1q−
n−1∑
p=1
δp1+
m−1∑
q=1
n−1∑
p=1
aqp
)
vm ⊗ vn.
It is not difficult to demonstrate that
KV |V vm ⊗ vn = q−l/(l+1)vm ⊗ vn, m = n,
KV |V vm ⊗ vn = q1/(l+1)vm ⊗ vn, m 6= n.
Hence, we come to the equation
KV |V = q−l/(l+1)
(
l+1∑
i=1
Eii ⊗ Eii + q
l+1∑
i,j=1
i 6=j
Eii ⊗ Ejj
)
. (3.23)
Following the Khoroshkin–Tolstoy procedure, described in Section 3.3.1, we obtain
ϕζ(eαij+nδ) = ζsij+ns(−1)inq(i+1)nEij , (3.24)
ϕζ(fαij+nδ) = ζ−sij−ns(−1)inq−(i+1)nEji, (3.25)
ϕζ(e(δ−αij)+nδ) = ζ(s−sij)+ns(−1)i(n+1)−1q(i+1)n+iEji, (3.26)
ϕζ(f(δ−αij)+nδ) = ζ−(s−sij)−ns(−1)i(n+1)−1q−(i+1)n−iEij , (3.27)
ϕζ(e
′
nδ,αij
) = ζns(−1)in−1q(i+1)n−1
(
Eii − q2Ejj
)
, (3.28)
ϕζ(f
′
nδ,αij
) = ζ−ns(−1)in−1q−(i+1)n+1
(
Eii − q−2Ejj
)
. (3.29)
Here and below we denote
sij =
j−1∑
k=i
sk.
Now we find expressions for ϕζ(enδ,αij ) and ϕζ(fnδ,αij ). Using (3.28) and (3.29), we obtain
for the corresponding generating functions the following expressions
1− κq
∞∑
n=1
ϕζ(e
′
nδ,αij
)un =
l+1∑
k=1
k 6=i,j
Ekk +
1− (−1)iqi−1ζsu
1− (−1)iqi+1ζsu
Eii +
1− (−1)iqi+3ζsu
1− (−1)iqi+1ζsu
Ejj ,
1 + κq
∞∑
n=1
ϕζ(f
′
nδ,αij
)un =
l+1∑
k=1
k 6=i,j
Ekk +
1− (−1)iq−i+1ζsu
1− (−1)iq−i−1ζsu
Eii +
1− (−1)iq−i−3ζsu
1− (−1)iq−i−1ζsu
Ejj ,
46 Kh.S. Nirov and A.V. Razumov
and come to the equations
ϕζ(enδ,αij ) = ζns(−1)in−1qin
[n]q
n
(
Eii − q2nEjj
)
,
ϕζ(fnδ,αij ) = ζ−ns(−1)in−1q−in
[n]q
n
(
Eii − q−2nEjj
)
.
Now, we find the expression
(ϕζ1 ⊗ ϕζ2)
−κq
∞∑
n=1
l∑
i,j=1
un,ijenδ,αi ⊗ fnδ,αj
= −
∞∑
n=1
qnl − q−nl
[l + 1]qn
ζns12
n
l+1∑
i=1
Eii ⊗ Eii
−
∞∑
n=1
q−n(l+2) − q−nl
[l + 1]qn
ζns12
n
l+1∑
i,j=1
i<j
Eii ⊗ Ejj −
∞∑
n=1
qnl − qn(l+2)
[l + 1]qn
ζns12
n
l+1∑
i,j=1
i>j
Eii ⊗ Ejj ,
which can be rewritten as
(ϕζ1 ⊗ ϕζ2)
−κq
∞∑
n=1
l∑
i,j=1
un,ijenδ,αi ⊗ fnδ,αj
= −
∞∑
n=1
qnl − q−nl
[l + 1]qn
ζns12
n
l+1∑
i,j=1
Eii ⊗ Ejj
−
∞∑
n=1
(
q−2n − 1
)ζns12
n
l+1∑
i,j=1
i<j
Eii ⊗ Ejj −
∞∑
n=1
(
1− q2n
)ζns12
n
l+1∑
i,j=1
i>j
Eii ⊗ Ejj .
Introducing the transcendental function
Fm(ζ) =
∞∑
n=1
1
[m]qn
ζn
n
,
and performing some summations, we obtain
(ϕζ1 ⊗ ϕζ2)
−κq
∞∑
n=1
l∑
i,j=1
(un)ijenδ,αi ⊗ fnδ,αj
=
(
Fl+1
(
q−lζs12
)
− Fl+1
(
qlζs12
)) l+1∑
i,j=1
Eii ⊗ Ejj
+ log
1− q−2ζs12
1− ζs12
l+1∑
i,j=1
i<j
Eii ⊗ Ejj + log
1− ζs12
1− q2ζs12
l+1∑
i,j=1
i>j
Eii ⊗ Ejj .
After all, we see that
(ϕζ1 ⊗ ϕζ2)(R∼δ) = e(Fl+1(q−lζs12)−Fl+1(qlζs12))
×
[
l+1∑
i=1
Eii ⊗ Eii +
1− q−2ζs12
1− ζs12
l+1∑
i,j=1
i<j
Eii ⊗ Ejj +
1− ζs12
1− q2ζs12
l+1∑
i,j=1
i>j
Eii ⊗ Ejj
]
. (3.30)
Proceed to the factor (ϕζ1⊗ϕζ2)(R≺δ). Using equations (3.24) and (3.25), we determine that
(ϕζ1 ⊗ ϕζ2)(Rαij+nδ) = expq2
(
−κqζsij+ns12 Eij ⊗ Eji
)
.
Vertex Models and Spin Chains in Formulas and Pictures 47
Since
(Eij ⊗ Eji)k = 0
for all 1 ≤ i < j ≤ l + 1 and k > 1 we can obtain
(ϕζ1 ⊗ ϕζ2)(Rαij+nδ) = 1− κqζsij+ns12 Eij ⊗ Eji.
Taking into account that
(Eij ⊗ Eji)(Ekm ⊗ Emk) = 0
for all 1 ≤ i < j ≤ l + 1 and 1 ≤ k < m ≤ l + 1, we see that the factors (ϕζ1 ⊗ ϕζ2)(Rαij+nδ) of
(ϕζ1 ⊗ ϕζ2)(R≺δ) can be taken in an arbitrary order and obtain the expression
(ϕζ1 ⊗ ϕζ2)(R≺δ) = 1− κq
∞∑
n=0
ζns12
l+1∑
i,j=1
i<j
ζ
sij
12 Eij ⊗ Eji
= 1− κq
1− ζs12
l+1∑
i,j=1
i<j
ζ
sij
12 Eij ⊗ Eji. (3.31)
In a similar way we come to the equation
(ϕζ1 ⊗ ϕζ2)(R�δ) = 1− κq
1− ζs12
l+1∑
i,j=1
i>j
ζ
s−sji
12 Eij ⊗ Eji. (3.32)
Finally, using equations (3.31), (3.30), (3.32) and (3.23) we obtain the following expression
for the R-operator
RV |V (ζ1|ζ2) =
l+1∑
i=1
Eii ⊗ Eii +
q(1− ζs12)
1− q2ζs12
l+1∑
i,j=1
i 6=j
Eii ⊗ Ejj
+
(1− q2)
1− q2ζs12
(
l+1∑
i,j=1
i<j
ζ
sij
12 Eij ⊗ Eji +
l+1∑
i,j=1
i>j
ζ
s−sji
12 Eij ⊗ Eji
)
,
where we assumed that the normalization factor has the form
ρV |V (ζ1|ζ2) = q−l/(l+1)eFl+1(q−lζs12)−Fl+1(qlζs12). (3.33)
It is common to use an R-operator depending on only one spectral parameter. To this end
one introduces the operator
RV |V (ζ) = RV |V (ζ|1)
so that
RV |V (ζ1|ζ2) = RV |V (ζ12).
With an appropriate choice of the integers si and normalization, we obtain the Bazhanov–Jimbo
R-operator [7, 39, 40].
48 Kh.S. Nirov and A.V. Razumov
3.3.4 Crossing and unitarity relations
It is convenient to put
ρV ∗|V (ζ1|ζ2) = ρV |V (ζ1|ζ2)−1, ρV |∗V (ζ1|ζ2) = ρV |V (ζ1|ζ2)−1, (3.34)
ρ∗V |V (ζ1|ζ2) = ρV |V (ζ1|ζ2)−1, ρV |V ∗(ζ1|ζ2) = ρV |V (ζ1|ζ2)−1, (3.35)
ρV ∗|V ∗(ζ1|ζ2) = ρV |V (ζ1|ζ2), ρ∗V |∗V (ζ1|ζ2) = ρV |V (ζ1|ζ2), (3.36)
where ρV |V (ζ1|ζ2) is defined by equation (3.33). In this case all coefficients D, entering the
crossing relations (2.42)–(2.47), are equal to 1. Moreover, all corresponding R-operators satisfy
unitarity relation with the coefficients C equal to 1.
Let us describe now the explicit form of the quantities entering the crossing relations (2.63)–
(2.66). Notice first that due to (3.2) and (3.3) one has
q−(θ|θ)hˇ/s = q−2(l+1)/s, q(θ|θ)hˇ/s = q2(l+1)/s.
Further, it follows from (2.57) and (3.22) that
λi = −(l + 1− i)i+ 2(l + 1− i)
i∑
j=1
jsj/s+ 2i
l∑
j=i+1
(l + 1− j)sj/s.
Now, taking into account (3.11), we obtain
XV = ϕζ
(
q
l∑
i=1
λihi)
=
l+1∑
i=1
qχiEii,
where
χi = λi − λi−1 = −(l + 2− 2i)− 2
i−1∑
j=1
jsj/s+ 2
l∑
j=i
(l + 1− j)sj/s.
It follows from the definition of Fm(ζ) that
Fm(qmζ)− Fm(q−mζ) = − log(1− qζ) + log
(
1− q−1ζ
)
.
At last, using equations (3.34) and (3.35), we obtain for the coefficients D(ζ1|ζ2) entering the
crossing relations (2.63) and (2.64) the expression
D(ζ1|ζ2) =
1− q−2ζs12
1− ζs12
1− q−2lζs12
1− q−2l−2ζs12
,
and for the coefficients D(ζ1|ζ2) entering the crossing relations (2.65) and (2.66) the expression
D(ζ1|ζ2) =
1− q2ζs12
1− ζs12
1− q2lζs12
1− q2l+2ζs12
.
3.3.5 Hamiltonian
It is easy to verify that the permutation operator P12 has the representation
P12 =
l+1∑
i,j=1
Eij ⊗ Eji.
Vertex Models and Spin Chains in Formulas and Pictures 49
Using this representation, we obtain
P12(Eij ⊗ Ekm) = Ekj ⊗ Eim,
and come to the equation
ŘV |V (ζ) =
l+1∑
i=1
Eii ⊗ Eii +
q(1− ζs)
1− q2ζs
l+1∑
i,j=1
i 6=j
Eji ⊗ Eij
+
(
1− q2
)
1− q2ζs
(
l+1∑
i,j=1
i<j
ζsijEjj ⊗ Eii +
l+1∑
i,j=1
i>j
ζs−sjiEjj ⊗ Eii
)
.
From the structure of the universal R-matrix [13, 28, 42, 43, 44, 72], it follows that the
dependence of a transfer operator on ζ is determined by the dependence on ζ of the elements
of the form ϕζ(a), where a is an element of the Hopf subalgebra of Uq(L(sll+1)) generated
by the elements ei, i ∈ [0 . . l], and qx, x ∈ h̃. Taking into account the form (3.7)–(3.9) of
Jimbo’s homomorphism, we see that ϕζ(a) for any such element equals π(A), where A is a linear
combination of monomials each of which is a product of Ei, i ∈ [1 . . l], F1,l+1 and qX for some
X ∈ kl+1. Let A be such a monomial. We have
qH1Aq−H1 = q2n1−n2−nA,
qHiAq−Hi = q−ni−1+2ni−ni+1A, i ∈ [2 . . l − 1],
qHlAq−Hl = q−nl−1+2nl−nA,
where ni, i ∈ [1 . . l], are the numbers of Ei, and n the number of F1,l+1 in A. Hence tr(A) can be
non-zero only if ni = n for any i ∈ [1 . . l]. Each Ei enters A with the factor ζsi , and each F1,l+1
with the factor ζs0 . Thus, for a monomial with non-zero trace we have the dependence on ζ
of the form ζns for some integer n. Therefore, assuming that the corresponding normalization
factor depends only on ζs, we see that transfer operator depends on ζ only via ζs. Thus, without
any loss of generality, finding the expression for the Hamiltonian, we can put sij = 0.
Choose the group-like element entering definition of the transfer operator as
a = q
l∑
i=1
φihi
.
Assuming that φ0 = φl+1 = 0, we have
AV = ϕ(a) =
l+1∑
i=1
qΦiEii, Φi = φi − φi−1.
Using equation (2.74), we obtain
HN = − s
κq
N−1∑
k=1
(
−
l+1∑
i,j=1
i 6=j
E(k)
ji E
(k+1)
ij + q
l+1∑
i,j=1
i<j
E(k)
jj E
(k+1)
ii + q−1
l+1∑
i,j=1
i>j
E(k)
jj E
(k+1)
ii
)
− s
κq
(
−
l+1∑
i,j=1
i 6=j
qΦi−ΦjE(N)
ji E(1)
ij + q
l+1∑
i,j=1
i<j
E(N)
jj E(1)
ii + q−1
l+1∑
i,j=1
i>j
E(N)
jj E(1)
ii
)
.
50 Kh.S. Nirov and A.V. Razumov
One can also write
HN = − s
κq
N∑
k=1
(
−
l+1∑
i,j=1
i 6=j
E(k)
ji E
(k+1)
ij + q
l+1∑
i,j=1
i<j
E(k)
jj E
(k+1)
ii + q−1
l+1∑
i,j=1
i>j
E(k)
jj E
(k+1)
ii
)
, (3.37)
where we assume that the following boundary condition
E(N+1)
ij = qΦi−ΦjE(1)
ij
is satisfied.
3.3.6 Case of Uq(L(sl2))
The fundamental representation π of Uq(sl2) is isomorphic to the representations π∗ and ∗π. The
corresponding representation ϕζ of Uq(L(sl2)) is isomorphic to the representations ϕ∗ζ and ∗ϕζ up
to a rescaling of the spectral parameter. In this case, in addition to the usual crossing relations,
we have some additional relations.
For example, we have
ϕ∗ζ(a) = Oϕq−2/sζ(a)O−1,
where
O = −q1−2s1/sE12 + E21.
It follows from this equation that
RV ∗|V (ζ1|ζ2) = ρV ∗|V (ζ1|ζ2)−1ρV |V (q−2/sζ1|ζ2)(O⊗ 1)RV |V
(
q−2/sζ1|ζ2
)
(O⊗ 1)−1.
Using the identity
F2(qζ) + F2
(
q−1ζ
)
= − log(1− ζ),
we find
ρV ∗|V (ζ1|ζ2)−1ρV |V
(
q−2/sζ1|ζ2
)
= q−1 1− ζs12
1− q−2ζs12
.
Since
RV ∗|V (ζ1|ζ2) =
(
RV |V (ζ1|ζ2)−1
)t1
we come to the equation
(
RV |V (ζ1|ζ2)−1
)t1 = q−1 1− ζs12
1− q−2ζs12
(O⊗ 1)RV |V
(
q−2/sζ1|ζ2
)
(O⊗ 1)−1.
For the representation ∗ϕζ we get
∗ϕζ(a) =
(
Ot
)−1
ϕq2/sζ(a)Ot
and come to the equation
(
RV |V (ζ1|ζ2)−1
)t2 = q−1 1− ζs12
1− q−2ζs12
(
1⊗Ot
)−1
RV |V
(
ζ1|q2/sζ2
)(
1⊗Ot
)
.
Vertex Models and Spin Chains in Formulas and Pictures 51
In a similar way, applying representations ϕ∗ζ and ∗ϕζ to other factors of the tensor product
Uq(L(sl2))⊗Uq(L(sl2)), we find that
(
RV |V (ζ1|ζ2)t1
)−1
= q−1 1− q2ζs12
1− ζs12
(
Ot ⊗ 1
)−1
RV |V
(
q2/sζ1|ζ2
)(
Ot ⊗ 1
)
,
(
RV |V (ζ1|ζ2)t2
)−1
= q−1 1− q2ζs12
1− ζs12
(1⊗O)RV |V
(
ζ1|q−2/sζ2
)
(1⊗O)−1.
The Hamiltonian HN given by (3.37) for the case of Uq(L(sl2)) is related to the well known
Hamiltonian of the XXZ-model
HXXZ = −
N∑
k=1
[
σ
(k)
+ σ
(k+1)
− + σ
(k)
− σ
(k+1)
+ +
q + q−1
4
(
σ(k)
z σ(k+1)
z − 1
)]
by the equation
HN = − s
κq
HXXZ
Here we use the standard notations
σ+ = E12, σ− = E21, σz = E11 − E22,
and assume the validity of the boundary conditions
σ
(N+1)
+ = qΦσ
(1)
+ σ
(N+1)
− = q−Φσ
(1)
− , σ(N+1)
z = σ(1)
z
with
Φ = Φ1 − Φ2.
4 Graphical description of open chains
4.1 Transfer operator
The transfer operator for an open chain is constructed in the following way. First we choose an
auxiliary space V and two quantum spaces
WR = W⊗m, WL = W⊗n.
With the space V we associate a spectral parameter ζ, and with the factors of WR and WL
the spectral parameters η1, . . . , ηm and ηm+1, . . . , ηm+n. Then we introduce two operators
KR
V |W (ζ|η1, . . . , ηm) and KL
V |W (ζ|ηm+1, . . . , ηm+n) acting on the spaces V ⊗WR and V ⊗WL
respectively. The depiction of their matrix elements can be seen in Figs. 4.1 and 4.2. It should
be noted that when an incoming line associated with the auxiliary space becomes an outgoing
one, the spectral parameter turns to its inverse. The case m = 0 or n = 0 is also allowed. Here
we have operators KL
V (ζ) and KR
V (ζ) acting on the auxiliary space V . Now, the transfer operator
is defined by the equation
TV |W (ζ|η1, . . . , ηm, ηm+1, . . . , ηm+n) (4.1)
= trV
(
KR
V |W (ζ|η1, . . . , ηm)KL
V |W (ζ|ηm+1, . . . , ηm+n)
)
. (4.2)
Here the expression under the partial trace in the right hand side means an operator acting on
V ⊗WR ⊗WL. In terms of matrix elements we have
TV |W (ζ|η1, . . . , ηm, ηm+1, . . . , ηm+n)i1...imim+1...im+n
j1...jmjm+1...jm+n
= KR
V |W (ζ|η1, . . . , ηm)α1i1...im
α2j1...jmK
L
V |W (ζ|ηm+1, . . . , ηm+n)α2im+1...im+n
α1jm+1...jm+n .
It is clear that the graphical analogue of this definition is the one given in Fig. 4.3.
52 Kh.S. Nirov and A.V. Razumov
KL
V|W
im+n im+1
jm+n jm+1
α1
α2
ζ−1
ζ
ηm+n ηm+1
KL
V|W(ζ|ηm+1, . . . , ηm+n)α1im+n ...im+1
α2 jm+n ...jm+1
Figure 4.1.
KR
V|W
i1im
j1jm
α1
α2
ζ−1
ζ
η1ηm
KR
V|W(ζ|η1, . . . , ηm)α2i1...im
α1 j1...jm
Figure 4.2.
KL
V|W
im+n im+1
jm+n jm+1
ζ−1
ηm+n ηm+1
KR
V|W
i1im
j1jm
ζ
η1ηm
TV|W(ζ|η1 . . . , ηm, ηm+1, . . . , ηm+n)i1...imim+1...im+n j1...jm jm+1...jm+n
Figure 4.3.
ηm+n ηm+1 ηm η1
KL
V1|W
KL
V2|W
KR
V1|W
KR
V2|W
ζ−1
1
ζ−1
2
ζ1
ζ2
Figure 4.4.
4.2 Commutativity of transfer operators
Let us derive sufficient conditions for the commutativity of the transfer operators for the case
under consideration.
The picture representing the product of two transfer operators is given in Fig. 4.4. We
subject it to the following transformations. First, we twist two horizontal lines in the middle of
the picture. One sees that these lines go in the opposite directions. Therefore, we use for our
purpose the graphical equation given in Fig. 2.10. The result is represented in Fig. 4.5. Then
we twist two upper lines of this figure. Now the lines go in the same direction, and we use for
Vertex Models and Spin Chains in Formulas and Pictures 53
∼
KL
V1|W
KL
V2|W
KR
V1|W
KR
V2|W
Figure 4.5.
∼
KL
V1|W
KL
V2|W
KR
V1|W
KR
V2|W
Figure 4.6.
ηm+n ηm+1 ηm η1
KL
V2|W
KL
V1|W
KR
V2|W
KR
V1|W
ζ−1
2
ζ−1
1
ζ2
ζ1
Figure 4.7.
twisting the graphical equation in Fig. 2.6 with the single and double lines interchanged.6 After
two transformations we come to the situation depicted in Fig. 4.6.
Proceed now to the opposite product of the same transfer operators represented graphically
in Fig. 4.7. Using graphical equation in Fig. 2.10 with the single and double lines interchanged,7
6See the comment to Figs. 2.6 and 2.7 on p. 14.
7See again the comment to Figs. 2.6 and 2.7 on p. 14.
54 Kh.S. Nirov and A.V. Razumov
∼
KL
V2|W
KL
V1|W
KR
V2|W
KR
V1|W
Figure 4.8.
KL
V1|W
KL
V2|W
KL
V2|W
KL
V1|W
=
Figure 4.9.
∼
KR
V1|W
KR
V2|W
∼
KR
V2|W
KR
V1|W
=
Figure 4.10.
we twist two horizontal lines in the middle of the picture. Then we use the graphical equation
given in Fig. 2.7 to twist two bottom lines. The result can be seen in Fig. 4.8. Compare Figs. 4.6
and 4.8. If we cut these figures vertically in the middle, then the types and directions of the
lines intersecting the cut will be the same. Therefore, it is consistent to equate the left and right
halves of the figures. Graphically it is represented by Figs. 4.9 and 4.10. It is clear that if the
equations given in these figures are satisfied, then the product of the transfer operators under
consideration does not depend on the order.
Vertex Models and Spin Chains in Formulas and Pictures 55
KL
V|W
Figure 4.11.
KR
V|W
Figure 4.12.
∼
KR
V1|W
KR
V2|W
Figure 4.13.
∼ = ∼
Figure 4.14.
=
Figure 4.15.
The remarkable fact is that if the operators depicted in Figs. 4.1 and 4.2 satisfy the graphical
equations presented in Figs. 4.9 and 4.10, then the ‘dressed’ operators depicted in Figs. 4.11
and 4.12 satisfy these equations as well. Let us demonstrate this first for the case of the opera-
tor KR
V |W . Insert the dressed operators KR
V1|W and KR
V2|W into the left hand side of the graphical
equation in Fig. 4.10. This gives the picture given in Fig. 4.13. Now, using the Yang–Baxter
equations depicted in Figs. 4.14 and 4.15, we move the leftmost wavy line to the left and come to
the situation which can be seen in Fig. 4.16. It should be noted that the Yang–Baxter equations
in Figs. 4.14 and 4.15 can be obtained without assuming the validity of the unitarity relations.
Then we apply the graphical equation in Fig. 4.10 to Fig. 4.16 and obtain Fig. 4.17. Finally,
using the Yang–Baxter equations in Figs. 4.18 and 4.19, we move the leftmost wavy line to the
right and come to the situation which can be seen in Fig. 4.20. The Yang–Baxter equations in
Figs. 4.18 and 4.19 can be again obtained without assuming the validity of the unitarity relations.
Comparing Figs. 4.13 and 4.20, we obtain the desired result. The case of the operator KL
V |W
can be analysed in the same way.
Thus, it is possible to take as the operators KL
V |W and KR
V |W the operators depicted in
Figs. 4.21 and 4.22. One can verify that they can be defined analytically by the equations
56 Kh.S. Nirov and A.V. Razumov
∼
KR
V1|W
KR
V2|W
Figure 4.16.
∼
KR
V2|W
KR
V1|W
Figure 4.17.
=
Figure 4.18.
∼ = ∼
Figure 4.19.
KL
V |W (ζ|ηm+1, . . . , ηm+n)
= MV |W
(
ζ−1|ηm+1, . . . , ηm+n
)−1
KL
V (ζ)MV |W (ζ|ηm+1, . . . , ηm+n), (4.3)
KR
V |W (ζ|η1, . . . , ηm)
=
((
MV |W
(
ζ−1|η1, . . . , ηm
)−1)tVKR
V (ζ)tVMV |W (ζ|η1, . . . , ηm)tV
)tV . (4.4)
The operators KL
V and KR
V act on the auxiliary space V and satisfy the graphical equations given
in Figs. 4.23 and 4.24. It is not difficult to find the analytical expression for these equations,
namely,
PV2|V1RV2|V1
(
ζ−1
2 |ζ−1
1
)
PV1|V2K
L
V1(ζ1)RV1|V2
(
ζ1|ζ−1
2
)
KL
V2(ζ2)
Vertex Models and Spin Chains in Formulas and Pictures 57
∼
KR
V2|W
KR
V1|W
Figure 4.20.
n
KL
V
Figure 4.21.
m
KR
V
Figure 4.22.
KL
V1
KL
V2
KL
V2
KL
V1
=
Figure 4.23.
∼
KR
V1
KR
V2
∼
KR
V2
KR
V1
=
Figure 4.24.
= KL
V2(ζ2)PV2|V1RV2|V1
(
ζ2|ζ−1
1
)
PV1|V2K
L
V1(ζ1)RV1|V2(ζ1|ζ2),
KR
V2(ζ2)R̃V1|V2
(
ζ1|ζ−1
2
)−1
KR
V1(ζ1)PV2|V1RV2|V1
(
ζ−1
2 |ζ−1
1
)−1
PV1|V2
= RV1|V2(ζ1|ζ2)−1KR
V1(ζ1)PV2|V1R̃V2|V1
(
ζ2|ζ−1
1
)−1
PV1|V2K
R
V2(ζ2).
58 Kh.S. Nirov and A.V. Razumov
N=m+n
KL
V KR
V
Figure 4.25.
KL
V KR
V
Figure 4.26.
There are many papers devoted to solving these equations with respect to the operators KL
V
and KR
V for a fixed R-operator, see, for example, paper [51] and references therein. The most
complete set of solutions for the case of the quantum loop algebras Uq(L(sll+1)) was obtained
in [65].
It is clear now that the graphical image of the transfer operator is the one given by Fig. 4.25
whose analytical expression is
TV |W (ζ|η1, . . . , ηN ) = trV
(
KR
V (ζ)MV |W
(
ζ−1|η1, . . . , ηN
)−1
KL
V (ζ)MV |W (ζ|η1, . . . , ηN )
)
.
It is rather tricky to obtain this expression from the definition (4.2) and equations (4.3) and (4.4),
see paper [68]. However, from the graphical point of view it is evident.
4.3 Hamiltonian
As in the case of a periodic chain we assume that the quantum space W coincides with the auxil-
iary space V and RV |V (1|1) coincides with the permutation operator P12. The Hamiltonian HN
for the chain of length N is again constructed from the homogeneous transfer operator
TV (ζ) = TV |V (ζ|1, 1, . . . , 1)
with the help of the equation
HN = ζ
d
dζ
log TV (ζ)
∣∣∣∣
ζ=1
=
dTV (ζ)
dζ
∣∣∣∣
ζ=1
TV (1)−1.
One can find the graphical image of the operator TV (1) in Fig. 4.26. Here and below gray border
means the value of an operator at ζ = 1. Thus to have an invertible operator TV (1), one should
assume that the operator KL
V (1) is invertible. Here one has
TV (1)−1 =
(
KL
V (1)−1
)(N)
/ trKR
V (1).
Vertex Models and Spin Chains in Formulas and Pictures 59
KL
V KR
V
Figure 4.27.
KL
V KR
V
Figure 4.28.
KL
V KR
V
Figure 4.29.
KL
V KR
V
Figure 4.30.
For the meaning of the superscript (N) see Appendix A.1. The graphical form of the various
possible summands which enters the expression for the derivative T ′V (1) are given in Figs. 4.27–
4.34, where, as above, a double line means the derivative at ζ = 1. Note that to draw Figs. 4.29,
4.31 and 4.33 we use the equation
ζ
dŘV |V
(
ζ−1|1
)
dζ
∣∣∣∣∣
ζ=1
= Ř′V |V (1|1).
Using Figs. 4.27–4.34, we come to the following analytical expression for the Hamiltonian
HN = trKR′
V (1)/ trKR
V (1) + 2 tr1′
(
H(1′,1)KR
V (1)(1′))/ trKR
V (1) + 2
N−2∑
i=1
H(i,i+1)
+ H(N−1,N) +KL
V (1)(N)H(N−1,N)
(
KL
V (1)−1
)(N)
+KL′
V (1)(N)
(
KL
V (1)−1
)(N)
,
where H(k,l) is defined by equation (2.75). The order of the terms in the above expression is
opposite to the order of the figures.
5 Conclusions
This paper has been devoted to systematisation and development of the graphical approach
to the investigation of the quantum integrable vertex models of statistical physics and the
60 Kh.S. Nirov and A.V. Razumov
KL
V KR
V
Figure 4.31.
KL
V KR
V
Figure 4.32.
KL
V KR
V
Figure 4.33.
KL
V KR
V
Figure 4.34.
corresponding spin chains. We hope that the usefulness and productivity of this approach was
clearly demonstrated. In fact, we have derived and graphically described much more relations
and equations than it was needed for the considered applications. We will use them in our future
works. The calculation of correlation functions is obviously one of such promising applications
of the graphical approach, as it was already commenced, for example, in papers [2, 22, 66] based
on qKZ equations.
Appendix. Some linear algebra
In this appendix we introduce notation for the operators acting in tensor products of vector
spaces and discuss some their properties, see also Appendix A of paper [75].
A.1 Tensor products and symmetric group
We mean by n-tuple a mapping from the interval [1 . . n] ⊂ N to a set or a class. It is common
for an n-tuple F to use the notation
F (i) = Fi, i ∈ [1 . . n],
and write
F = (F1, . . . , Fn) = (Fi)i∈[1 . . n].
We define the range of an n-tuple as the range of the corresponding mapping.
Vertex Models and Spin Chains in Formulas and Pictures 61
Let A = (Ai)i∈[1 . . n] be an n-tuple of unital associative algebras. Consider the tensor product
A⊗ = A1 ⊗A2 ⊗ · · · ⊗An.
For any element σ of the symmetric group Sn denote
Aσ = ((Aσ)i)i∈[1 . . n] = (Aσ−1(i))i∈[1 . . n],
so that
(Aσ)⊗ = Aσ−1(1) ⊗Aσ−1(2) ⊗ · · · ⊗Aσ−1(n).
It is clear that for any two elements σ, τ ∈ Sn one has
(Aσ)τ = Aστ .
Define also a linear mapping Πσ : A⊗ → (Aσ)⊗ acting on a monomial a1 ⊗ a2 ⊗ · · · ⊗ an in
accordance with the rule
Πσ(a1 ⊗ a2 ⊗ · · · ⊗ an) = aσ−1(1) ⊗ aσ−1(2) ⊗ · · · ⊗ aσ−1(n).
It is not difficult to demonstrate that for any σ, τ ∈ Sn one obtains
Πσ ◦Πτ = Πστ .
It is evident that Πσ is an isomorphic mapping from the algebra A⊗ to the algebra (Aσ)⊗.
Now, let B be one more unital associative algebra, and b ∈ B. If for some i ∈ [1 . . n] we have
B = Ai, then we denote by b(i) the element of A⊗ defined as
b(i) = 1⊗ · · · ⊗ 1
i−1
⊗ b⊗ 1⊗ · · · ⊗ 1
n−i
.
One can easily get convinced that
Πσ
(
b(i)
)
= b(σ(i))
for any σ ∈ Sn.
More generally, given 0 < k ≤ n, let B = (Bi)i∈[1 . . k] be a k-tuple of unital associative
algebras. Further, let i = (i1, i2, . . . , ik) be a k-tuple of distinct positive integers from the
interval [1 . . n]. If Bl = Ail for all l ∈ [1 . . k] and b = b1 ⊗ b2 ⊗ · · · ⊗ bk is an element of the
algebra
B⊗ = B1 ⊗B2 ⊗ · · · ⊗Bk,
we denote by bi the element of the algebra A defined as
bi = b(i1,i2,...,ik) = b
(i1)
1 b
(i2)
2 · · · b(ik)
k .
We extend this rule to all elements of B by linearity. Further, given σ ∈ Sn, we denote
σi = (σ(i1), σ(i2), . . . , σ(ik)),
so that
στ i = σ(τ i)
62 Kh.S. Nirov and A.V. Razumov
for any σ, τ ∈ Sn. Here one has
Πσ(bi) = b σi,
or, more explicitly,
Πσ
(
b(i1,i2,...,ik)
)
= b(σ(i1),σ(i2),...,σ(ik)).
Sometimes, if it does not lead to a misunderstanding, one simplifies the notation b(i1,i2,...,ik) to
bi1,i2,...,ik or even to bi1i2...ik .
Now, let W = (Wi)i∈[1 . . n] be an n-tuple of vector spaces, and
A = (Ai)i∈[1 . . n] = (End(Wi))i∈[1 . . n]
an n-tuple of unital associative algebras. Similarly as above, we denote
W⊗ = W1 ⊗W2 ⊗ · · · ⊗Wn, (A.1)
and
A⊗ = End(W1)⊗ End(W2)⊗ · · · ⊗ End(Wn) ∼= End
(
W⊗
)
.
Given σ ∈ Sn, define
(
W⊗
)
σ
= Wσ−1(1) ⊗Wσ−1(2) ⊗ · · · ⊗Wσ−1(n),
so that for any two elements σ, τ ∈ Sn one has
(Wσ)τ = Wστ .
Now, define a linear mapping Pσ : W⊗ → (W⊗)σ by the equation
Pσ(w1 ⊗ w2 ⊗ · · · ⊗ wn) = wσ−1(1) ⊗ wσ−1(2) ⊗ · · · ⊗ wσ−1(n).
Here for any σ, τ ∈ Sn one has
Pσ ◦ Pτ = Pστ .
Let M ∈ End(V ), and V = Wi for some i ∈ [1 . . n]. It means that End(V ) = End(Wi) and
one can define M i ∈ End(W ). One can show that
Pσ ◦M i = Mσ(i) ◦ Pσ
for any σ ∈ Sn. It follows from this equation that
Πσ
(
M i
)
= Pσ ◦M i ◦ (Pσ)−1.
More generally, given k ≤ n, let W = (Wi)i∈[1 . . k] be a k-tuple of vector spaces, and i =
(i1, i2, . . . , ik) a k-tuple of distinct positive integers from the interval [1 . . n]. If M ∈ End
(
W⊗
)
,
and Wl = Vil for all l ∈ [1 . . k], one can define M i ∈ End
(
V ⊗
)
. Here for any σ ∈ Sn one has
Pσ ◦M i = Mσi ◦ Pσ
and
Πσ
(
M i
)
= Pσ ◦M i ◦ Pσ−1 ,
or, more explicitly,
Pσ ◦M i1i2...ik = Mσ(i1)σ(i2)...σ(ik) ◦ Pσ
and
Πσ
(
M i1i2...ik
)
= Pσ ◦M i1i2...ik ◦ (Pσ)−1.
If σ ∈ Sn is a transposition (ij) one writes Πij and Pij instead of Πσ and Pσ respectively.
Furthermore, if n = 2 one denotes Π = Π12 and P = P12.
If vector spaces V and U belong to the range of W and there is only one i and one j such
that V = Wi and U = Wj , we also write PV |U instead of Pij .
Vertex Models and Spin Chains in Formulas and Pictures 63
A.2 Partial transpose
Let again W⊗ be defined as in (A.1). For any monomial
M = M1 ⊗M2 ⊗ · · · ⊗Mn, (A.2)
where Mi ∈ End(Wi), i ∈ [1 . . n], we define the partial transpose M tl of M with respect to Wl
by the equation
M tl = M1 ⊗ · · · ⊗Ml−1 ⊗ (Ml)
t ⊗Ml+1 ⊗ · · · ⊗Mn,
and extend this definition to all M ∈ End
(
W⊗
)
by linearity. By definition, for any M ∈
End
(
W⊗
)
the partial transpose M tl is an element of the space8
End(W1)⊗ · · · ⊗ End(Wl−1)⊗ End((Wl)
?)⊗ End(Wl+1)⊗ · · · ⊗ End(Wn)
∼= End(W1)⊗ · · · ⊗ End(Wl−1)⊗ (End(Wl))
? ⊗ End(Wl+1)⊗ · · · ⊗ End(Wn).
If a vector space V belongs to the range of W and there is only one i such that V = Wi we also
write W tV instead of W ti .
It is evident that
(
M tl
)tl = M
and
(
M tl
)tm =
(
M tm
)tl
for any distinct l and m. The relation of the partial transposes to the usual transpose is described
by the equation
(
. . .
(
M t1
)t2 . . .
)tn = M t.
Given 0 < k1, k2 ≤ n, let
V1 = ((V1)1, (V1)2, . . . , (V1)k1)
and
V2 = ((V2)1, (V2)2, . . . , (V2)k2)
be a k1-tuple and a k2-tuple of vector spaces, i1 = ((i1)1, (i1)2, . . . , (i1)k1) and i2 = ((i2)1, (i2)2,
. . . , (i2)k2) be a k1-tuple and a k2-tuple of distinct positive integers from the interval [1 . . n].
Let M1 ∈ End
(
(V1)⊗
)
and M2 ∈ End
(
(V2)⊗
)
. Assume that V1l = W(i1)l for all l ∈ [1 . . k1] and
V2l = W(i2)l for all l ∈ [1 . . k2], and define the corresponding operators (M1)i1 ∈ End
(
W⊗
)
and
(M2)i2 ∈ End
(
W⊗
)
.
Now, if range i1 ∩ range i2 = {l} we have
(
(M1)i1(M2)i2
)tl =
(
(M2)i2
)tl((M1)i1
)tl . (A.3)
Furthermore,
(
(M1)i1(M2)i2
)tl = (M1)i1
(
(M2)i2
)tl (A.4)
if l does not belong to the range of i1, and
(
(M1)i1(M2)i2
)tl =
(
(M1)i1
)tl(M2)i2 (A.5)
if l does not belong to the range of i2.
8We assume that the vector spaces under consideration are Uq(L(g))-modules in the category O and denote
by V ? the restricted dual of V , see Section 2.2.4.
64 Kh.S. Nirov and A.V. Razumov
A.3 Partial trace
Let again W = (Wi)i∈[1 . . n] be an n-tuple of vector spaces. Given l ∈ [1 . . n], denote by V the
(n− 1)-tuple defined as
Vi = Wi, i ∈ [1 . . l − 1], Vi = Wi+1, i ∈ [l . . n− 1].
We define the partial trace trl with respect to Wl as the mapping from End
(
W⊗
)
to End
(
V ⊗
)
in the following evident way. Let M be a monomial of the form (A.2). We define
trlM = trMl(M1 ⊗ · · · ⊗Ml−1 ⊗Ml+1 ⊗ · · · ⊗Mn),
and extend this definition to the case of an arbitrary M ∈ End
(
V ⊗
)
by linearity. If a vector
space V belongs to the range of W and there is only one i such that V = Wi we also write
trV W instead of triW .
Let N be an element of V and V = Wi, then for any M ∈ End(W ) one has
tri
(
MN i
)
= tr
(
N iM
)
.
One can demonstrate that9
trl trk = trk trl+1
for l ≥ k, and
trl trk = trk−1 trl
for l < k.
Let V1, V2 be two vector spaces, and M1 ∈ End
(
V1 ⊗W⊗
)
, M2 ∈ End
(
V2 ⊗W⊗
)
. Define
two (n+ 1)-tuples of vector spaces, W1 defined by the rules
(W1)1 = V1, (W1)i = Wi−1, i ∈ [2 . . n+ 1],
and W2 defined by the rules
(W2)1 = V2, (W2)i = Wi−1, i ∈ [2 . . n+ 1].
It is clear that M1 ∈ End
(
(W1)⊗
)
and M2 ∈ End
(
(W2)⊗
)
. Further, let W̃ be an (n+ 2)-tuple
of vector spaces given by the equations
W̃1 = V1, W̃2 = V2, W̃i = Wi+2, i ∈ [3 . . n+ 1].
Consider two elements of End
(
W̃⊗
)
defined as
M̃1 = M1,3,...,n+2
1 , M̃2 = M2,3,...,n+2
2 .
One can see that
tr1 tr2
(
M̃1M̃2
)
= (tr1M1)(tr1M2).
Finally, we give some examples of interplay between partial traces and partial transposes.
First of all, one has
tri
(
M tiN ti
)
= tri(MN)
for any M,N ∈ End
(
W⊗
)
. Further,
triM
tj = (triM)tj−1
for all M ∈W⊗ and i < j, and
triM
tj = (triM)tj
for all M ∈W⊗ and i > j.
9We draw the reader’s attention to the partial shift in the numbering of the factors of the tensor product after
taking the trace.
Vertex Models and Spin Chains in Formulas and Pictures 65
Acknowledgments
This work was supported in part by the Russian Foundation for Basic Research grant # 16-01-
00473. KhSN was also supported by the DFG grant # BO3401/31 and by the Russian Academic
Excellence Project ‘5-100’; results obtained in Section 3 were funded by the HSE Faculty of
Mathematics. We thank our colleagues and coauthors H. Boos, F. Göhmann and A. Klümper
for numerous fruitful discussions. AVR thanks the Max Plank Institute for Mathematics in
Bonn, where this work was finished, for the warm hospitality.
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1 Introduction
2 Quantum loop algebras and integrability objects
2.1 Quantum loop algebras
2.1.1 Some information on loop algebras
2.1.2 Definition of a quantum loop algebra
2.1.3 Poincaré–Birkhoff–Witt basis
2.1.4 Universal R-matrix
2.1.5 Modules and representations
2.1.6 Spectral parameter
2.2 Integrability objects and their graphical representations
2.2.1 Introductory words
2.2.2 R-operators
2.2.3 Unitarity relations
2.2.4 Crossing relations
2.2.5 Double duals
2.2.6 Yang–Baxter equation
2.2.7 Monodromy operators
2.2.8 Transfer operators and Hamiltonians
3 Integrability objects for the case of quantum loop algebra Uq(L(sll + 1))
3.1 Quantum group Uq(gll + 1) and some its representations
3.1.1 Definition
3.1.2 Representation
3.1.3 Representation
3.2 Representations of Uq(L(sll + 1))
3.2.1 Jimbo's homomorphism
3.2.2 Representation
3.2.3 Representation
3.2.4 Representations * and *
3.3 Integrability objects
3.3.1 Poincaré–Birkhoff–Witt basis
3.3.2 Monodromy operators
3.3.3 Explicit form of R-operator
3.3.4 Crossing and unitarity relations
3.3.5 Hamiltonian
3.3.6 Case of Uq(L(sl2))
4 Graphical description of open chains
4.1 Transfer operator
4.2 Commutativity of transfer operators
4.3 Hamiltonian
5 Conclusions
A.1 Tensor products and symmetric group
A.2 Partial transpose
A.3 Partial trace
References
|
| id | nasplib_isofts_kiev_ua-123456789-210227 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:50Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Nirov, K.S. Razumov, A.V. 2025-12-04T13:02:47Z 2019 Vertex Models and Spin Chains in Formulas and Pictures / K.S. Nirov, A.V. Razumov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 77 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 17B80; 16T05; 16T25 arXiv: 1811.09401 https://nasplib.isofts.kiev.ua/handle/123456789/210227 https://doi.org/10.3842/SIGMA.2019.068 We systematise and develop a graphical approach to the investigations of quantum integrable vertex statistical models and the corresponding quantum spin chains. The graphical forms of the unitarity and various crossing relations are introduced. Their explicit analytical forms for the case of integrable systems associated with the quantum loop algebra Uq(L(slₗ₊₁)) are given. The commutativity conditions for the transfer operators of lattices with a boundary are derived by the graphical method. Our consideration reveals useful advantages of the graphical approach for certain problems in the theory of quantum integrable systems. This work was supported in part by the Russian Foundation for Basic Research grant # 16-01-00473. KhSN was also supported by the DFG grant # BO3401/31 and by the Russian Academic Excellence Project ‘5-100’; results obtained in Section 3 were funded by the HSE Faculty of Mathematics. We thank our colleagues and coauthors H. Boos, F. Göhmann, and A. Klümper for numerous fruitful discussions. AVR thanks the Max Planck Institute for Mathematics in Bonn, where this work was finished, for the warm hospitality. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Vertex Models and Spin Chains in Formulas and Pictures Article published earlier |
| spellingShingle | Vertex Models and Spin Chains in Formulas and Pictures Nirov, K.S. Razumov, A.V. |
| title | Vertex Models and Spin Chains in Formulas and Pictures |
| title_full | Vertex Models and Spin Chains in Formulas and Pictures |
| title_fullStr | Vertex Models and Spin Chains in Formulas and Pictures |
| title_full_unstemmed | Vertex Models and Spin Chains in Formulas and Pictures |
| title_short | Vertex Models and Spin Chains in Formulas and Pictures |
| title_sort | vertex models and spin chains in formulas and pictures |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210227 |
| work_keys_str_mv | AT nirovks vertexmodelsandspinchainsinformulasandpictures AT razumovav vertexmodelsandspinchainsinformulasandpictures |