Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems
The main aim of this work is to develop a method of constructing higher Hamiltonians of quantum integrable systems associated with the solution of the Zamolodchikov tetrahedral equation. As opposed to the result of V.V. Bazhanov and S.M. Sergeev the approach presented here is effective for generic s...
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irk-123456789-1486252019-02-19T01:29:20Z Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems Talalaev, D.V. The main aim of this work is to develop a method of constructing higher Hamiltonians of quantum integrable systems associated with the solution of the Zamolodchikov tetrahedral equation. As opposed to the result of V.V. Bazhanov and S.M. Sergeev the approach presented here is effective for generic solutions of the tetrahedral equation without spectral parameter. In a sense, this result is a two-dimensional generalization of the method by J.-M. Maillet. The work is a part of the project relating the tetrahedral equation with the quasi-invariants of 2-knots. 2017 Article Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems / D.V. Talalaev // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 14 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16T25 DOI:10.3842/SIGMA.2017.031 http://dspace.nbuv.gov.ua/handle/123456789/148625 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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The main aim of this work is to develop a method of constructing higher Hamiltonians of quantum integrable systems associated with the solution of the Zamolodchikov tetrahedral equation. As opposed to the result of V.V. Bazhanov and S.M. Sergeev the approach presented here is effective for generic solutions of the tetrahedral equation without spectral parameter. In a sense, this result is a two-dimensional generalization of the method by J.-M. Maillet. The work is a part of the project relating the tetrahedral equation with the quasi-invariants of 2-knots. |
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Talalaev, D.V. |
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Talalaev, D.V. Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems Symmetry, Integrability and Geometry: Methods and Applications |
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Talalaev, D.V. |
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Talalaev, D.V. |
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Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems |
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Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems |
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Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems |
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Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems |
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Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems |
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zamolodchikov tetrahedral equation and higher hamiltonians of 2d quantum integrable systems |
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Інститут математики НАН України |
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Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems / D.V. Talalaev // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 14 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT talalaevdv zamolodchikovtetrahedralequationandhigherhamiltoniansof2dquantumintegrablesystems |
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2025-07-12T19:48:42Z |
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1837471870507548672 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 031, 14 pages
Zamolodchikov Tetrahedral Equation and Higher
Hamiltonians of 2d Quantum Integrable Systems
Dmitry V. TALALAEV
Geometry and Topology Department, Faculty of Mechanics and Mathematics,
Moscow State University, Moscow, 119991 Russia
E-mail: dtalalaev@yandex.ru
URL: http://higeom.math.msu.su/people/talalaev/english.htm
Received January 17, 2017, in final form May 13, 2017; Published online May 22, 2017
https://doi.org/10.3842/SIGMA.2017.031
Abstract. The main aim of this work is to develop a method of constructing higher Hamil-
tonians of quantum integrable systems associated with the solution of the Zamolodchikov
tetrahedral equation. As opposed to the result of V.V. Bazhanov and S.M. Sergeev the ap-
proach presented here is effective for generic solutions of the tetrahedral equation without
spectral parameter. In a sense, this result is a two-dimensional generalization of the method
by J.-M. Maillet. The work is a part of the project relating the tetrahedral equation with
the quasi-invariants of 2-knots.
Key words: Zamolodchikov tetrahedral equation; quantum integrable systems; star-triangle
transformation
2010 Mathematics Subject Classification: 16T25
1 Introduction
1.1 Yang–Baxter equation and its generalizations
This work is mainly focusing on the matrix Yang–Baxter equation
R12R13R23 = R23R13R12 ∈ End
(
V ⊗3
)
, R ∈ End
(
V ⊗2
)
, (1.1)
and the matrix tetrahedral Zamolodchikov equation [13]
Φ123Φ145Φ246Φ356 = Φ356Φ246Φ145Φ123 ∈ End
(
V ⊗6
)
, Φ ∈ End
(
V ⊗3
)
. (1.2)
In both cases V is a finite-dimensional vector space, the indices denote the numbers of the
space copies in which linear operators act non-trivially. We should also mention the universal
description of the n-simplex equation (see, e.g., [6, 12]), generalizing the Yang–Baxter and the
Zamolodchikov equations.
The work is aimed to generalize one of the existing applications of the theory of the Yang–
Baxter equation to the theory of quantum integrable systems. It should be noted that this
equation and subsequently the theory of quantum groups marked a new era in the field of exactly-
solvable models of mathematical physics. This concerned both models of statistical physics in
dimension 2, and one-dimensional quantum mechanical models of the theory of magnets. The
established connection turned out to be effective in the purely physical issues like the effect
of the spontaneous magnetization as like as in many mathematical problems. The language of
Hopf algebras not only expanded the machine of modern algebra but also allowed to explore
new patterns, for example, in topology.
This paper is a contribution to the Special Issue on Recent Advances in Quantum Integrable Systems. The
full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2016.html
mailto:dtalalaev@yandex.ru
http://higeom.math.msu.su/people/talalaev/english.htm
https://doi.org/10.3842/SIGMA.2017.031
http://www.emis.de/journals/SIGMA/RAQIS2016.html
2 D.V. Talalaev
The mentioning of such diverse areas of modern mathematical physics here is not accidental.
The transition from the Yang–Baxter equations to the tetrahedral equation appears natural
from many points of view: low-dimensional topology, combinatorics, statistical models, topolo-
gical field theories, homotopy algebraic structures. Here we present a brief table showing some
heredity in subjects related to the Yang–Baxter equation and the tetrahedral one.
Table 1. Relations.
Yang–Baxter equation Zamolodchikov equation
statistical models d = 2 d = 3
spin chains d = 1 d = 2
homotopy Lie algebras Lie algebras 2-Lie algebras (e.g., thesis by
A.S. Crans [4])
topological invariants Turaev–Reshetikhin-type knot
invariants
2-knot quasi-invariants in [9, 10]
Hopf algebras quasi-triangular Hopf algebras ?
1.2 Universal integrability in d = 1
Besides the equation (1.1)
R12R13R23 = R23R13R12
we consider the structural equations in the combinatorial form
R′12R
′
23R
′
12 = R′23R
′
12R
′
23, (1.3)
R′L⊗ L = L⊗ LR′, (1.4)
where R′ ∈ End(V )⊗2 and L ∈ End(V ) ⊗ End(Vq), here Vq is the quantum vector space (i.e.,
one another vector space, distinguished from V .) The transition from (1.1) to (1.3) can be
realized as follows: if R satisfies (1.1) then R′12 = R12P12 as like as R′′12 = P12R12 satisfy (1.3),
here P12 is the transposition linear operator in V ⊗V . The equation (1.4) is traditionally called
the RLL-relation, it plays a substantial role in Faddeev–Reshetikhin–Takhtadzhyan algebras [5].
A solution example of (1.4) is provided by
L = R1i1R1i2 · · ·R1ik ,
where L is an operator in the quantum space Vq = Vi1 ⊗ · · · ⊗ Vik , here Vij is just a copy of the
space V .
The work [11] presents a construction of a commutative family in the algebra End(Vq) con-
taining the trace I1 = TrV L.
Lemma 1.1 ([11]). Let us introduce a notation Li for the corresponding element in End(Vi)⊗
End(Vq). Then the operators
Ik = Tr1···k L1 · · ·LkR′12R
′
23 · · ·R′k−1,k (1.5)
commute in End(Vq). The trace is meant with respect to the auxiliary spaces Vi.
This statement has an important role in the technique of constructing quantum-mechanical
integrable systems. This is applicable in quantum Gaudin systems, Ruijenaars–Schneider system
and others. Moreover, this is directly associated with the theory of exactly-solvable models of
statistical physics on 2-dimensional lattices. The study of the spectrum of the transfer-matrix is
a key ingredient in the problem of finding the partition function asymptotics of some statistical
models (e.g., [1]).
Tetrahedral Equation and Higher Hamiltonians of Quantum Integrable Systems 3
1.3 Tetrahedral equation
As like as in the Yang–Baxter case for many purposes it is more convenient to consider the
set-theoretic tetrahedral equation (STTE) which may be defined as follows: let X be a finite set,
we say that there is a solution for STTE on X if there is a map
X ×X ×X Φ−→ X ×X ×X,
satisfying the relation (graphically coinciding with (1.2))
Φ123 ◦ Φ145 ◦ Φ246 ◦ Φ356 = Φ356 ◦ Φ246 ◦ Φ145 ◦ Φ123 : X×6 → X×6.
Here, however, unlike (1.2), X×6 denotes the Cartesian 6-th power of X and the subscripts
denote the number of factors to which Φ is applied, in other factors the map acts identically.
For example
Φ356(a1, a2, a3, a4, a5, a6) = (a1, a2,Φ1(a3, a5, a6), a4,Φ2(a3, a5, a6),Φ3(a3, a5, a6))
= (a1, a2, a
′
3, a4, a
′
5, a
′
6),
where
Φ(x, y, z) = (Φ1(x, y, z),Φ2(x, y, z),Φ3(x, y, z)) = (x′, y′, z′).
One distinguishes the functional tetrahedral equation (FTE), for example the so-called electric
solution [7, 8], represented as a transformation acting on the space of functions of three variables
Φ(x, y, z) = (x1, y1, z1),
x1 =
xy
x+ z + xyz
, y1 = x+ z + xyz, z1 =
yz
x+ z + xyz
.
This solution is relevant to the well-known star-triangle relation in the theory of electric circuits.
There is another one interpretation of the tetrahedral equation in the task of coloring of
2-faces of the 4-dimensional cube with elements of the set X. Let Φ: X ×X ×X → X ×X ×X
be a map. A coloring is called admissible if the colors of the incoming faces of all 3-cubes x, y, z
are related with the colors of outgoing faces x′, y′, z′ by the action of the map
Φ: (x′, y′, z′) = Φ(x, y, z).
At Fig. 1 we design a projection of the 4-cube to a 3-space and at Fig. 2 – two alternative
sequences of coloring steps. It turns out that the condition of equivalence of the coloring obtained
by these two ways is equivalent to the tetrahedral equation on Φ. The problem is described in
more details in [9].
The N -cube 2-faces coloring problem allows us to construct a complex analogous to those
calculating the Yang–Baxter cohomology for the case of set-theoretic tetrahedral equations in [9].
The 3-cocycles of the complex play a special role in this subject, they are determined by the
condition
ϕ(a1, a2, a3)ϕ(a′1, a4, a5)ϕ(a′2, a
′
4, a6)ϕ(a′3, a
′
5, a
′
6)
= ϕ(a3, a5, a6)ϕ(a2, a4, a
′
6)ϕ(a1, a
′
4, a
′
5)ϕ(a′1, a
′
2, a
′
3)
in the notation of Fig. 2. In particular the following lemma fulfills:
4 D.V. Talalaev
Figure 1. Tesseract.
Figure 2. Coloring.
Tetrahedral Equation and Higher Hamiltonians of Quantum Integrable Systems 5
Lemma 1.2 ([9, Theorem 4]). Let Φ be a solution for the STTE, ϕ be a 3-cocycle of the
tetrahedral complex. Let V be the vector space generated by the elements of the set X. It has
basis {ex : x ∈ X} marked by elements of X. Then let us define a linear operator A on V ⊗3
specifying its values on tensor products of basis vectors. We say that
A(s)(ex ⊗ ey ⊗ ez) = ϕ(x, y, z)s(ex′ ⊗ ey′ ⊗ ez′)
if and only if Φ(x, y, z) = (x′, y′, z′). In this case A(s) provides a solution for the matrix
tetrahedral equation.
Remark 1.3. To construct such matrix solution from the electric solution for the FTE one
should realize the finite-dimensional reduction of the latter. In [9, Lemma 6] it is demonstrated
that the electric solution reduces correctly to the set X obtained as follows: consider the residue
ring Z/pkZ, where p is either a prime number of the form p = 4l + 1, or p = 2, and k is an
integer ≥ 2 (the Legendre symbol
(
−1
p
)
equals 1). We fix one square root of −1 and call it
ε ∈ Z/pZ. Our set X will be the following subset of Z/pkZ
X =
{
x ∈ Z/pkZ : x = ε mod p
}
.
2 Commutative family
2.1 The commutativity demonstration in d = 1
Let us present here our own proof of the Maillet result in Lemma 1.1. It demonstrates the
main technique and suggests the path for generalization. In this section we denote by R and L
solutions for the equations (1.3) and (1.4) (do not confuse with equation (1.1)). We assume that
R is invertible. We present here the Maillet generators (1.5) in a slightly more algebraic way
Ik = TrV1⊗···⊗Vk L
⊗kR12 · · ·Rk−1,k,
where the tensor product is taken with respect to the auxiliary spaces Vi, the product in quantum
space Vq is implied. To prove the commutativity of Ik we demonstrate that there exists a linear
operator A ∈ End(V )⊗k+l such that
AL⊗k+lR12 · · ·Rk−1,kRk+1,k+2 · · ·Rk+l−1,k+lA
−1
= L⊗k+lR12 · · ·Rl−1,lRl+1,l+2 · · ·Rk+l−1,k+l.
This yields that the traces of these expressions with respect to the auxiliary spaces coincide.
This in turn produce the identity
[Ik, Il] = 0.
In what follows AdgX means the group adjoint action gXg−1. Let us introduce some accessory
notations: Rlm = Rl,l+1Rl+1,l+2 · · ·Rm−1,m for l < m. This expression is subject to some
relations
Lemma 2.1.
AdR1k
Rm,m+1 = Rm+1,m+2, 1 ≤ m ≤ k − 2.
Proof.
R12 · · ·Rm−1,mRm,m+1Rm+1,m+2 · · ·Rk−1,kRm,m+1R
−1
k−1,k · · ·
6 D.V. Talalaev
×R−1
m+1,m+2R
−1
m,m+1R
−1
m−1,m · · ·R
−1
12
= R12 · · ·Rm−1,mRm,m+1Rm+1,m+2Rm,m+1R
−1
m+1,m+2R
−1
m,m+1R
−1
m−1,m · · ·R
−1
12
= R12 · · ·Rm−1,mRm,m+1R
−1
m,m+1Rm+1,m+2Rm,m+1R
−1
m,m+1R
−1
m−1,m · · ·R
−1
12
= R12 · · ·Rm−1,mRm+1,m+2R
−1
m−1,m · · ·R
−1
12 = Rm+1,m+2. �
Another relation is expressed by the
Lemma 2.2.
AdR2k
AdR1,k−1
Rk−1,k = R12.
Proof. We demonstrate the statement by induction. For k = 3 we have
R23R12R23R
−1
12 R
−1
23 = R23R
−1
23 R12R23R
−1
23 = R12.
Let the statement be true for k − 1, then
AdR2k
AdR1,k−1
Rk−1,k = R23 · · ·Rk−1,kR12 · · ·Rk−3,k−2Rk−2,k−1Rk−1,kR
−1
k−2,k−1
×R−1
k−3,k−2 · · ·R
−1
12 R
−1
k−1,k · · ·R
−1
23
= R23 · · ·Rk−1,kR12 · · ·Rk−3,k−2R
−1
k−1,kRk−2,k−1Rk−1,kR
−1
k−3,k−2 · · ·R
−1
12 R
−1
k−1,k · · ·R
−1
23
= R23 · · ·Rk−2,k−1R12 · · ·Rk−3,k−2Rk−2,k−1R
−1
k−3,k−2 · · ·R
−1
12 R
−1
k−2,k−1 · · ·R
−1
23
= AdR2,k−1
AdR1,k−2
Rk−2,k−1 = R12. �
We may now fabricate an operator A by the formula
A = Rl,k+lRl−1,k+l−1 · · ·R1,k+1.
Lemma 2.3.
AdA(L⊗k+lR12 · · ·Rk−1,kRk+1,k+2 · · ·Rk+l−1,k+l)
= L⊗k+lR12 · · ·Rl−1,lRl+1,l+2 · · ·Rk+l−1,k+l.
Proof. Let us note that in virtue of (1.4) we need to demonstrate only
AdA(R12 · · ·Rk−1,kRk+1,k+2 · · ·Rk+l−1,k+l) = R12 · · ·Rl−1,lRl+1,l+2 · · ·Rk+l−1,k+l.
On the other hand Lemma 2.1 yields
AdA(R12 · · ·Rk−1,k) = Rl+1,l+2 · · ·Rk+l−1,k+l. (2.1)
Lemma 2.2 provides
AdA(Rk+1,k+2 · · ·Rk+l−1,k+l) = R12R23 · · ·Rl−1,l. (2.2)
The commutativity argument of the right hand sides of (2.1) and (2.2) finishes the proof. �
This reasoning can be generalized for generating functions of integrals. Let us introduce a
notation
S(t) = L⊗N
(
1 + tR12 + t2R12R23 + · · ·+ tN−1R12 · · ·RN−1,N
)
.
Tetrahedral Equation and Higher Hamiltonians of Quantum Integrable Systems 7
Then
Q(t) = TrS(t) =
N−1∑
k=0
tkIkI
N−k
1 .
If one now considers the conjugation operator
A = RN,2NRN−1,2N−1 · · ·R1,N+1,
then
AdA(S(t)⊗ S(u)) = S(u)⊗ S(t).
This immediately implies
[Q(t), Q(u)] = 0.
2.2 Regular 3d lattices and statistical models
Consider a periodic three-dimensional lattice of size K × L ×M , we mark the edges incoming
to the node (i, j, k) as xi,j,k, yi,j,k, zi,j,k. The periodicity conditions imply ∗K+1,j,k = ∗1,j,k, and
similar identities for other indexes. Consider a statistical model with the Boltzmann weights
in the nodes of the lattice sites determined by the value of the 3-cocycle ϕ of the tetrahedral
complex. The states are defined as admissible colorings of the edges, i.e., such that in each node
the condition fulfills
Φ(xi,j,k, yi,j,k, zi,j,k) = (xi+1,j,k, yi,j+1,k, zi,j,k+1).
A partition function is defined as follows
Z(s) =
∑
col
∏
i,j,k
ϕ(xi,j,k, yi,j,k, zi,j,k)
s.
To explore the “integrability” of the subsidiary quantum problem one needs to recognize the
layer-to-layer transfer-matrix. In order to determine what it is, we need another interpretation
of the partition function.
We associate a copy of the space V to each line of the lattice. For convenience, we denote
the vertical spaces by characters Vik and the horizontal ones – by Ei and Nk. We construct
an operator Aik(s) acting in Ei ⊗ Vik ⊗ Nk with the chosen 3-cocycle according to Lemma 1.2
pattern. Let us define the transfer-matrix by a 1-layer product
T (s) = Tr
∏
−→
i
∏
−→
k
Aik(s) = Tr(B1(s) · · ·BK(s)),
where
Bi(s) = Ai1(s) · · ·AiM (s).
The formula implies the trace of matrices with respect to horizontal spaces. This operator acts
on the tensor product of vertical spaces. It turns out that the partition function takes the form
Z(s) = TrVjk T (s)L.
Such issues as the asymptotic behavior of partition functions with respect to increasing the
size of the lattice may be solved by the study of the spectrum of the transfer-matrix. The inte-
grability condition, i.e., the possibility of including the transfer-matrix in a large commutative
family, simplifies the problem of finding the spectrum.
8 D.V. Talalaev
Figure 3. 1-layer configuration.
2.3 Some consequences of the tetrahedral equation
In this section we give a generalization of the Maillet construction in the case of the three-
dimensional lattice and the transfer-matrix associated with the solution of the matrix tetrahedral
equation Φ. Consider a lattice of size K × L×M and several forms of 1-layer product
Φ(i)∗(j) = Φ(i1...iK)∗(j1...jM ) =
t∈
−−−−−→
(1,...,M)∏
s∈
−−−−−→
(1,...,K)
Φislst jt
=
∏
s∈
−−−−−→
(1,...,K)
Φis(ls1...l
s
M )(j1...jM ) =
∏
s∈
−−−−−→
(1,...,K)
Φis(ls)(j). (2.3)
The arrows over indexing sets mean the direction in products: for example∏
s∈
−−−−−→
(1,...,K)
As = A1 · · ·AK .
In the last two parts of (2.3) we use notations
Φk(i)(j) = Φk(i1...it)(j1...jt) =
∏
s∈
−−−−→
(1,...,t)
Φkisjs ,
Φ(i)(j)k = Φ(i1...it)(j1...jt)k =
∏
s∈
−−−−→
(1,...,t)
Φisjsk,
where different letters in indices correspond to different copies of the space V .
Remark 2.4. Let us make some comments on the indices rule. Here (i) is a multiindex
(i1, . . . , is). In what follows we use sequences of multiindices, we denote by superscript the
number of a multiindex, for example by (il) we denote the multiindex (il1, . . . , i
l
s) remarking
each time the dimension s. The dimension depends on the lattice parameters K, L. The con-
struction at all does, but for the readability reasons we omit this parameters in the definition
of the commutative family and point out the values for each instance explicitly.
The transfer-matrix can be represented as the trace of the expression (2.3)
T = I1 = Tr(i)(j) Φ(i)∗(j),
here (i) is a K-multiindex, (j) is an M -one. Let us write down several identities which are
straightforward consequences of the tetrahedral equation. They can be considered as generali-
zations of the RLL-relations.
Tetrahedral Equation and Higher Hamiltonians of Quantum Integrable Systems 9
Lemma 2.5.
Φ123Φ1(i)(j)Φ2(i)(l)Φ3(j)(l) = Φ3(j)(l)Φ2(i)(l)Φ1(i)(j)Φ123, (2.4)
Φ(i)(j)1Φ(i)(l)2Φ(j)(l)3Φ123 = Φ123Φ(j)(l)3Φ(i)(l)2Φ(i)(j)1, (2.5)
Φ(i)(i′)0Φ(i)∗(j)Φ(i′)∗(j′)Φ0(j)(j′) = Φ0(j)(j′)Φ(i′)∗(j′)Φ(i)∗(j)Φ(i)(i′)0, (2.6)
for s-multiindices (i), (j), (l) in (2.4), (2.5) and K-multiindices (i), (i′) and M -multiindi-
ces (j), (j′) in (2.6).
The proof is placed in Appendix A.
We introduce also some twisted versions of solutions for the tetrahedral equation
ΦL
123 = P12Φ123, Φ̃L
123 = P23Φ123, ΦR
123 = Φ123P23, Φ̃R
123 = Φ123P12.
The identities fulfill
ΦL
123Φ145Φ246ΦR
356 = ΦR
356Φ145Φ246ΦL
123, (2.7)
Φ̃L
αβγΦR
α12Φβ23Φγ13 = Φβ23Φγ13ΦR
α12Φ̃L
αβγ , (2.8)
ΦR
145ΦR
123ΦL
356ΦL
246 = ΦL
356ΦL
246ΦR
145ΦR
123, (2.9)
Φ̃L
123Φ̃L
145Φ̃R
246Φ̃R
356 = Φ̃R
246Φ̃R
356Φ̃L
123Φ̃L
145. (2.10)
They perform a role similar to the Yang–Baxter equation in combinatorial notation. These
equalities are also simple consequences of the tetrahedral equation, we do not provide here the
proofs with the only purpose to simplify the exposition.
2.4 Two families
In this section we always consider the slice of the regular cubic lattice with K ×L×M vertices.
The 2-dimensional layer is defined by fixing the 2-nd coordinate and has the dimension K×M .
The main constructing block of what follows is Φ(i)∗(j) defined by (2.3).
First we pay attention to the connection of the tetrahedral equation and the Yang–Baxter
equations of the following type: for an invertible Φ the formula (2.4) can be transformed to the
kind
Φ123Φ1(i)(j)Φ2(i)(l)Φ3(j)(l)Φ
−1
123 = Φ3(j)(l)Φ2(i)(l)Φ1(i)(j),
where (i), (j) and (l) are multiindices of the same dimension s. Taking the trace of both parts
with respect to indices 1, 2, 3, one deduces that the expression R(i)(j) = Tr1 Φ1(i)(j) satisfies the
Yang–Baxter equation (1.1) in the tensor product V ⊗s. Moreover as a consequence of (2.6) we
have
Φ(i)(i′)0Φ(i)∗(j)Φ(i′)∗(j′)Φ0(j)(j′)Φ
−1
(i)(i′)0 = Φ0(j)(j′)Φ(i′)∗(j′)Φ(i)∗(j),
here (i) and (i′) are of dimension K, (j), (j′) are of dimension M . If one takes the trace of both
parts with respect to the spaces (i), (i′), 0 it turns out that the expression L∗(j) = Tr(i) Φ(i)∗(j)
satisfies the RLL relation
L∗(j)L∗(j′)R(j)(j′) = R(j)(j′)L∗(j′)L∗(j).
Thus the data
L∗(j) = Tr(i) Φ(i)∗(j), R′(i)(j) = Tr1 ΦR
1(i)(j)
10 D.V. Talalaev
meets Lemma 1.1 conditions. The immediate consequence of this is that the following set of
operators
I0,k = Tr(jl),l=1,...,k
∏
l=
−−−→
1,...,k
L∗(jl)
∏
m=
−−−−−→
1,...,k−1
R′(jm)(jm+1)
is commutative and contains I1 = Tr(j) L∗(j) = Tr(i)(j) Φ(i)∗(j). Here multiindices (jm) have
dimension M . There is a version of the formula for I0,k
I0,k = Tr(il)(jl)(s)
∏
l=
−−−→
1,...,k
Φ(il)∗(jl)
∏
m=
−−−−−→
1,...,k−1
ΦR
sm(jm)(jm+1)
with multiindices (il) of dimension K, (jl) of dimension M and (s) of dimension k−1. A similar
argument can show that, due to Lemma 1.1 the family
In,0 = Tr(il)(jl)(t)
∏
l=
−−−→
1,...,n
Φ(il)∗(jl)
∏
m=
−−−−−→
1,...,n−1
ΦL
(im)(im+1)tm
is also commutative and includes I1, here the multiindices have the same dimension except (m)
which is n−1. The main accomplishment of the paper is that these two families commute among
themselves. In order to give a precise formulation let us introduce some notation:
Bk = Φ(i1)∗(j1) · · ·Φ(ik)∗(jk) =
∏
α∈
−−−−→
(1,...,k)
Φ(iα)∗(jα),
Rsk = ΦR
s1(j1)(j2) · · ·Φ
R
sk−1(jk−1)(jk) =
∏
α∈
−−−−−−−→
(1,...,k−1)
ΦR
sα(jα)(jα+1),
Lsl = ΦL
(i1)(i2)s1
· · ·ΦL
(il−1)(il)sl−1
=
∏
α∈
−−−−−−→
(1,...,l−1)
ΦL
(iα)(iα+1)sα
,
ApL =
∏
−→α
←−
β
Φ(iα)(iβ)pαβ
, ApR =
∏
−→
β←−α
Φpαβ(jα)(jβ).
Here (iα) is a K-multiindex, (jβ) is an M -multiindex. pαβ is the set of accessory indices in last
two formulas. In addition, we need two auxiliary elements
Φp =
∏
−→α
←−
β
Φ̃L
sαpα+1,βpα,β
, Φ∗p =
∏
−→
β←−α
Φ̃R
pα,β+1pα,βtβ
.
2.5 The commutativity proof in d = 2
Definition 2.6. Let us call a solution of the tetrahedral equation generic if it is invertible and
the operator Trp(A
p
RΦ∗p) is invertible too.
Theorem 2.7. For any generic solution Φ of the tetrahedral equation the following is true
[I0,k, In,0] = 0.
Proof. Let us consider an expression
ApL︸︷︷︸
iα,iβ ,p
Φp︸︷︷︸
p,s
Bk︸︷︷︸
iα,jα
⊗ Bl︸︷︷︸
iβ ,jβ
Rsk︸︷︷︸
jα,s
Ltl︸︷︷︸
iβ ,t
ApR︸︷︷︸
jα,jβ ,p
Φ∗p︸︷︷︸
p,t
.
Tetrahedral Equation and Higher Hamiltonians of Quantum Integrable Systems 11
The indices under each multiplier indicate the spaces in which it acts. Next, we shall omit the
tensor product sign, considering the indices of the corresponding spaces. This expression can be
transmuted to the form
ApLBkBl ΦpR
s
kA
p
R LtlΦ
∗
p
Lemma 2.8
= ApLBkBl A
p
RR
s
kΦp L
t
lΦ
∗
p
= ApLBkBlA
p
R RskΦpL
t
lΦ
∗
p
Lemma 2.9
= ApRBlBkA
p
L R
s
kΦpL
t
lΦ
∗
p
= ApRBlBkA
p
LR
s
kL
t
l ΦpΦ
∗
p
Lemma 2.10
= ApRBlBkA
p
LR
s
kL
t
l Φ∗pΦp
= ApRBlBkR
s
k A
p
LL
t
lΦ
∗
p Φp
Lemma 2.11
= ApRBlBkR
s
k Φ∗pL
t
lA
p
L Φp
= ApRΦ∗pBlBkR
s
kL
t
lA
p
LΦp.
Then we deduce
ApLΦpBkBlR
s
kL
t
lA
p
RΦ∗p(A
p
LΦp)
−1 = ApRΦ∗pBlBkR
s
kL
t
l .
In particular this implies
Tr(iα)(iβ)(p),sα Bk+lR
s
kL
t
lA
p
RΦ∗p = Tr(iα)(iβ)(p)sα A
p
RΦ∗pBlBkR
s
kL
t
l .
Let us note that the trace procedure factorizes
Tr(iα)(iβ)sα
(
BkBlR
s
kL
t
l
)
Tr(p)
(
ApRΦ∗p
)
= Tr(p)
(
ApRΦ∗p
)
Tr(iα)(iβ)sα
(
BlBkR
s
kL
t
l
)
.
One may notice that(
Tr(p)
(
ApRΦ∗p
))−1
Tr(iα)(iβ)sα
(
BkBlR
s
kL
t
l
)
Tr(p)
(
ApRΦ∗p
)
= Tr(iα)(iβ)sα
(
BlBkR
s
kL
t
l
)
.
Taking a trace over the remaining auxiliary spaces completes the proof
I0kIl0 = Tr(jα)(jβ)tβ
(
Tr(iα)(iβ)sα
(
Bk+lR
s
kL
t
l
))
= Tr(jα)(jβ)tβ
(
Tr(iα)(iβ)sα
(
BlBkR
s
kL
t
l
))
= Il0I0k. �
Proceed now to the proof of lemmas.
Lemma 2.8.
ΦpR
s
kA
p
R = ApRR
s
kΦp. (2.11)
Proof. Let us deduce a useful formula
Φ̃L
αβγΦR
α(i)(j)Φβ(j)(k)Φγ(i)(k) = Φβ(j)(k)Φγ(i)(k)Φ
R
α(i)(j)Φ̃
L
αβγ . (2.12)
It is a direct consequence of equation (2.8). Let us write more thoroughly (2.11)∏
−→α
←−
β
Φ̃L
sαpα+1,βpα,β
∏
−→α
ΦR
sα(jα)(jα+1)
∏
−→
β←−α
Φpα,β(jα)(jβ).
Note that the left and right multipliers commute at different values of β so it is enough to check
the relation at fixed β = β0 performing further dressing starting with the multipliers closest to
the center∏
−→α
Φ̃L
sαpα+1,β0
pα,β0
∏
−→α
ΦR
sα(jα)(jα+1)
∏
←−α
Φpα,β0 (jα)(jβ0 )
=
∏
−→α
(
Φ̃L
sαpα+1,β0
pα,β0
ΦR
sα(jα)(jα+1)
)∏
←−α
Φpα,β0 (jα)(jβ0 ).
12 D.V. Talalaev
Now we will focus attention on how the elements of the left product are transferred through the
right one. This can be done consecutively: fix the index α = α0 in the left product
Φ̃L
sα0pα0+1,β0
pα0,β0
ΦR
sα(jα0 )(jα0+1)
∏
←−α
Φpα,β0 (jα)(jβ0 ).
The only multipliers of the right product which do not commute with the elements on the left
have indices α = α0 and α = α0 + 1. When moving through them we use equality (2.12)
Φ̃L
sα0pα0+1,β0
pα0,β0
ΦR
sα(jα0 )(jα0+1)Φpα0+1,β0
(jα0+1)(jβ0 )Φpα0,β0 (jα0 )(jβ0 )
= Φpα0+1,β0
(jα0+1)(jβ0 )Φpα0,β0 (jα0 )(jβ0 )Φ
R
sα(jα0 )(jα0+1)Φ̃
L
sα0pα0+1,β0
pα0,β0
. �
Lemma 2.9.
ApLBkBlA
p
R = ApRBlBkA
p
L.
Proof. The statement is an immediate consequence of equation (2.5). We will illustrate the
basic techniques of generalization. We write the left part of the expression∏
−→α
←−
β
Φ(iα)(iβ)pαβ
∏
−→α
Φ(iα)∗(jα)
∏
−→
β
Φ(iβ)∗(jβ)
∏
←−α
−→
β
Φpαβ(jα)(jβ).
Here (iα) and (iβ) are K-multiindices, and (jα) and (jβ) are M -ones. We are going to move the
multipliers of the third product through the multipliers of the second one. For doing this we
use a twisting multipliers of the first and fourth factors. Note that the multipliers of the first
and fourth factors commute except those with both coinciding indices. It remains to verify that
the multiplier order is suitable. We will move the elements of the third product consequently
from the left through the second product. To do this, the order relation on β in the first
product should be such that the junior members stand on the right, and in the fourth on the
left. Similarly, one can check that the order relation on α in the first product should be such
that the senior members are on the right, and in the fourth on the left. �
Lemma 2.10.
ΦpΦ
∗
p = Φ∗pΦp.
Proof. Let us write the left side of the expression∏
−→α
←−
β
Φ̃L
sαpα+1,βpα,β
∏
−→
β←−α
Φ̃R
pα,β+1pα,βtβ
.
We move the right product through the left one component-wise, for fixed α on the left and
fixed β on the right. Let us consider these multipliers∏
←−
β
Φ̃L
sα0pα0+1,βpα0,β
∏
←−α
Φ̃R
pα,β0+1pα,β0 tβ0
.
The only non-commutative elements of these products have neighboring indices β = β0, β0 + 1
and α = α0, α0 + 1. We write only them
Φ̃L
sα0pα0+1,β0+1pα0,β0+1
Φ̃L
sα0pα0+1,β0
pα0,β0
Φ̃R
pα0+1,β0+1pα0+1,β0
tβ0
Φ̃R
pα0,β0+1pα0,β0 tβ0
.
Note that the right pair can be moved through the left pair of multipliers according to (2.9). �
Lemma 2.11.
ApLL
t
lΦ
∗
p = Φ∗pL
t
lA
p
L.
Proof. Analogously to Lemma 2.9. �
Tetrahedral Equation and Higher Hamiltonians of Quantum Integrable Systems 13
3 Conclusion
The choice of the notations Il,0 and I0,k used in the main part of the work pursued a definitive
goal. We hope that the following hypothesis is correct
Hypothesis 3.1. For any generic solution of the tetrahedral equation it can be constructed
a two-parameter commutative family Il,k, which includes two families presented above.
Remark 3.2. Note that in the work [2] it is constructed a two-parameter family of operators
commuting with a layer-by-layer transfer-matrix related to the special solution of the tetrahedral
equation. It is curious to compare the result in the q-oscillator realization case and the result
presented here.
However, the main result of this work reveals some important perspectives:
• Primarily, it provides the opportunity to study the spectra of the corresponding quantum
models and models of statistical physics in dimension d = 3. We hope that in this case
the Bethe ansatz technique also can be applied.
• Another interesting direction is the generalization of the notion of integrability in the
case of 2-dimensional surfaces comprising the classical case. It is interesting to juxtapose
the approach of this work with the language of paper [14] on the theory of the Hitchin
systems on surfaces. We would also be enthusiastic in the study of moduli spaces of 2-
bundles on surfaces and an appropriate analogue of the Hitchin theory. In this case one
has a significant difficulty in constructing nonabelian gerbs.
• In [9, 10] we present a construction of quasi-invariant of 2-knots in the form of a par-
tition function on the graph of double points of the 2-knot diagram. This structure is
resembling to the approach of [3]. The similarity of the partition function expressions is
intriguing and gives hope for further development activities in low-dimensional topology
and topological quantum field theory. Presumably a certain connection of this method
with four-dimensional quantum field theories, like the BF-theory, can be established.
A Proof of Lemma 2.5
Proof. Let us prove the formula (2.4). The rest is similar. The left hand side of (2.4) can
expressed as follows
Φ123Φ1(i)(j)Φ2(i)(l)Φ3(j)(l) = Φ123
∏
s∈
−−−−→
(1,...,t)
Φ1isjs
∏
u∈
−−−−→
(1,...,t)
Φ2iulu
∏
v∈
−−−−→
(1,...,t)
Φ3jvlv .
The last product can be rearranged due to the commutativity of operators in different tensor
components
Φ123
∏
s∈
−−−−→
(1,...,t)
Φ1isjsΦ2islsΦ3jsls .
Then we apply the Zamoolodchkov equation with Φ123 to each factor consecutively from the left∏
s∈
−−−−→
(1,...,t)
Φ3jslsΦ2islsΦ1isjsΦ123
and again rearrange the product∏
v∈
−−−−→
(1,...,t)
Φ3jvlv
∏
u∈
−−−−→
(1,...,t)
Φ2iulu
∏
s∈
−−−−→
(1,...,t)
Φ1isjsΦ123 = Φ3(j)(l)Φ2(i)(l)Φ1(i)(j)Φ123. �
14 D.V. Talalaev
Acknowledgements
The work was partially supported by the RFBR grant of Russian Foundation of Basic Research
17-01-00366A, grant of the support of scientific schools 4833.2014.1, and the grant of the Dy-
nasty Foundation. I would like to thank the referees for careful reading of the manuscript and
constructive remarks on the material exposition.
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https://doi.org/10.1088/0305-4470/39/13/009
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http://arxiv.org/abs/math.QA/0409602
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https://doi.org/10.1016/0370-2693(90)90677-X
https://doi.org/10.1016/0370-2693(89)91466-4
https://doi.org/10.1134/S0001434607030236
http://arxiv.org/abs/math.AG/0610364
1 Introduction
1.1 Yang–Baxter equation and its generalizations
1.2 Universal integrability in d=1
1.3 Tetrahedral equation
2 Commutative family
2.1 The commutativity demonstration in d=1
2.2 Regular 3d lattices and statistical models
2.3 Some consequences of the tetrahedral equation
2.4 Two families
2.5 The commutativity proof in d=2
3 Conclusion
A Proof of Lemma 2.5
References
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