# On extension of the Yang-Baxter equation and the fermionic $R$-operator

A<EMAIL_ADDRESS>Instituto de Física
Universidade de Brasília
70910-900, Brasília, DF, Brasil

###### Abstract

We consider the fermionic $R$-operator based on Bazhanov-Stroganov’s three-
parameter elliptic parametrization of the free fermion model, and find the
most general solution of the related tetrahedral Zamolodchikov algebra in the
trigonometric limit for an arbitrary set of parameters. This allows to
construct an extension of the $R$-operator and the corresponding Yang-Baxter
equation, which are of the difference type in one of the spectral parameters.

#### Keywords:

Exactly Solvable Models, Bethe Ansatz; Continuum models; Integration of
Completely integrable systems by inverse spectral and scattering methods;

##  1 Introduction

In [1] we considered an elliptic parametrization of the free fermion model on
the lattice [2, 3, 4], with the $R$-matrices being of the difference type in
one of the spectral parameters, i.e., the dependence on the spectral
parameters $u_{j}$ is through the differences $\Delta u_{jk}:=u_{j}-u_{k}$,
and found a particular solution of the tetrahedral Zamolodchikov algebra. The
original motivation came from investigating the integrable properties of the
string theory on $AdS_{5}\times S^{5}$ (see [5] for a review), and, in
particular, its relation to the one-dimensional Hubbard model [6], the
$R$-matrix of which has been proposed to be related to the $S$-matrix of the
string theory via $AdS/CFT$ correspondence (see [7, 8, 9] and the references
therein). It has been long known that the $R$-matrix of the one-dimensional
Hubbard model can be obtained, as shown by Shastry [10, 11], by pairing two
$R$-matrices corresponding to the free fermion model. The construction, which
is based on the use of the so-called decorated Yang-Baxter equation ($DYBE$),
in addition to the standard Yang-Baxter equation ($YBE$), leads to an
$R$-matrix which is not of the difference type.

There is another interesting integrable fermionic model which appears in the
$su(1|1)$ subsector of the string theory on $AdS_{5}\times S^{5}$, which is
described by a $(1+1)$-relativistic fermion model [12].222 The Lagrangian of
the model has the following explicit form:  $\displaystyle\mathscr{L}$
$\displaystyle=\mathrm{i}\,\bar{\psi}\gamma_{\mu}\partial^{\mu}\psi\>-m\bar{\psi}\psi+\frac{g_{2}}{4m}\epsilon^{\alpha\beta}\left(\bar{\psi}\partial_{\alpha}\psi\;\bar{\psi}\>\gamma^{3}\partial_{\beta}\psi-\partial_{\alpha}\bar{\psi}\psi\;\partial_{\beta}\bar{\psi}\>\gamma^{3}\psi\right)-\frac{g_{3}}{16m}\epsilon^{\alpha\beta}\left(\bar{\psi}\psi\right)^{2}\partial_{\alpha}\bar{\psi}\>\gamma^{3}\partial_{\beta}\psi.$
(1.1) It can be shown to be a completely integrable classically, and its
quantum integrability properties has been investigated from various points of
view [13, 14, 15, 16, 17, 18]. Nevertheless, due to the non-ultralocal nature
of the algebra of Lax operators,333This remains the case if one sets
$g_{2}=g_{3}=0$, i.e., the non-ultralocal nature is also present for the free
fermion model. The explicit forms of the Lax operators and the corresponding
algebra can be found in [14, 15]. the quantization of this model by standard
methods of integrable systems remains an open problem, and, as a consequence,
the lattice version of the model is not known. One can, however, start with
the lattice formulation of a model - in particular, given the same non-
ultralocal nature for the free fermion and the model in (1.1) - with the free
fermion model on the lattice and trace the appearance of the non-ultralocal
terms in the continuous limit.444A partial progress towards this program was
reported in [19]. For the interacting theory, the one-dimensional Hubbard
model is an obvious candidate, given, as mentioned above, its relation to
string theory on $AdS_{5}\times S^{5}$ on one hand, and to the free fermion
model, by means Shastry’s procedure of pairing two corresponding $R$-matrices,
on the other hand (see, however, [20, 21]). Thus, it would be quite
advantageous to start with the $R$-matrix for the free fermion model of the
difference type, and investigate the consequences when coupling the spins, as,
for example, in the case of the one-dimensional Hubbard model, which would in
principle allow obtaining the lattice formulations of interacting continuous
$(1+1)$-relativistic fermion models such as the model in (1.1).

To this end, we consider the fermionic $R$-operator formalism due to Umeno et
al. [22, 23, 24], and the three-parameter elliptic parametrization of the free
fermion model due to Bazhanov and Stroganov [2, 3, 4], which allow a
formulation of both Yang-Baxter and decorated Yang-Baxter relations in the
form where the dependence of $R$-matrix is of the difference type with respect
to one of the spectral parameters. We give the main formulas and results of
this construction in section 2. Next, in section 3, we give the most general
solution of the tetrahedral Zamolodchikov algebra [25, 26, 27, 28] in the
trigonometric limit for an arbitrary set of parameters. Using these results,
in section 4, we construct an extension of the original fermionic $R$-operator
and obtain a family of the corresponding Yang-Baxter equations, each being of
the difference type in one of the spectral parameters. In appendices we
collect many explicit formulas and useful expressions: In Appendix A we give
some useful identities used in the text, in Appendix B we list the explicit
form of the equations arising in the tetrahedral Zamolodchikov algebra for the
full elliptic case, and, finally, in Appendix C we present our solution to the
tetrahedral Zamolodchikov algebra in the trigonometric limit for an arbitrary
set of spectral parameters.

##  2  Three-parameter elliptic parametrization of the free fermion model

In [2, 3, 4] (see also [29]) an interesting general solution to an
inhomogeneous eight-vertex model was obtained, with the $R$-matrix of the
form:

$\displaystyle\hat{R}=\begin{pmatrix}a&0&0&d\\\ 0&b&c&0\\\
0&c^{\prime}&b^{\prime}&0\\\ d^{\prime}&0&0&a\end{pmatrix},$ (2.1)

satisfying the free fermion condition [30]:

$\displaystyle aa^{\prime}+bb^{\prime}-cc^{\prime}-dd^{\prime}=0$ (2.2)

The coefficients in (2.1) are parametrized in terms of by three parameters:
$\left\\{\zeta_{0}\equiv u,\zeta_{1},\zeta_{2}\right\\}$, and have the form:

$\displaystyle
a(u;\zeta_{1},\zeta_{2})=\rho\left[1-\mathrm{e}(u)\mathrm{e}(\zeta_{1})\mathrm{e}(\zeta_{2})\right];\quad
a^{\prime}(u;\zeta_{1},\zeta_{2})=\rho\left[\mathrm{e}(u)-\mathrm{e}(\zeta_{1})\mathrm{e}(\zeta_{2})\right]$
(2.3) $\displaystyle
b(u;\zeta_{1},\zeta_{2})=\rho\left[\mathrm{e}(\zeta_{1})-\mathrm{e}(u)\mathrm{e}(\zeta_{2})\right];\quad
b^{\prime}(u;\zeta_{1},\zeta_{2})=\rho\left[\mathrm{e}(\zeta_{2})-\mathrm{e}(u)\mathrm{e}(\zeta_{1})\right]$
(2.4) $\displaystyle
c(u;\zeta_{1},\zeta_{2})=c^{\prime}(u;\zeta_{1},\zeta_{2})=\rho\>\mathrm{sn}^{-1}\left(\frac{u}{2}\right)\left[1-\mathrm{e}(u)\right]\left[\mathrm{e}(\zeta_{1})e(\zeta_{2})\mathrm{sn}(\zeta_{1})\mathrm{sn}(\zeta_{2})\right]^{1/2},$
(2.5) $\displaystyle
d(u;\zeta_{1},\zeta_{2})=d^{\prime}(u;\zeta_{1},\zeta_{2})=-\mathrm{i}\,k\rho\>\mathrm{sn}\left(\frac{u}{2}\right)\left[1+\mathrm{e}(u)\right]\left[\mathrm{e}(\zeta_{1})e(\zeta_{2})\mathrm{sn}(\zeta_{1})\mathrm{sn}(\zeta_{2})\right]^{1/2}.$
(2.6)

Here the functions $\mathrm{sn}(x)$ and $\mathrm{cn}(x)$ are the Jacobi
elliptic functions of modulus $k$ [31], $\mathrm{e}(x)$ is the elliptic
exponential $\mathrm{e}(x):=\mathrm{cn}(x)+\mathrm{i}\,\mathrm{sn}(x)$, and
$\rho$ is an arbitrary factor. The $R$-matrix in (2.1) satisfies the Yang-
Baxter equation [2]:

$\displaystyle\hat{R}_{12}(u_{12};\zeta_{1},\zeta_{2})\hat{R}_{13}(u_{13};\zeta_{1},\zeta_{3})\hat{R}_{23}(u_{23};\zeta_{2},\zeta_{3})=\hat{R}_{23}(\eta_{23};\zeta_{2},\zeta_{3})\hat{R}_{13}(u_{13};\zeta_{1},\zeta_{3})\hat{R}_{12}(u_{12};\zeta_{1},\zeta_{2}),$
(2.7)

where we have used the shorthand notation $u_{jk}\equiv u_{j}-u_{k}$.

Following [22, 23], we next introduce the fermionic $R$-operator corresponding
to the $R$-matrix (2.1), which is more convenient to use from the beginning in
order, e.g., to make a connection with the Lax operators for the relativistic
$(1+1)$ free fermion model or the model defined by (1.1), which are written in
terms of fermionic variables. To this end, one has to apply the Jordan-Wigner
transformation (see [6] for an extensive treatment) to the above
$\hat{R}$-matrix as well as the the Yang-Baxter equation (2.7). The essential
technical steps are explained in details in [22, 23] and are omitted here.

The final result, when applied to the $R$-matrix (2.1) is as follows: The
associated fermionic $R$-operator, written in terms of the fermionic variables
$(c^{\dagger}_{k},c_{k})$, satisfying the usual anti-commutation relations:
$\\{c^{\dagger}_{j},c_{k}\\}=\delta_{jk}$, takes the from:

$\displaystyle R_{jk}(u;\zeta_{j},\zeta_{k})$
$\displaystyle=a(u;\zeta_{j},\zeta_{k})\left[-n_{j}n_{k}\right]+a^{\prime}(u;\zeta_{j},\zeta_{k})\left[(1-n_{j})(1-n_{k})\right]+b(u;\zeta_{j},\zeta_{k})\left[n_{j}(1-n_{k})\right]$
$\displaystyle+b^{\prime}(u;\zeta_{j},\zeta_{k})\left[n_{k}(1-n_{j})\right]+c(u;\zeta_{j},\zeta_{k})\left[\Delta_{jk}+\Delta_{kj}\right]+d(u;\zeta_{j},\zeta_{k})\left[-\tilde{\Delta}^{(\dagger)}_{jk}-\tilde{\Delta}_{jk}\right].$
(2.8)

Here we have denoted:

$\displaystyle n_{k}$
$\displaystyle:=c^{\dagger}_{k}c_{k},\quad\Delta_{jk}:=c^{\dagger}_{j}c_{k},\quad\tilde{\Delta}^{(\dagger)}_{jk}=c^{\dagger}_{j}c^{\dagger}_{k};\quad\tilde{\Delta}_{jk}=c_{j}c_{k}.$
(2.9)

Furthermore, it can be shown that the fermionic $R$-operator (2.8) satisfies
the Yang-Baxter equation:

$\displaystyle
R_{12}(u_{12};\zeta_{1},\zeta_{2})R_{13}(u_{13};\zeta_{1},\zeta_{3})R_{23}(u_{23};\zeta_{2},\zeta_{3})=R_{23}(u_{23};\zeta_{2},\zeta_{3})R_{13}(u_{13};\zeta_{1},\zeta_{3})R_{12}(u_{12};\zeta_{1},\zeta_{2}),$
(2.10)

The corresponding fermionic monodromy operator is defined in the usual manner:

$\displaystyle
T_{a}(u;\zeta_{a},\\{\zeta_{j}\\})=R_{aN}(u;\zeta_{a},\zeta_{N})R_{a,N-1}(u;\zeta_{a},\zeta_{N-1})\cdot\ldots\cdot
R_{a1}(u;\zeta_{a},\zeta_{1}),$ (2.11)

and satisfies the $RTT=TTR$ relation:

$\displaystyle
R_{ab}(u_{a}-u_{b};\zeta_{a},\zeta_{b})T_{a}(u_{a};\zeta_{a},\\{\zeta_{j}\\})T_{b}(u_{b};\zeta_{b},\\{\zeta_{j}\\})=T_{b}(u_{b};\zeta_{b},\\{\zeta_{j}\\})T_{a}(u_{a};\zeta_{a},\\{\zeta_{j}\\})R_{ab}(u_{a}-u_{b};\zeta_{a},\zeta_{b}).$
(2.12)

Defining also the transfer operator:

$\displaystyle\tau(u;\zeta_{a},\\{\zeta_{j}\\})=\mathrm{Str}_{a}\left[T_{a}(u;\zeta_{a},\\{\zeta_{j}\\})\right],$
(2.13)

where the supertrace $\mathrm{Str}_{a}\left[F\right]$ over the auxiliary Fock
space $a$ is defined by [22]:

$\displaystyle\mathrm{Str}_{a}\left[F\right]$
$\displaystyle:={}_{a}\bra{0}F\ket{0}_{a}-{}_{a}\bra{1}F\ket{1}_{a}$ (2.14)
$\displaystyle c_{a}\ket{0}_{a}$
$\displaystyle=0,\quad\ket{1}_{a}:=c^{\dagger}_{a}\ket{0}_{a}.$ (2.15)

one obtains the commuting quantities:

$\displaystyle\left[\tau(u;\zeta_{a},\\{\zeta_{j}\\}),\tau(v;\zeta_{b},\\{\zeta_{j}\\})\right]=0.$
(2.16)

As an example, we apply the fermionic Yang-Baxter relation (2.10) for the case
of equal parameters $\zeta_{i}\equiv\zeta$, and compute the Hamiltonian via
the standard formula [6]:

$\displaystyle\hat{\mathcal{H}}=\tau^{-1}(0;\zeta)\frac{d}{du}\tau(u;\zeta)|_{u=0}.$
(2.17)

Using the explicit form of the coefficients (2.3)-(2.6) it is easy to check
the relations:

$\displaystyle R_{jk}(0;\zeta)=\beta\mathcal{P}_{jk}$ (2.18)
$\displaystyle\mathcal{P}_{jk}R^{\prime}(0;\zeta)=a^{\prime}_{0}(\zeta)+\left\\{c_{0}(\zeta)-a^{\prime}_{0}(\zeta)\right\\}\left[n_{j}+n_{k}\right]+b_{0}(\zeta)\left[\Delta_{jk}+\Delta_{kj}\right]+d_{0}(\zeta)\left[\tilde{\Delta}^{(\dagger)}_{jk}-\tilde{\Delta}_{jk}\right]$
(2.19)
$\displaystyle\tau(0;\zeta)=\beta^{N}\mathcal{P}_{12}\mathcal{P}_{23}\cdot\ldots\cdot\mathcal{P}_{N,N-1},$
(2.20)

where we denoted $R_{jk}(0;\zeta)=R_{jk}(0;\zeta,\zeta)$,
$\beta=(-2\mathrm{i}\,\rho)\mathrm{e}(\zeta)\mathrm{sn}(\zeta)$,
$\mathcal{P}_{jk}:=1-n_{j}-n_{k}+\Delta_{jk}+\Delta_{kj}$ is the permutation
operator, and $a^{\prime}_{0}(\zeta),c_{0}(\zeta),b_{0}(\zeta),d_{0}(\zeta)$
in the right-hand side of (2.19) stand for the derivatives of the
corresponding coefficients (2.3)-(2.6) at $u=0$. Using (2.17)-(2.20), one
immediately obtains the Hamiltonian for the fermionic $XY$ model in an
external field:555Note, that the parameter $\zeta$ controls the external field
in (2.21).

$\displaystyle\hat{\mathcal{H}}^{{(XY)}}=\frac{1}{2\,\mathrm{sn}(\zeta)}\sum_{j=1}^{N}\left[\left(\Delta_{j,j+1}+\Delta_{j+1,j}\right)+k\,\mathrm{sn}(\zeta)\left(\tilde{\Delta}^{(\dagger)}_{j,j+1}-\tilde{\Delta}_{j+1,j}\right)+2\,\mathrm{cn}(\zeta)\left(n_{j}-\nicefrac{{1}}{{2}}\right)\right].$
(2.21)

To extend the above construction for the spinless fermions to the models with
spin, one introduces two copies of the $R^{(s)}$-operator for each spin
$s=\ket{\uparrow},\ket{\downarrow}$. The generalization is straightforward,
and the free fermion model with spin $s=\nicefrac{{1}}{{2}}$ is obtained from
two copies of the fermionic $R$-matrix (2.8):

$\displaystyle\mathcal{R}_{jk}(u_{j}-u_{k};\zeta_{j},\zeta_{k};\hat{\zeta}_{j},\hat{\zeta}_{k}):=R^{(\uparrow)}_{jk}(u_{j}-u_{k};\zeta_{j},\zeta_{k})R^{(\downarrow)}_{jk}(u_{j}-u_{k};\hat{\zeta}_{j},\hat{\zeta}_{k}).$
(2.22)

The fermionic operator $\mathcal{R}_{jk}(u;\zeta_{j},\zeta_{k})$ in (2.22)
satisfies the same Yang-Baxter equation (2.10), and the corresponding
monodromy operator and commuting quantities can be constructed in the same
manner as above. Starting from
$\mathcal{R}_{jk}(u;\zeta_{j},\zeta_{k};\hat{\zeta}_{j},\hat{\zeta}_{k})$,
corresponding to the case of equal parameters
$\zeta_{i}\equiv\zeta,\hat{\zeta}_{i}\equiv\hat{\zeta}$ considered above, one
obtains two copies of non-interacting fermionic $XY$ models in external
fields, with the $\mathcal{R}$-matrix (2.22) being of the difference type in
one of the spectral parameters, unlike the case in [22, 23].666In this case,
the definition of the supertrace over the auxiliary Fock space $a$ (c.f.
(2.14)) is as follows [22]:  $\displaystyle\mathrm{Str}_{a}\left[F\right]$
$\displaystyle:={}_{a}\bra{0}F\ket{0}_{a}-{}_{a}\bra{\uparrow}F\ket{\uparrow}_{a}-{}_{a}\bra{\downarrow}F\ket{\downarrow}_{a}+{}_{a}\bra{\downarrow\uparrow}F\ket{\uparrow\downarrow}_{a}$
(2.23) $\displaystyle c_{a,(s)}\ket{0}_{a}$
$\displaystyle=0,\quad\bra{\uparrow}_{a}:=c^{\dagger}_{a,\uparrow}\ket{0}_{a},\quad\bra{\downarrow}_{a}:=c^{\dagger}_{a,\downarrow}\ket{0}_{a},\quad\bra{\uparrow\downarrow}_{a}:=c^{\dagger}_{a,\uparrow}c^{\dagger}_{a,\downarrow}\ket{0}_{a}.$
The details of obtaining the corresponding Lax pair were given in [19], and
the problem of taking the continuous limit will be presented in a future
publication.

##  3 Decorated Yang-Baxter equation and tetrahedral Zamolodchikov algebra

We now turn our attention to the interacting case. First, we derive the so-
called decorated Yang-Baxter equation [10, 11, 22]. In our case, it follows
from the relation:777Changing the sign of the modulus $k\to-k$ only changes
the sign of $d(u;\zeta_{1},\zeta_{2})$ in (2.6).

$\displaystyle(2n_{j,(s)}-1)R^{(s)}_{jl}(u;\zeta_{j},\zeta_{l};k)(2n_{l,(s)}-1)=R^{(s)}_{jl}(u;\zeta_{j}+2\mathrm{K}(k),\zeta_{l}+2\mathrm{K}(k);-k);\quad
s=\ket{\uparrow},\ket{\downarrow}$ (3.1)

where we have written the dependence on the modulus $k$ in
$R^{(s)}_{jl}(u;\zeta_{j},\zeta_{k};k)$ explicitly, and $\mathrm{K}(k)$ is the
complete elliptic integral of the first kind [31]. The above formula can be
checked using the formulas given in Appendix A, as well as the explicit
expressions for the functions (2.3)-(2.6). Then, the decorated Yang-Baxter
equation is a relation that is derived from the Yang-Baxter equation (2.10),
together with the above identity (3.1). It has the form:

$\displaystyle
R^{(s)}_{12}(u_{12};\zeta_{1},\zeta_{2}-2\mathrm{K}(k);k)(2n_{1,(s)}-1)R^{(s)}_{13}(u_{13};\zeta_{1},\zeta_{3}-2\mathrm{K}(k);-k)R^{(s)}_{23}(u_{23};\zeta_{2},\zeta_{3};k)$
$\displaystyle=R^{(s)}_{23}(u_{23};\zeta_{2},\zeta_{3};k)R^{(s)}_{13}(u_{13};\zeta_{1},\zeta_{3}-2\mathrm{K}(k);-k)(2n_{1,(s)}-1)R^{(s)}_{12}(u_{12};\zeta_{1},\zeta_{2}-2\mathrm{K}(k);k)$
(3.2)

Unlike the previously considered cases [6, 10, 11, 22], the decorated Yang-
Baxter relation (3.2) depends on the differences of the spectral parameters
$u_{jk}\equiv u_{j}-u_{k}$, taking an asymmetrical form with respect to the
other parameters $\zeta_{i}$. Using (3.1) and taking the product of two copies
of (3.2) for $s=\ket{\uparrow},\ket{\downarrow}$ one can readily arrive at a
similar identity for $\mathcal{R}_{jk}(u;\zeta_{j},\zeta_{k})$ defined in
(2.22). In what follows, it is convenient to introduce the following
notations:

$\displaystyle\mathrm{L}^{0,(s)}_{jk}$
$\displaystyle=R^{(s)}_{jk}(u_{jk};\zeta_{j},\zeta_{k};k)$ (3.3)
$\displaystyle\mathrm{L}^{1,(s)}_{jk}$
$\displaystyle=R^{(s)}_{jk}(u_{jk};\zeta_{j},\zeta_{k}-2\mathrm{K}(k);-k)(2n_{j,s}-1),$
(3.4)

The explicit form of the $\mathrm{L}^{1,(s)}_{jk}$-operator is given in (A.4),
Appendix A. With these notation, the Yang-Baxter and decorated Yang-Baxter
equations take the form:

$\displaystyle\mathrm{L}^{0,(s)}_{12}\mathrm{L}^{0,(s)}_{13}\mathrm{L}^{0,(s)}_{23}$
$\displaystyle=\mathrm{L}^{0,(s)}_{23}\mathrm{L}^{0,(s)}_{13}\mathrm{L}^{0,(s)}_{12},$
(3.5)
$\displaystyle\mathrm{L}^{0,(s)}_{12}\mathrm{L}^{1,(s)}_{13}\mathrm{L}^{1,(s)}_{23}$
$\displaystyle=\mathrm{L}^{1,(s)}_{23}\mathrm{L}^{1,(s)}_{13}\mathrm{L}^{0,(s)}_{12}.$
(3.6)

The tetrahedral Zamolodchikov algebra is an algebraic expression of the form:

$\displaystyle\mathrm{L}^{\alpha_{1},(s)}_{12}\mathrm{L}^{\alpha_{2},(s)}_{13}\mathrm{L}^{\alpha_{3},(s)}_{23}=\sum_{\beta_{i}=0,1}S^{\alpha_{1}\alpha_{2}\alpha_{3}}_{\beta_{1}\beta_{2}\beta_{3}}\mathrm{L}^{\beta_{1},(s)}_{23}\mathrm{L}^{\beta_{2},(s)}_{13}\mathrm{L}^{\beta_{3},(s)}_{12};\quad\alpha_{1,2,3}=\\{0,1\\},$
(3.7)

generalizing the above two relations (3.5) and (3.6). It was introduced by
Korepanov in [32, 26, 28, 33, 27] to investigate Zamolodchikov’s tetrahedron
equation, which underlines the symmetry of a three-dimensional integrable
model, generalizing the relation of the Yang-Baxter equation to two-
dimensional integrable models. As was shown in [34, 35, 22] the tetrahedral
Zamolodchikov algebra can be used in order to construct an interacting model
of spin $s=\nicefrac{{1}}{{2}}$ fermions, and, in particular, to obtain the
one-dimensional Hubbard model (see [6] for a review, and there references
therein). Thus, our first goal is to obtain the solutions to tetrahedral
algebra (3.7). To obtain the coefficients
$S^{\alpha_{1}\alpha_{2}\alpha_{3}}_{\beta_{1}\beta_{2}\beta_{3}}$ in (3.7)
one has to evaluate generic tensor products
$\mathrm{L}^{\alpha_{1},(s)}_{12}\mathrm{L}^{\alpha_{1},(s)}_{13}\mathrm{L}^{\alpha_{3},(s)}_{23}$
and
$\mathrm{L}^{\beta_{1},(s)}_{23}\mathrm{L}^{\beta_{1},(s)}_{13}\mathrm{L}^{\beta_{3},(s)}_{12}$
that appear in the left and right hand sides of (3.7). To this end, we note
that both $\mathrm{L}^{0,(s)}_{jk}$ and $\mathrm{L}^{1,(s)}_{jk}$ can be
written in the following general form:

$\displaystyle\Gamma_{jk}(u;\zeta_{j},\zeta_{k})$
$\displaystyle=a_{0}(u;\zeta_{j},\zeta_{k})+a_{1}(u;\zeta_{j},\zeta_{k})n_{j}+a_{2}(u;\zeta_{j},\zeta_{k})n_{k}+a_{3}(u;\zeta_{j},\zeta_{k})n_{j}n_{k}+c_{1}(u;\zeta_{j},\zeta_{k})\Delta_{jk}$
$\displaystyle+c_{2}(u;\zeta_{j},\zeta_{k})\Delta_{kj}+d_{1}(u;\zeta_{j},\zeta_{k})\tilde{\Delta}^{(\dagger)}_{jk}+d_{2}(u;\zeta_{j},\zeta_{k})\tilde{\Delta}_{jk},$
(3.8)

One then obtains either $\mathrm{L}^{0,(s)}_{jk}$ or $\mathrm{L}^{1,(s)}_{jk}$
by fixing accordingly the functions
$\Bigl{(}a_{0}(u;\zeta_{j},\zeta_{k}),\ldots,\\\
a_{3}(u;\zeta_{j},\zeta_{k}),c_{1,2}(u;\zeta_{j},\zeta_{k}),d_{1,2}(u;\zeta_{j},\zeta_{k})\Bigr{)}$
in terms of the coefficients of the $R$-operator (2.3)-(2.6). Thus, one only
needs to evaluate the general tensor products of the form:

$\displaystyle\Omega$
$\displaystyle:=\Gamma_{12}(w;\zeta^{\prime\prime}_{1},\zeta^{\prime\prime}_{2})\Gamma_{13}(u;\zeta_{1},\zeta_{3})\Gamma_{23}(v;\zeta^{\prime}_{2},\zeta^{\prime}_{3})$
(3.9) $\displaystyle\tilde{\Omega}$
$\displaystyle:=\Gamma_{23}(\hat{v};\hat{\zeta}_{2},\hat{\zeta}_{3})\Gamma_{13}(\hat{u};\hat{\zeta}^{\prime}_{1},\hat{\zeta}^{\prime}_{3})\Gamma_{12}(\hat{w};\hat{\zeta}^{\prime\prime}_{1},\hat{\zeta}^{\prime\prime}_{2})$
(3.10)

to take into account all possible permutations of indices $\alpha_{1,2,3}$ and
$\beta_{1,2,3}$ in (3.7). These expressions are also needed to verify the
linear dependency of the $\mathrm{L}^{(0,1),(s)}_{jk}$-operators products. To
this end, it is convenient to represent the explicit expressions for $\Omega$,
and, similarly, for $\tilde{\Omega}$, as sums of terms corresponding to all
possible independent products of operators
$\mathcal{O}_{a}=\\{\mathbb{1},n_{i},\Delta_{jk},\tilde{\Delta}_{jk},\tilde{\Delta}^{(+)}_{jk}\\}$:

$\displaystyle\Omega$
$\displaystyle=\sum_{ab}\left[\Gamma_{\mathcal{O}_{a}\mathcal{O}_{b}}\right]\mathcal{O}_{a}\mathcal{O}_{b}=\Gamma^{(0)}_{0}+\left[\Gamma^{(0)}_{n_{1}}\right]n_{1}+\left[\Gamma^{(0)}_{n_{1}n_{2}}\right]n_{1}n_{2}+\ldots+\left[\Gamma^{(1)}_{\Delta_{12}}\right]\Delta_{12}+\left[\Gamma^{(1)}_{\tilde{\Delta}_{12}}\right]\tilde{\Delta}_{12}+\ldots,$
(3.11) $\displaystyle\tilde{\Omega}$
$\displaystyle=\sum_{ab}\left[\tilde{\Gamma}_{\mathcal{O}_{a}\mathcal{O}_{b}}\right]\mathcal{O}_{a}\mathcal{O}_{b}=\tilde{\Gamma}^{(0)}_{0}+\left[\tilde{\Gamma}^{(0)}_{n_{1}}\right]n_{1}+\left[\tilde{\Gamma}^{(0)}_{n_{1}n_{2}}\right]n_{1}n_{2}+\ldots+\left[\tilde{\Gamma}^{(1)}_{\Delta_{12}}\right]\Delta_{12}+\left[\tilde{\Gamma}^{(1)}_{\tilde{\Delta}_{12}}\right]\tilde{\Delta}_{12}+\ldots\,,$
(3.12)

where we use the following notations for the coefficients
$\left[{\Gamma}^{A}_{B}\right]$ in the above expansion: the subscript $B$
indicates the operator which multiplies $\left[{\Gamma}^{A}_{B}\right]$, and
the superscript $A$ in indicates the presence ($A=1$) or absence $(A=0)$ of
$\\{\Delta_{jk},\tilde{\Delta}_{jk},\tilde{\Delta}^{(+)}_{jk}\\}$. The result
of this very lengthy calculation is given in Appendix B where we collect all
the coefficients in the above expansions for $\Omega$ and $\tilde{\Omega}$. We
stress that the formulas in Appendix B are general for any modulus $k\neq 0$.
Using these explicit formulas, one can, for example, readily check the $YBE$
(2.10) by appropriately choosing the coefficients
$\Bigl{(}a_{0}(u;\zeta_{j},\zeta_{k}),\ldots,a_{3}(u;\zeta_{j},\zeta_{k}),c_{1,2}(u;\zeta_{j},\zeta_{k}),d_{1,2}(u;\zeta_{j},\zeta_{k})\Bigr{)}$
which reduce the generic operator (3.8) to the fermionic $R$-operator (2.8).

We omit everywhere below the superscript $(s)$ everywhere to simplify the
notations, and consider the trigonometric limit corresponding to the modulus
$k=0$. We stress, however, that we let the other parameters $\zeta_{i}$ to be
completely arbitrary. To further simplify our notations we also denote:

$\displaystyle\mathcal{L}^{\alpha_{1}\alpha_{2}\alpha_{3}}\equiv\mathrm{L}^{\alpha_{1}}_{12}\mathrm{L}^{\alpha_{2}}_{13}\mathrm{L}^{\alpha_{3}}_{23};\quad\tilde{\mathcal{L}}^{\beta_{1}\beta_{2}\beta_{3}}\equiv\mathrm{L}^{\beta_{1}}_{23}\mathrm{L}^{\beta_{2}}_{13}\mathrm{L}^{\beta_{3}}_{12}.$
(3.13)

In these notations the tetrahedral relations (3.7) take the form:

$\displaystyle\mathcal{L}^{\alpha_{1},\alpha_{2},\alpha_{3}}=\sum_{\beta_{i}=0,1}S^{\alpha_{1}\alpha_{2}\alpha_{3}}_{\beta_{1}\beta_{2}\beta_{3}}\tilde{\mathcal{L}}^{\beta_{1},\beta_{2},\beta_{3}};\quad\alpha_{1,2,3}=\\{0,1\\},$
(3.14)

One of the main results of this paper is the following solution to the
tetrahedral Zamolodchikov algebraic relations (3.14):

$\displaystyle\mathcal{L}^{000}$ $\displaystyle=\tilde{\mathcal{L}}^{000},$
$\displaystyle\mathcal{L}^{011}$ $\displaystyle=\tilde{\mathcal{L}}^{110},$
$\displaystyle\mathcal{L}^{110}$ $\displaystyle=\tilde{\mathcal{L}}^{011},$
$\displaystyle\mathcal{L}^{101}$ $\displaystyle=\tilde{\mathcal{L}}^{101},$
$\displaystyle\mathcal{L}^{111}$
$\displaystyle=\left(S^{111}_{010}\right)\tilde{\mathcal{L}}^{010}+\left(S^{111}_{001}\right)\tilde{\mathcal{L}}^{001}+\left(S^{111}_{100}\right)\tilde{\mathcal{L}}^{100}+\left(S^{111}_{110}\right)\tilde{\mathcal{L}}^{110}+\left(S^{111}_{011}\right)\tilde{\mathcal{L}}^{011}+\left(S^{111}_{111}\right)\tilde{\mathcal{L}}^{111},$
$\displaystyle\mathcal{L}^{001}$
$\displaystyle=\left(S^{001}_{010}\right)\tilde{\mathcal{L}}^{010}+\left(S^{001}_{001}\right)\tilde{\mathcal{L}}^{001}+\left(S^{001}_{100}\right)\tilde{\mathcal{L}}^{100}+\left(S^{001}_{110}\right)\tilde{\mathcal{L}}^{110}+\left(S^{001}_{011}\right)\tilde{\mathcal{L}}^{011}+\left(S^{001}_{111}\right)\tilde{\mathcal{L}}^{111},$
$\displaystyle\mathcal{L}^{010}$
$\displaystyle=\left(S^{010}_{010}\right)\tilde{\mathcal{L}}^{010}+\left(S^{010}_{001}\right)\tilde{\mathcal{L}}^{001}+\left(S^{010}_{100}\right)\tilde{\mathcal{L}}^{100}+\left(S^{010}_{110}\right)\tilde{\mathcal{L}}^{110}+\left(S^{010}_{011}\right)\tilde{\mathcal{L}}^{011}+\left(S^{010}_{111}\right)\tilde{\mathcal{L}}^{111},$
$\displaystyle\mathcal{L}^{100}$
$\displaystyle=\left(S^{100}_{001}\right)\tilde{\mathcal{L}}^{001}+\left(S^{100}_{001}\right)\tilde{\mathcal{L}}^{001}+\left(S^{100}_{100}\right)\tilde{\mathcal{L}}^{100}+\left(S^{100}_{110}\right)\tilde{\mathcal{L}}^{110}+\left(S^{100}_{011}\right)\tilde{\mathcal{L}}^{011}+\left(S^{100}_{111}\right)\tilde{\mathcal{L}}^{111},$

where the coefficients
$S^{\alpha_{1}\alpha_{2}\alpha_{3}}_{\beta_{1}\beta_{2}\beta_{3}}$ are given
in Appendix C.

In addition, we note that the set of operators
$\\{\mathcal{L}^{\alpha_{1}\alpha_{2}\alpha_{3}}\\}$ and
$\\{\tilde{\mathcal{L}}^{\beta_{1}\beta_{2}\beta_{3}}\\}$ are linearly
dependent. One can check, for example, the following relation between
$\tilde{\mathcal{L}}^{\alpha_{1}\alpha_{2}\alpha_{3}}$:

$\displaystyle\tilde{\mathcal{L}}^{111}$
$\displaystyle=\left(Y^{111}_{110}\right)\tilde{\mathcal{L}}^{110}+\left(Y^{111}_{011}\right)\tilde{\mathcal{L}}^{011}+\left(Y^{111}_{101}\right)\tilde{\mathcal{L}}^{101}+\left(Y^{111}_{001}\right)\tilde{\mathcal{L}}^{001}+\left(Y^{111}_{010}\right)\tilde{\mathcal{L}}^{010}+\left(Y^{111}_{100}\right)\tilde{\mathcal{L}}^{100}.$
(3.15)

The coefficients $Y^{111}_{\beta_{1}\beta_{2}\beta_{3}}$ are also given in
Appendix C.

To summarize this section, we have found a solution which depends only on the
differences $u_{ij}=u_{j}-u_{k}$ in one of the spectral parameters, while
leaving the other parameters $\zeta_{i}$ completely arbitrary. This is of
course the consequence of the decorated Yang-Baxter equations (3.2) having the
same dependence on $u_{ij}$, and is in contrast to the solution of the
tetrahedral Zamolodchikov algebra given in [34, 35, 22] where the solution for
the coefficients
$S^{\alpha_{1}\alpha_{2}\alpha_{3}}_{\beta_{1}\beta_{2}\beta_{3}}$ is not of
the difference type. We also note that in our case the linear space spanned by
$\tilde{\mathcal{L}}^{\beta_{1},\beta_{2},\beta_{3}}$ is 6-dimensional, as
opposed to 3-dimensional case of [34, 35, 22] (see also [6]). As we explained
in the introduction, the solution that depends on the differences of the
spectral parameters is natural for taking the continuous limit and obtaining
an integrable model of relativistic fermions.

It is an interesting problem whether the solution to the tetrahedral
Zamolodchikov algebraic relations (3.14) given in Appendix C can be reduced to
other known solutions, e.g., given in [34, 35, 22]. One can show, that for
some specific numerical values of the extra parameters $\zeta_{i}$, the
general 6-dimensional linear space spanned by
$\tilde{\mathcal{L}}^{\beta_{1},\beta_{2},\beta_{3}}$ indeed reduces to a
3-dimensional linear space. However, the coefficients
$S^{\alpha_{1}\alpha_{2}\alpha_{3}}_{\beta_{1}\beta_{2}\beta_{3}}$ do not
immediately match, being of the difference type for the solution given in this
paper, as opposed to the solutions given in the works cited above.

##  4 Extended Yang-Baxter equation

An immediate application of our solution is the possibility, as we show below,
to extend the fermionic $R$-operator and the corresponding Yang-Baxter
equation, which in turn will generalize the transfer operator in (2.13). The
latter, as we explain below, will depend not only on
$\\{\zeta_{1},\ldots,\zeta_{N}\\}$, but also on some arbitrary functions of
these parameters.

To this end, we look for an extension of the fermionic
$\mathrm{L}^{0,(s)}_{jk}$-operator (3.3) (i.e., the original fermionic
$R$-operator (2.8)) in the form:888Recall that we are considering the case
$k=0$; we omitted the dependence on the modulus $k$ in the left-hand side of
(4.1).

$\displaystyle\widetilde{\mathcal{R}}^{(s)}(u_{jk};\zeta_{j},\zeta_{k}):=\mathrm{L}^{0,(s)}_{jk}+c_{jk}\mathrm{L}^{1,(s)}_{jk}.$
(4.1)

For the moment, the coefficients $c_{jk}$ in (4.1) are arbitrary functions of
$u_{jk},\zeta_{j}$ and $\zeta_{k}$. The motivation to look for an
$\widetilde{\mathcal{R}}$-operator of the form (4.1) comes from the one-
dimensional Hubbard model. Recall, that the latter corresponds to two copies
of $XX$-models, together with an interaction term between the spins. In the
notations (2.9) it has the form[6]:

$\displaystyle\hat{\mathcal{H}}=-\sum_{j=1}^{N}\sum_{\,\,\,s=\uparrow\downarrow}\left(\Delta_{j+1,j(s)}+\Delta_{j,j+1(s)}\right)+\frac{\textrm{U}}{4}\sum_{j=1}^{N}\left(2n_{j\uparrow}-1\right)\left(2n_{j\downarrow}-1\right),$
(4.2)

where U is a constant describing the interaction strength. The key point is
the structure of the $\mathrm{L}^{1,(s)}_{jk}$ (3.4) operator: it is readily
verified (see (A.4)) that $\mathrm{L}^{1,(s)}_{jk}$ (3.4) produces a term
$\sim\mathcal{P}^{(s)}_{jk}(2n_{j(s)}-1)$.999Recall that
$\mathcal{P}^{(s)}_{jk}:=1-n_{j(s)}-n_{k(s)}+\Delta_{jk(s)}+\Delta_{kj(s)}$ is
the permutation operator for spin $s$. Thus, provided the extended
$\widetilde{\mathcal{R}}$-operator (4.1), one then can couple the spins as,
for example, in (2.22), leading to the term
$\sim\mathcal{P}^{(\uparrow)}_{jk}\mathcal{P}^{(\downarrow)}_{jk}(2n_{j(\uparrow)}-1)(2n_{(\downarrow)}-1)$,
which is precisely the term needed to obtain the interaction term in the one-
dimensional Hubbard model (4.2).

We next set the problem of fixing the functions
$c_{jk}(u_{jk},\zeta_{j},\zeta_{k})$ in (4.1), by requiring the $YBE$ to be
satisfied for the extended fermionic operator
$\widetilde{\mathcal{R}}^{(s)}(u_{jk};\zeta_{j},\zeta_{k})$:

$\displaystyle\widetilde{R}^{(s)}_{12}(u_{12};\zeta_{1},\zeta_{2})\widetilde{R}^{(s)}_{13}(u_{13};\zeta_{1},\zeta_{3})\widetilde{R}^{(s)}_{23}(u_{23};\zeta_{2},\zeta_{3})=\widetilde{R}^{(s)}_{23}(u_{23};\zeta_{2},\zeta_{3})\widetilde{R}^{(s)}_{13}(u_{13};\zeta_{1},\zeta_{3})\widetilde{R}^{(s)}_{12}(u_{12};\zeta_{1},\zeta_{2}).$
(4.3)

Substituting (4.1) into (4.3) and using the solution for the tetrahedral
Zamolodchikov algebra from the previous section one obtains, in principle, a
set of complex equations on the $c_{jk}$-functions - a brief look at Appendix
C shows the daunting task one is facing. We note, however, that the
coefficients
$S^{\alpha_{1}\alpha_{2}\alpha_{3}}_{\beta_{1}\beta_{2}\beta_{3}}$ are related
as follows:
$\displaystyle S^{111}_{001}$ $\displaystyle=S^{010}_{001}+Y^{111}_{001};$
$\displaystyle S^{111}_{011}$ $\displaystyle=S^{010}_{011}+Y^{111}_{011};$
$\displaystyle S^{111}_{100}$ $\displaystyle=S^{010}_{100}+Y^{111}_{100};$
$\displaystyle S^{111}_{101}$ $\displaystyle=S^{010}_{101}+Y^{111}_{101};$
$\displaystyle S^{111}_{110}$ $\displaystyle=S^{010}_{110}+Y^{111}_{110};$
$\displaystyle S^{111}_{010}$ $\displaystyle=S^{010}_{010}+Y^{111}_{010}-1;$
$\displaystyle S^{100}_{010}$ $\displaystyle=S^{001}_{010};$ $\displaystyle
S^{100}_{001}$ $\displaystyle=S^{001}_{001}+1;$ $\displaystyle S^{100}_{011}$
$\displaystyle=S^{001}_{011};$ $\displaystyle S^{100}_{101}$
$\displaystyle=S^{001}_{101};$ $\displaystyle S^{100}_{110}$
$\displaystyle=S^{001}_{110};$ $\displaystyle S^{100}_{100}$
$\displaystyle=S^{001}_{100}-1;$ $\displaystyle S^{001}_{001}$
$\displaystyle=-S^{010}_{001};$ $\displaystyle S^{001}_{010}$
$\displaystyle=1-S^{010}_{010};$ $\displaystyle S^{001}_{100}$
$\displaystyle=1-S^{010}_{100};$ $\displaystyle S^{001}_{011}$
$\displaystyle=-S^{010}_{011};$ $\displaystyle S^{001}_{101}$
$\displaystyle=-S^{010}_{101};$ $\displaystyle S^{001}_{110}$
$\displaystyle=-S^{010}_{110}.$
These relations suffice to show that the resulting set of conditions on the
$c_{jk}$-functions in (4.1) are reduced to a single equation:

$\displaystyle c_{23}=\frac{c_{12}-c_{13}}{c_{12}c_{13}-1}.$ (4.4)

The general solution of (4.4) is given by:

$\displaystyle
c_{jk}=\tanh\left[h_{j}(u_{j},\zeta_{j})-h_{k}(u_{k},\zeta_{k})\right],$ (4.5)

where $h_{j}(u_{j},\zeta_{j})$ are arbitrary functions.101010This fact is a
consequence of the modulus $k=0$ in our solution; we expect that in general,
for the case $k\neq 0$, a restriction on $h_{j}(u_{j},\zeta_{j})$ should
appear. Thus, we find the following solution for the extended
$\widetilde{\mathcal{R}}^{(s)}(u_{jk};\zeta_{j},\zeta_{k})$-operator (4.1):

$\displaystyle\widetilde{\mathcal{R}}^{(s)}(u_{jk};\zeta_{j},\zeta_{k}):=\mathrm{L}^{0,(s)}_{jk}+\tanh\left[h_{j}(u_{j},\zeta_{j})-h_{k}(u_{k},\zeta_{k})\right]\mathrm{L}^{1,(s)}_{jk}.$
(4.6)

If the dependence on the difference $u_{jk}=u_{j}-u_{k}$ is of no interest,
then one can consider the generic form of the solution in (4.5) and (4.6) and
investigate the consequences by considering the monodromy operator (4.7), the
transfer matrix (4.8), commuting quantities (4.9) and coupling the spins in
this general setting - for each choice of $h_{j}(u_{j},\zeta_{j})$.111111It
will likely be the case for the one-dimensional Hubbard model. The monodromy
and transfer operators are constructed in the usual manner (c.f. (2.11) -
(2.16)):

$\displaystyle\widetilde{\mathcal{T}}_{a}(u_{a},\\{u_{j}\\};\zeta_{a},\\{\zeta_{j}\\})=\widetilde{\mathcal{R}}_{aN}(u_{aN};\zeta_{a},\zeta_{N})\widetilde{\mathcal{R}}_{a,N-1}(u_{a,N-1};\zeta_{a},\zeta_{N-1})\cdot\ldots\cdot\widetilde{\mathcal{R}}_{a1}(u_{a1};\zeta_{a},\zeta_{1}),$
(4.7)
$\displaystyle\widetilde{{\tau}}\left(u_{a},\\{u_{j}\\};\zeta_{a},\\{\zeta_{j}\\}\right)=\mathrm{Str}_{a}\left[\widetilde{\mathcal{T}}_{a}\left(u_{a},\\{u_{j}\\};\zeta_{a},\\{\zeta_{j}\\}\right)\right],$
(4.8)

with:

$\displaystyle\left[\widetilde{\tau}\left(u_{a},\\{u_{j}\\};\zeta_{a},\\{\zeta_{j}\\}\right),\widetilde{\tau}\left(u_{b},\\{u_{j}\\};\zeta_{b},\\{\zeta_{j}\\}\right)\right]=0.$
(4.9)

We also note here that the transfer operator
$\widetilde{\tau}\left(u_{a},\\{u_{j}\\};\zeta_{a},\\{\zeta_{j}\\}\right)$
depends on two free parameters: $u_{a}$ and $\zeta_{a}$. Thus, one can explore
this freedom to construct the commuting quantities by using a more general
formula:

$\displaystyle\hat{{I}}^{(n,k)}$
$\displaystyle\sim\frac{d^{(n)}}{du_{a}^{(n)}}\frac{d^{(k)}}{d\zeta_{a}^{(k)}}\widetilde{\tau}\left(u_{a},\\{u_{j}\\};\zeta_{a},\\{\zeta_{j}\\}\right)\Big{|}_{\begin{subarray}{c}u_{j}\to
u_{a}\\\ \zeta_{j}\to\zeta_{a}\end{subarray}}.$ (4.10)

On the other hand, for the extended
$\widetilde{\mathcal{R}}^{(s)}(u_{jk};\zeta_{j},\zeta_{k})$-operator (4.1) to
be of the difference type, one must, however, further restrict the
$h_{j}(u_{j},\zeta_{j})$ functions and the solution (4.4) as follows:

$\displaystyle
c_{jk}=\tanh\left[\alpha(u_{j}-u_{k})-\left\\{g_{j}(\zeta_{j})-g_{k}(\zeta_{k})\right\\}\right],$
(4.11)

where $\alpha$ is an arbitrary constant, and $g_{j}(\zeta_{j});\,j=1,2,3$ are
still arbitrary functions of $\zeta_{j}$. Thus, in this case the extended
$\widetilde{\mathcal{R}}^{(s)}(u_{jk};\zeta_{j},\zeta_{k})$-operator (4.1)
takes the form:121212Note, that by using the inverse Jordan-Wigner
transformation one can show that the fermionic
$\widetilde{\mathcal{R}}^{(s)}(u_{jk};\zeta_{j},\zeta_{k})$-operator satisfies
the free-fermion condition (2.2).

$\displaystyle\widetilde{\mathcal{R}}^{(s)}(u_{jk};\zeta_{j},\zeta_{k}):=\mathrm{L}^{0,(s)}_{jk}+\tanh\left[\alpha(u_{j}-u_{k})-\left\\{g_{j}(\zeta_{j})-g_{k}(\zeta_{k})\right\\}\right]\mathrm{L}^{1,(s)}_{jk}.$
(4.12)

The above comments are also valid for the systems with coupled spins, which
can be obtained by using the
$\widetilde{\mathcal{R}}^{(s)}(u_{jk};\zeta_{j},\zeta_{k})$-operator (4.6) and
the corresponding $YBE$ (4.3). The simplest case is to proceed as in (2.22)
using the general extended :

$\displaystyle{\bm{R}}_{jk}(u_{jk};\zeta_{j},\zeta_{k};\hat{\zeta}_{j},\hat{\zeta}_{k};h_{1},h_{2},\hat{h}_{1},\hat{h}_{2}):=\widetilde{\mathcal{R}}^{(\uparrow)}_{jk}(u_{jk};\zeta_{j},\zeta_{k};h_{1},h_{2})\widetilde{\mathcal{R}}^{(\downarrow)}_{jk}(u_{jk};\hat{\zeta}_{j},\hat{\zeta}_{k};\hat{h}_{1},\hat{h}_{2}).$
(4.13)

We have written explicitly, in the above formula, the dependencies on the
arbitrary functions $h_{1},h_{2}$ (for spin $\uparrow$) and
$\hat{h}_{1},\hat{h}_{2}$ (for spin $\downarrow$). The
${\bm{R}}_{jk}(u_{jk};\zeta_{j},\zeta_{k};\hat{\zeta}_{j},\hat{\zeta}_{k};h_{1},h_{2},\hat{h}_{1},\hat{h}_{2})$-operator
satisfies the corresponding $YBE$ obtained by taking the product of (4.3) for
spin $\uparrow$ and spin $\downarrow$. The transfer operator
$\widetilde{{\bm{\tau}}}(u_{a},\\{u_{j}\\};\zeta_{a},\\{\zeta_{j}\\})$ can
then be defined and the local conserved quantities may be extracted as in the
above case for the spinless fermions by means of the formula (4.10).

To summarize, by finding the most general solution to the tetrahedral
Zamolodchikov algebra in the trigonometric limit, we have found an extension
for the fermionic $R$-operator of the difference type which led to an extended
Yang-Baxter equation and the transfer operator depending on two free
parameters.

One can also try to construct different extensions, for example of the
following type: (c.f. [32, 23, 22, 6]):

$\displaystyle\mathfrak{R}_{jk}=L^{0(\uparrow)}_{jk}L^{0(\downarrow)}_{jk}+\sigma_{jk}L^{1(\uparrow)}_{jk}L^{1(\downarrow)}_{jk}.$
(4.14)

This should be now possible to analyze, using the explicit solution for the
tetrahedral Zamolodchikov algebra given in Appendix C.

Another open problem is the elliptic case with $k\neq 0$. Despite being
technically more complex, the task is now more feasible, given the explicit
formulas in Appendix B. These and related questions will be investigated in a
future publication.

##  Appendix A Appendix

In this appendix we list some useful identities and formulas used to calculate
the tetrahedral Zamolodchikov algebra. Starting from the fermionic
$R$-operator (2.8):

$\displaystyle R_{jk}(u;\zeta_{j},\zeta_{k})$
$\displaystyle=a(u;\zeta_{j},\zeta_{k})\left[-n_{j}n_{k}\right]+a(u;\zeta_{j},\zeta_{k})\left[(1-n_{j})(1-n_{k})\right]+b(u;\zeta_{j},\zeta_{k})\left[n_{j}(1-n_{k})\right]$
$\displaystyle+b^{\prime}(u;\zeta_{j},\zeta_{k})\left[n_{k}(1-n_{j})\right]+c(u;\zeta_{j},\zeta_{k})\left[\Delta_{jk}+\Delta_{kj}\right]+d(u;\zeta_{j},\zeta_{k})\left[-\tilde{\Delta}^{(\dagger)}_{jk}-\tilde{\Delta}_{jk}\right],$
(A.1)

one can show the following identities used to derive the relation (3.1), as
well as the decorated Yang-Baxter equation (3.2):

$\displaystyle(2n_{j}-1)R_{jk}(u;\zeta_{j},\zeta_{k})(2n_{k}-1)$
$\displaystyle=(2n_{k}-1)R_{jk}(u;\zeta_{j},\zeta_{k})(2n_{j}-1),$ (A.2)
$\displaystyle(2n_{j}-1)R_{jk}(u;\zeta_{j},\zeta_{k})(2n_{j}-1)$
$\displaystyle=(2n_{k}-1)R_{jk}(u;\zeta_{j},\zeta_{k})(2n_{k}-1).$ (A.3)

In addition, one has the following relations:

$\displaystyle R_{jk}(u;\zeta_{j},\zeta_{k})(2n_{j}-1)$
$\displaystyle=a(u;\zeta_{j},\zeta_{k})\left[-n_{j}n_{k}\right]-a(u;\zeta_{j},\zeta_{k})\left[(1-n_{j})(1-n_{k})\right]+b(u;\zeta_{j},\zeta_{k})\left[n_{j}(1-n_{k})\right]$
$\displaystyle-b^{\prime}(u;\zeta_{j},\zeta_{k})\left[n_{k}(1-n_{j})\right]+c(u;\zeta_{j},\zeta_{k})\left[-\Delta_{jk}+\Delta_{kj}\right]+d(u;\zeta_{j},\zeta_{k})\left[\tilde{\Delta}^{(\dagger)}_{jk}-\tilde{\Delta}_{jk}\right],$
(A.4) $\displaystyle(2n_{j}-1)R_{jk}(u;\zeta_{j},\zeta_{k})$
$\displaystyle=a(u;\zeta_{j},\zeta_{k})\left[-n_{j}n_{k}\right]-a(u;\zeta_{j},\zeta_{k})\left[(1-n_{j})(1-n_{k})\right]+b(u;\zeta_{j},\zeta_{k})\left[n_{j}(1-n_{k})\right]$
$\displaystyle-b^{\prime}(u;\zeta_{j},\zeta_{k})\left[n_{k}(1-n_{j})\right]+c(u;\zeta_{j},\zeta_{k})\left[\Delta_{jk}-\Delta_{kj}\right]+d(u;\zeta_{j},\zeta_{k})\left[-\tilde{\Delta}^{(\dagger)}_{jk}+\tilde{\Delta}_{jk}\right],$
(A.5)

##  Appendix B The equation for the tetrahedral Zamolodchikov algebra

Below we list the set of the coefficients in the expansions for $\Omega$
(3.11) and $\tilde{\Omega}$ (3.12), which are necessary to obtain the set of
equations for the tetrahedral Zamolodchikov algebra. We use here the following
shorthand notations:

$\displaystyle u$ $\displaystyle:=(u;\zeta_{1},\zeta_{3});\quad
v:=(u;\zeta^{\prime}_{2},\zeta^{\prime}_{3});\quad
w:=(w;\zeta^{\prime\prime}_{1},\zeta^{\prime\prime}_{2}).$

###  B.1 $\Omega:$

$\displaystyle\Gamma^{(0)}_{0}$
$\displaystyle=a_{0}(w)a_{0}(u)a_{0}(v)+d_{2}(w)c_{1}(u)d_{1}(v);$
$\displaystyle\Gamma^{(0)}_{n_{1}}$
$\displaystyle=a_{0}(v)\left\\{a_{0}(w)a_{1}(u)+a_{1}(w)a_{0}(u)+a_{1}(w)a_{1}(u)\right\\}+c_{1}(w)d_{2}(u)d_{1}(v)-d_{2}(w)c_{1}(u)d_{1}(v);$
$\displaystyle\Gamma^{(0)}_{n_{2}}$
$\displaystyle=a_{0}(u)\left\\{a_{0}(w)a_{1}(v)+a_{2}(w)a_{0}(v)+a_{2}(w)a_{1}(v)\right\\}+c_{2}(w)c_{1}(u)c_{2}(v)-d_{2}(w)c_{1}(u)d_{1}(v);$
$\displaystyle\Gamma^{(0)}_{n_{3}}$
$\displaystyle=a_{0}(w)\left\\{a_{0}(u)a_{2}(v)+a_{2}(u)a_{0}(v)+a_{2}(u)a_{2}(v)\right\\}-d_{2}(w)c_{1}(u)d_{1}(v)+d_{2}(w)d_{1}(u)c_{1}(v);$
$\displaystyle\Gamma^{(0)}_{n_{1}n_{2}}$
$\displaystyle=a_{0}(w)a_{1}(u)a_{1}(v)+a_{1}(w)a_{1}(v)\left\\{a_{0}(u)+a_{1}(u)\right\\}-c_{1}(w)d_{2}(u)d_{1}(v)-c_{2}(w)c_{1}(u)c_{2}(v)$
$\displaystyle\left\\{a_{2}(w)a_{1}(u)+a_{3}(w)a_{0}(u)+a_{3}(w)a_{1}(u)\right\\}\left\\{a_{0}(v)+a_{1}(v)\right\\}-d_{1}(w)d_{2}(u)c_{2}(v)+d_{2}(w)c_{1}(u)d_{1}(v);$
$\displaystyle\Gamma^{(0)}_{n_{1}n_{3}}$
$\displaystyle=\left\\{a_{0}(w)+a_{1}(w)\right\\}\left\\{a_{1}(u)a_{2}(v)+a_{3}(u)a_{0}(v)+a_{3}(u)a_{2}(v)\right\\}-c_{1}(w)c_{2}(u)c_{1}(v)-c_{1}(w)d_{2}(w)d_{1}(v)$
$\displaystyle+a_{1}(w)\left\\{a_{0}(u)a_{2}(v)+a_{2}(u)a_{0}(v)+a_{2}(u)a_{2}(v)\right\\}-d_{2}(w)c_{1}(u)d_{1}(v)-d_{2}(w)d_{1}(u)c_{1}(v);$
$\displaystyle\Gamma^{(0)}_{n_{2}n_{3}}$
$\displaystyle=\left\\{a_{0}(w)+a_{2}(w)\right\\}\left\\{a_{0}(u)a_{3}(v)+a_{2}(u)a_{1}(v)+a_{2}(u)a_{3}(v)\right\\}-c_{2}(w)c_{1}(u)c_{2}(v)-c_{2}(w)d_{1}(u)d_{2}(v)$
$\displaystyle+a_{2}(w)\left\\{a_{0}(u)a_{2}(v)+a_{2}(u)a_{0}(v)+a_{2}(u)a_{2}(v)\right\\}+d_{2}(w)c_{1}(u)d_{1}(v)-d_{2}(w)d_{1}(u)c_{1}(v);$
$\displaystyle\Gamma^{(0)}_{n_{1}n_{2}n_{3}}$
$\displaystyle=a_{0}(w)\left\\{a_{1}(u)a_{3}(v)+a_{3}(u)a_{1}(v)+a_{3}(u)a_{3}(v)\right\\}+a_{1}(w)\left\\{a_{0}(u)a_{3}(v)+a_{2}(u)a_{1}(v)+a_{2}(u)a_{3}(v)\right.$
$\displaystyle\left.+a_{1}(u)a_{3}(v)+a_{3}(u)a_{1}(v)+a_{3}(u)a_{3}(v)\right\\}+a_{2}(w)\left\\{a_{1}(u)a_{2}(v)+a_{3}(u)a_{0}(v)+a_{3}(u)a_{2}(v)\right.$
$\displaystyle\left.+a_{1}(u)a_{3}(v)+a_{3}(u)a_{1}(v)+a_{3}(u)a_{3}(v)\right\\}+a_{3}(w)\left\\{a_{0}(u)a_{2}(v)+v)a_{2}(u)a_{0}(v)+a_{2}(u)a_{2}(v)\right.$
$\displaystyle\left.+a_{1}(u)a_{2}(v)+a_{3}(u)a_{0}(v)+a_{3}(u)a_{2}(v)+a_{0}(u)a_{3}(v)+a_{2}(u)a_{1}(v)+a_{2}(u)a_{3}(v)+a_{1}(u)a_{3}(v)\right.$
$\displaystyle\left.+a_{3}(u)a_{1}(v)+a_{3}(u)a_{3}(v)\right\\}+c_{1}(w)c_{2}(u)c_{1}(v)+c_{1}(w)d_{2}(u)d_{1}(v)+c_{2}(w)c_{1}(u)c_{2}(v)$
$\displaystyle+c_{2}(w)d_{1}(u)d_{2}(v)+d_{1}(w)d_{2}(u)c_{2}(v)-d_{1}(w)c_{2}(u)d_{2}(v)+d_{2}(w)d_{1}(u)c_{1}(v)$
$\displaystyle-d_{2}(w)c_{1}(u)c_{1}(v);$
$\displaystyle\Gamma^{(1)}_{\Delta_{23}}$
$\displaystyle=a_{0}(u)c_{1}(v)\left\\{a_{0}(w)+a_{2}(w)\right\\}+c_{2}(w)c_{1}(u)\left\\{a_{0}(v)+a_{2}(v)\right\\};$
$\displaystyle\Gamma^{(1)}_{\tilde{\Delta}^{(+)}_{23}}$
$\displaystyle=d_{1}(v)\left\\{a_{0}(u)+a_{2}(u)\right\\}\left\\{a_{0}(w)+a_{2}(w)\right\\}+c_{2}(w)d_{1}(u)a_{0}(v);$
$\displaystyle\Gamma^{(1)}_{\tilde{\Delta}_{23}}$
$\displaystyle=a_{0}(w)a_{0}(u)d_{2}(v)-d_{2}(w)c_{1}(u)\left\\{a_{0}(v)+a_{1}(v)+a_{2}(v)+a_{3}(v)\right\\};$
$\displaystyle\Gamma^{(1)}_{{\Delta}_{13}}$
$\displaystyle=c_{1}(u)\left\\{a_{0}(w)+a_{1}(w)\right\\}\left\\{a_{0}(v)+a_{2}(v)\right\\}+c_{1}(w)a_{0}(u)c_{1}(v);$
$\displaystyle\Gamma^{(1)}_{{\Delta}_{31}}$
$\displaystyle=a_{0}(w)c_{2}(u)a_{0}(v)-d_{2}(w)d_{1}(v)\left\\{a_{0}(u)+a_{1}(u)+a_{2}(u)+a_{3}(u)\right\\};$
$\displaystyle\Gamma^{(1)}_{\tilde{\Delta}^{(+)}_{13}}$
$\displaystyle=a_{0}(v)d_{1}(u)\left\\{a_{0}(w)+a_{1}(w)\right\\}+c_{1}(w)d_{1}(v)\left\\{a_{0}(u)+a_{2}(u\right\\};$
$\displaystyle\Gamma^{(1)}_{\tilde{\Delta}_{13}}$
$\displaystyle=a_{0}(w)d_{2}(u)\left\\{a_{0}(v)+a_{2}(v)\right\\}+d_{2}(w)c_{1}(v)\left\\{a_{0}(u)+a_{1}(u)\right\\};$
$\displaystyle\Gamma^{(1)}_{n_{1}{\Delta}_{23}}$
$\displaystyle=a_{1}(u)c_{1}(v)\left\\{a_{0}(w)+a_{1}(w)+a_{2}(w)+a_{3}(w)\right\\}+a_{0}(u)c_{1}(v)\left\\{a_{1}(w)+a_{3}(w)\right\\}$
$\displaystyle-\left\\{c_{2}(w)c_{1}(u)+d_{1}(w)d_{2}(u)\right\\}\left\\{a_{0}(v)+a_{2}(v)\right\\};$
$\displaystyle\Gamma^{(1)}_{n_{1}{\tilde{\Delta}}_{23}}$
$\displaystyle=\left\\{a_{0}(w)+a_{1}(w)\right\\}a_{1}(u)d_{2}(v)+a_{1}(w)a_{0}(u)d_{2}(v)+\left\\{d_{2}(w)c_{1}(u)-c_{1}(w)d_{2}(u)\right\\}\times$
$\displaystyle\times\left\\{a_{0}(v)+a_{1}(v)+a_{2}(v)+a_{3}(v)\right\\};$
$\displaystyle\Gamma^{(1)}_{n_{1}{\tilde{\Delta}}^{(+)}_{23}}$
$\displaystyle=d_{1}(v)\left\\{a_{1}(u)+a_{3}(u)\right\\}\left\\{a_{0}(w)+a_{1}(w)+a_{2}(w)+a_{3}(w)\right\\}+d_{1}(w)c_{2}(u)a_{0}(v)$
$\displaystyle+d_{1}(v)\left\\{a_{0}(u)+a_{2}(u)\right\\}\left\\{a_{1}(w)+a_{3}(w)\right\\}-c_{2}(w)d_{1}(u)a_{0}(v);$
$\displaystyle\Gamma^{(1)}_{n_{1}{{\Delta}}_{32}}$
$\displaystyle=c_{2}(v)\left\\{a_{1}(u)+a_{3}(u)\right\\}\left\\{a_{0}(w)+a_{1}(w)\right\\}+a_{1}(w)c_{2}(v)\left\\{a_{0}(u)+a_{2}(u)\right\\}$
$\displaystyle-\left\\{c_{1}(w)c_{2}(u)+d_{2}(w)d_{1}(u)\right\\}\left\\{a_{0}(v)+a_{1}(v)\right\\};$
$\displaystyle\Gamma^{(1)}_{n_{2}{{\Delta}}_{13}}$
$\displaystyle=c_{1}(u)\left\\{a_{0}(w)+a_{1}(w)+a_{2}(w)+a_{3}(w)\right\\}+c_{1}(u)\left\\{a_{0}(v)+a_{2}(v)\right\\}\left\\{a_{2}(w)+a_{3}(w)\right\\}$
$\displaystyle-c_{1}(w)a_{0}(u)c_{1}(v)+d_{1}(w)a_{0}(u)d_{2}(v);$
$\displaystyle\Gamma^{(1)}_{n_{2}{{\Delta}}_{31}}$
$\displaystyle=c_{2}(u)a_{1}(v)\left\\{a_{0}(w)+a_{2}(w)\right\\}+a_{2}(w)c_{2}(u)a_{0}(v)+\left\\{d_{2}(w)d_{1}(v)-c_{2}(w)c_{2}(v)\right\\}\times$
$\displaystyle\times\left\\{a_{0}(u)+a_{1}(u)+a_{2}(u)+a_{3}(u)\right\\};$
$\displaystyle\Gamma^{(1)}_{{n_{2}\tilde{\Delta}}^{(+)}_{13}}$
$\displaystyle=d_{1}(u)a_{1}(v)\left\\{a_{0}(w)+a_{1}(w)+a_{2}(w)+a_{3}(w)\right\\}+d_{1}(u)a_{0}(v)\left\\{a_{2}(w)+a_{3}(w)\right\\}$
$\displaystyle-
c_{1}(w)d_{1}(v)\left\\{a_{0}(u)+a_{2}(u)\right\\}-d_{1}(w)c_{2}(v)\left\\{a_{0}(u)+a_{2}(u)\right\\};$
$\displaystyle\Gamma^{(1)}_{{n_{2}\tilde{\Delta}}_{13}}$
$\displaystyle=d_{2}(u)\left\\{a_{1}(v)+a_{3}(v)\right\\}\left\\{a_{0}(w)+a_{2}(w)\right\\}+a_{2}(w)d_{2}(u)\left\\{a_{0}(v)+a_{2}(v)\right\\}$
$\displaystyle-\left\\{c_{2}(w)d_{2}(v)+d_{2}(w)c_{1}(v)\right\\}\left\\{a_{0}(u)+a_{1}(u)\right\\};$
$\displaystyle\Gamma^{(1)}_{{\Delta}_{12}}$
$\displaystyle=c_{1}(u)c_{2}(v)\left\\{a_{0}(w)+a_{1}(w)\right\\}+c_{1}(w)a_{0}(u)\left\\{a_{0}(v)+a_{1}(v)\right\\};$
$\displaystyle\Gamma^{(1)}_{{n_{3}{\Delta}}_{12}}$
$\displaystyle=-\left\\{c_{1}(u)c_{2}(v)+d_{1}(u)d_{2}(v)\right\\}\left\\{a_{0}(w)+a_{1}(w)\right\\}+c_{1}(w)a_{0}(u)\left\\{a_{2}(v)+a_{3}(v)\right\\}$
$\displaystyle+c_{1}(w)a_{2}(u)\left\\{a_{0}(v)+a_{1}(v)+a_{2}(v)+a_{3}(v)\right\\};$
$\displaystyle\Gamma^{(1)}_{{\tilde{\Delta}}^{(+)}_{12}}$
$\displaystyle=-c_{1}(u)d_{1}(v)\left\\{a_{0}(w)+a_{1}(w)+a_{2}(w)+a_{3}(w)\right\\}+a_{0}(u)a_{0}(v)d_{1}(w);$
$\displaystyle\Gamma^{(1)}_{n_{3}{\tilde{\Delta}}^{(+)}_{12}}$
$\displaystyle=\left\\{c_{1}(u)d_{1}(v)-d_{1}(u)c_{1}(v)\right\\}\left\\{a_{0}(w)+a_{1}(w)+a_{2}(w)+a_{3}(w)\right\\}+d_{1}(w)a_{0}(u)a_{2}(v)$
$\displaystyle+d_{1}(w)a_{2}(u)\left\\{a_{0}(v)+a_{2}(v)\right\\};$
$\displaystyle\Gamma^{(1)}_{{\tilde{\Delta}}_{12}}$
$\displaystyle=a_{0}(w)d_{2}(u)c_{2}(v)+\left\\{d_{2}(w)a_{0}(v)+d_{2}(w)a_{1}(v)\right\\}\left\\{a_{0}(u)+a_{1}(v)\right\\};$
$\displaystyle\Gamma^{(1)}_{n_{3}{\tilde{\Delta}}_{12}}$
$\displaystyle=a_{0}(w)\left\\{c_{2}(u)d_{2}(v)-d_{2}(u)c_{2}(v)\right\\}+d_{2}(w)\left\\{a_{0}(v)+a_{1}(v)\right\\}\left\\{a_{2}(u)+a_{3}(u)\right\\}$
$\displaystyle+d_{2}(w)\left\\{a_{2}(v)+a_{3}(v)\right\\}\left\\{a_{0}(u)+a_{1}(u)+a_{2}(u)+a_{3}(u)\right\\};$
$\displaystyle\Gamma^{(1)}_{{{\Delta}}_{21}}$
$\displaystyle=d_{2}(u)d_{1}(v)\left\\{a_{0}(w)+a_{2}(w)\right\\}+c_{2}(w)a_{0}(v)\left\\{a_{0}(u)+a_{1}(u)\right\\};$
$\displaystyle\Gamma^{(1)}_{{n_{3}{\Delta}}_{21}}$
$\displaystyle=-\left\\{d_{2}(u)d_{1}(v)+c_{2}(u)c_{1}(v)\right\\}\left\\{a_{0}(w)+a_{2}(w)\right\\}+c_{2}(w)a_{2}(v)\left\\{a_{0}(u)+a_{1}(u)\right\\}$
$\displaystyle+c_{2}(w)\left\\{a_{0}(v)+a_{2}(v)\right\\}\left\\{a_{2}(u)+a_{3}(v)\right\\};$
$\displaystyle\Gamma^{(1)}_{{{\Delta}}_{32}}$
$\displaystyle=a_{0}(w)c_{2}(v)\left\\{a_{0}(u)+a_{2}(u)\right\\}+d_{2}(w)d_{1}(u)\left\\{a_{0}(v)+a_{1}(v)\right\\}.$

###  B.2 $\tilde{\Omega}:$

$\displaystyle\tilde{\Gamma}^{(0)}_{0}$
$\displaystyle=a_{0}(v)a_{0}(u)a_{0}(w)+d_{2}(v)c_{2}(u)d_{1}(w);$
$\displaystyle\tilde{\Gamma}^{(0)}_{n_{1}}$
$\displaystyle=a_{0}(v)\left\\{a_{1}(u)a_{0}(w)+a_{0}(u)a_{1}(w)+a_{1}(u)a_{1}(w)\right\\}+d_{2}(v)d_{1}(u)c_{2}(w)-d_{2}(v)c_{2}(u)d_{1}(w);$
$\displaystyle\tilde{\Gamma}^{(0)}_{n_{2}}$
$\displaystyle=a_{0}(u)\left\\{a_{1}(v)a_{0}(w)+a_{0}(v)a_{2}(w)+a_{1}(v)a_{2}(w)\right\\}+c_{1}(v)c_{2}(u)c_{1}(w)-d_{2}(v)c_{2}(u)d_{1}(w);$
$\displaystyle\tilde{\Gamma}^{(0)}_{n_{3}}$
$\displaystyle=a_{0}(w)\left\\{a_{0}(v)a_{2}(u)+a_{2}(v)a_{0}(u)+a_{2}(v)a_{2}(u)\right\\}-d_{2}(v)c_{2}(u)d_{1}(w)+c_{2}(w)d_{2}(u)d_{1}(w);$
$\displaystyle\tilde{\Gamma}^{(0)}_{n_{1}n_{2}}$
$\displaystyle=a_{1}(v)a_{1}(u)a_{0}(w)+a_{1}(v)a_{1}(w)\left\\{a_{0}(u)+a_{1}(u)\right\\}-c_{1}(v)c_{2}(u)c_{1}(w)-c_{1}(v)d_{1}(u)d_{2}(w)$
$\displaystyle\left\\{a_{0}(v)+a_{1}(v)\right\\}\left\\{a_{1}(u)a_{2}(w)+a_{0}(u)a_{3}(w)+a_{1}(u)a_{3}(w)\right\\}+d_{2}(v)c_{2}(u)d_{1}(w)-d_{2}(v)d_{1}(u)c_{2}(w);$
$\displaystyle\tilde{\Gamma}^{(0)}_{n_{1}n_{3}}$
$\displaystyle=\left\\{a_{0}(w)+a_{1}(w)\right\\}\left\\{a_{0}(v)a_{3}(u)+a_{2}(v)a_{1}(u)+a_{2}(v)a_{3}(u)\right\\}-c_{2}(v)c_{1}(u)c_{2}(w)-c_{2}(v)d_{2}(u)d_{1}(w)$
$\displaystyle+a_{1}(w)\left\\{a_{0}(v)a_{2}(u)+a_{2}(v)a_{0}(u)+a_{2}(v)a_{2}(u)\right\\}+d_{2}(v)c_{2}(u)d_{1}(w)-d_{2}(v)d_{1}(u)c_{2}(w);$
$\displaystyle\tilde{\Gamma}^{(0)}_{n_{2}n_{3}}$
$\displaystyle=\left\\{a_{0}(w)+a_{2}(w)\right\\}\left\\{a_{1}(v)a_{2}(u)+a_{3}(v)a_{0}(u)+a_{3}(u)a_{2}(v)\right\\}-c_{1}(v)c_{2}(u)c_{1}(w)-c_{2}(v)d_{2}(u)d_{1}(w)$
$\displaystyle+a_{2}(w)\left\\{a_{0}(v)a_{2}(u)+a_{2}(v)a_{0}(u)+a_{2}(v)a_{2}(u)\right\\}-d_{1}(v)d_{2}(u)c_{1}(w)+d_{2}(v)c_{2}(u)d_{1}(w);$
$\displaystyle\tilde{\Gamma}^{(0)}_{n_{1}n_{2}n_{3}}$
$\displaystyle=a_{0}(w)\left\\{a_{1}(v)a_{3}(u)+a_{3}(v)a_{1}(u)+a_{3}(v)a_{3}(u)\right\\}+a_{1}(w)\left\\{a_{1}(v)a_{2}(u)+a_{3}(v)a_{0}(u)+a_{3}(v)a_{2}(u)\right.$
$\displaystyle\left.+a_{1}(v)a_{3}(u)+a_{3}(v)a_{1}(u)+a_{3}(v)a_{3}(u)\right\\}+a_{2}(w)\left\\{a_{0}(v)a_{3}(u)+a_{2}(v)a_{1}(u)+a_{2}(v)a_{3}(u)\right.$
$\displaystyle\left.+a_{1}(v)a_{3}(u)+a_{3}(v)a_{1}(u)+a_{3}(v)a_{3}(u)\right\\}+a_{3}(w)\left\\{a_{0}(v)a_{2}(u)+a_{2}(v)a_{0}(u)+a_{2}(v)a_{2}(u)\right.$
$\displaystyle\left.+a_{1}(v)a_{2}(u)+a_{3}(v)a_{0}(u)+a_{3}(v)a_{2}(u)+a_{0}(v)a_{3}(u)+a_{2}(v)a_{1}(u)+a_{2}(v)a_{3}(u)+a_{1}(v)a_{3}(u)\right.$
$\displaystyle\left.+a_{3}(v)a_{1}(u)+a_{3}(v)a_{3}(u)\right\\}+c_{1}(v)c_{2}(u)c_{1}(w)+d_{1}(v)d_{2}(u)c_{1}(w)+d_{2}(v)d_{1}(u)c_{2}(w)$
$\displaystyle+c_{2}(v)c_{1}(u)c_{2}(w)+c_{2}(v)d_{2}(u)d_{1}(w)-d_{2}(v)c_{2}(u)d_{1}(w)+c_{1}(v)d_{1}(u)d_{2}(w)$
$\displaystyle-d_{1}(v)c_{1}(u)d_{2}(w);$
$\displaystyle\tilde{\Gamma}^{(1)}_{\Delta_{23}}$
$\displaystyle=a_{0}(w)c_{1}(v)\left\\{a_{0}(u)+a_{2}(u)\right\\}+d_{2}(u)d_{1}(w)\left\\{a_{0}(v)+a_{1}(v)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{\tilde{\Delta}^{(+)}_{23}}$
$\displaystyle=d_{1}(v)a_{0}(u)a_{0}(w)-d_{1}(w)c_{2}(u)\left\\{a_{0}(v)+a_{1}(v)+a_{2}(v)+a_{3}(v)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{\tilde{\Delta}_{23}}$
$\displaystyle=d_{2}(v)\left\\{a_{0}(u)+a_{2}(u)\right\\}\left\\{a_{0}(w)+a_{2}(w)\right\\}+a_{0}(v)d_{2}(u)c_{1}(w);$
$\displaystyle\tilde{\Gamma}^{(1)}_{{\Delta}_{13}}$
$\displaystyle=a_{0}(v)c_{1}(u)a_{0}(w)-d_{2}(v)d_{1}(w)\left\\{a_{0}(u)+a_{1}(u)+a_{2}(u)+a_{3}(u)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{{\Delta}_{31}}$
$\displaystyle=c_{2}(u)\left\\{a_{0}(v)+c_{2}(v)\right\\}\left\\{a_{0}(w)+a_{1}(w)\right\\}+c_{2}(v)a_{0}(u)c_{2}(w);$
$\displaystyle\tilde{\Gamma}^{(1)}_{\tilde{\Delta}^{(+)}_{13}}$
$\displaystyle=a_{0}(w)d_{1}(u)\left\\{a_{0}(v)+a_{2}(v)\right\\}+c_{2}(v)d_{1}(w)\left\\{a_{0}(u)+a_{1}(u)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{\tilde{\Delta}_{13}}$
$\displaystyle=a_{0}(v)d_{2}(u)\left\\{a_{0}(w)+a_{1}(w)\right\\}+d_{2}(v)c_{2}(w)\left\\{a_{0}(u)+a_{2}(u)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{n_{1}{\Delta}_{23}}$
$\displaystyle=c_{1}(v)\left\\{a_{1}(u)+a_{3}(u)\right\\}\left\\{a_{0}(w)+a_{1}(w)\right\\}-c_{1}(u)c_{2}(w)\left\\{a_{0}(v)+a_{1}(v)\right\\}$
$\displaystyle-
d_{2}(u)d_{1}(w)\left\\{a_{0}(v)+a_{1}(v)\right\\}+c_{1}(v)a_{1}(w)\left\\{a_{0}(u)+a_{2}(u)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{n_{1}{\tilde{\Delta}}_{23}}$
$\displaystyle=d_{2}(v)\left\\{a_{1}(u)+a_{3}(u)\right\\}\left\\{a_{0}(w)+a_{1}(w)+a_{2}(w)+a_{3}(w)\right\\}+a_{0}(v)c_{1}(u)d_{2}(w)$
$\displaystyle+d_{2}(v)\left\\{a_{0}(u)+a_{2}(u)\right\\}\left\\{a_{1}(w)+a_{3}(w)\right\\}-a_{0}(v)d_{2}(u)c_{1}(w);$
$\displaystyle\tilde{\Gamma}^{(1)}_{n_{1}{\tilde{\Delta}}^{(+)}_{23}}$
$\displaystyle=d_{1}(v)a_{1}(u)\left\\{a_{0}(w)+a_{1}(w)\right\\}+d_{1}(v)a_{0}(u)a_{1}(w)$
$\displaystyle+\left\\{d_{1}(w)c_{2}(u)-c_{2}(w)d_{1}(u)\right\\}\left\\{a_{0}(v)+a_{1}(v)+a_{2}(v)+a_{3}(v)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{n_{1}{{\Delta}}_{32}}$
$\displaystyle=c_{2}(v)a_{1}(u)\left\\{a_{0}(w)+a_{1}(w)+a_{2}(w)+a_{3}(w)\right\\}+c_{2}(v)a_{0}(u)\left\\{a_{1}(w)+a_{3}(w)\right\\}$
$\displaystyle-\left\\{c_{2}(u)c_{1}(w)+d_{1}(u)d_{2}(w)\right\\}\left\\{a_{0}(v)+a_{2}(v)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{{n_{2}{\Delta}}_{13}}$
$\displaystyle=a_{1}(v)c_{1}(u)\left\\{a_{0}(w)+a_{2}(w)\right\\}+a_{2}(w)a_{0}(v)c_{1}(u)$
$\displaystyle+\left\\{d_{1}(w)d_{2}(v)-c_{1}(v)c_{1}(w)\right\\}\left\\{a_{0}(u)+a_{1}(u)+a_{2}(u)+a_{3}(u)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{{n_{2}{\Delta}}_{31}}$
$\displaystyle=c_{2}(u)\left\\{a_{1}(v)+a_{3}(v)\right\\}\left\\{a_{0}(w)+a_{1}(w)+a_{2}(w)+a_{3}(w)\right\\}-c_{2}(v)a_{0}(u)c_{2}(w)$
$\displaystyle+c_{2}(u)\left\\{a_{0}(v)+a_{2}(v)\right\\}\left\\{a_{2}(w)+a_{3}(w)\right\\}+d_{1}(v)a_{0}(u)d_{2}(w);$
$\displaystyle\tilde{\Gamma}^{(1)}_{n_{2}{\tilde{\Delta}}^{(+)}_{13}}$
$\displaystyle=d_{1}(u)\left\\{a_{1}(v)+a_{3}(v)\right\\}\left\\{a_{0}(w)+a_{2}(w)\right\\}+d_{1}(u)a_{2}(w)\left\\{a_{0}(v)+a_{2}(v)\right\\}$
$\displaystyle-\left\\{c_{1}(w)d_{1}(v)+d_{1}(w)c_{2}(v)\right\\}\left\\{a_{0}(u)+a_{1}(u)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{n_{2}{\tilde{\Delta}}_{13}}$
$\displaystyle=a_{1}(v)d_{2}(u)\left\\{a_{0}(w)+a_{1}(w)+a_{2}(w)+a_{3}(w)\right\\}+a_{0}(v)d_{2}(u)\left\\{a_{2}(w)+a_{3}(w)\right\\}$
$\displaystyle-\left\\{d_{2}(v)c_{2}(w)+c_{1}(v)d_{2}(w)\right\\}\left\\{a_{0}(u)+a_{2}(u)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{{{\Delta}}_{12}}$
$\displaystyle=d_{2}(v)d_{1}(u)\left\\{a_{0}(w)+a_{2}(w)\right\\}+c_{1}(w)a_{0}(v)\left\\{a_{0}(u)+a_{1}(u)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{{n_{3}{\Delta}}_{12}}$
$\displaystyle=-\left\\{c_{2}(v)c_{1}(u)+d_{2}(v)d_{1}(u)\right\\}\left\\{a_{0}(w)+a_{2}(w)\right\\}+c_{1}(w)a_{0}(v)\left\\{a_{2}(u)+a_{3}(u)\right\\}$
$\displaystyle+c_{1}(w)a_{2}(v)\left\\{a_{0}(u)+a_{1}(u)+a_{2}(u)+a_{3}(u)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{{\tilde{\Delta}^{(+)}}_{12}}$
$\displaystyle=c_{1}(v)d_{1}(u)a_{0}(w)+d_{1}(w)\left\\{a_{0}(u)+a_{1}(u)\right\\}\left\\{a_{0}(v)+a_{1}(v)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{n_{3}{\tilde{\Delta}^{(+)}}_{12}}$
$\displaystyle=a_{0}(w)\left\\{d_{1}(v)c_{1}(u)-c_{1}(v)d_{1}(u)\right\\}+d_{1}(w)\left\\{a_{0}(u)+a_{1}(u)\right\\}\left\\{a_{2}(v)+a_{3}(v)\right\\}$
$\displaystyle+d_{1}(w)a_{2}(u)\left\\{a_{0}(v)+a_{1}(v)+a_{2}(v)+a_{3}(v)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{{\tilde{\Delta}}_{12}}$
$\displaystyle=a_{0}(v)a_{0}(u)d_{2}(w)-d_{2}(v)c_{2}(u)\left\\{a_{0}(w)+a_{1}(w)+a_{2}(w)+a_{3}(w)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{n_{3}{\tilde{\Delta}}_{12}}$
$\displaystyle=\left\\{d_{2}(v)c_{2}(u)-c_{2}(v)d_{2}(u)\right\\}\left\\{a_{0}(w)+a_{1}(w)+a_{2}(w)+a_{3}(w)\right\\}+d_{2}(w)a_{0}(v)a_{2}(u)$
$\displaystyle+d_{2}(w)a_{2}(v)\left\\{a_{0}(u)+a_{2}(u)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{{{\Delta}}_{21}}$
$\displaystyle=c_{1}(v)c_{2}(u)\left\\{a_{0}(w)+a_{1}(w)\right\\}+c_{2}(w)a_{0}(u)\left\\{a_{0}(v)+a_{1}(v)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{n_{3}{{\Delta}}_{21}}$
$\displaystyle=-\left\\{c_{1}(v)c_{2}(u)+d_{1}(v)d_{2}(u)\right\\}\left\\{a_{0}(w)+a_{1}(w)\right\\}+c_{2}(w)a_{0}(u)\left\\{a_{2}(v)+a_{3}(v)\right\\}$
$\displaystyle+c_{2}(w)a_{2}(u)\left\\{a_{0}(v)+a_{1}(v)+a_{2}(v)+a_{3}(v)\right\\};$
$\displaystyle\tilde{\Gamma}^{(1)}_{n_{3}{{\Delta}}_{21}}$
$\displaystyle=c_{2}(v)a_{0}(u)\left\\{a_{0}(w)+a_{2}(w)\right\\}+c_{2}(u)c_{1}(w)\left\\{a_{0}(v)+a_{2}(v)\right\\}.$

##  Appendix C The list of coefficients for the Zamolodchikov algebra

In this appendix we list the set of all non-zero coefficients of the
tetrahedral Zamolodchikov algebra $S^{abc}_{ijk}$, as well as the coefficients
for the linear dependency $Y^{111}_{ijk}$. To make our formulas more readable,
we use the following shorthand notations:
$\mathrm{c}(x)\equiv\cos(x);\,\mathrm{s}(x)=\sin(x)$:

$\displaystyle S^{010}_{001}$
$\displaystyle=-\frac{1}{4\mathcal{N}}\Bigl{[}\mathrm{s}(2(u_{23}-{\zeta_{1}}))-\mathrm{s}(2({u_{23}}+{\zeta_{1}}))+8\mathrm{c}({\zeta_{3}})\mathrm{s}(u_{12})\left\\{\mathrm{s}({\zeta_{2}})\mathrm{s}u_{13}-\mathrm{s}({\zeta_{1}})\mathrm{s}(u_{23})\right\\}-2\mathrm{s}(2{\zeta_{2}})\mathrm{c}(2u_{13})$
$\displaystyle\left.+4\mathrm{s}({\zeta_{1}}+{\zeta_{2}})\Bigl{(}\mathrm{c}(u_{13}+u_{23})+\mathrm{c}(2{\zeta_{3}})\left\\{\mathrm{c}({\zeta_{1}}-{\zeta_{2}})-\mathrm{c}(u_{12})\right\\}-2\mathrm{s}({\zeta_{2}})\mathrm{s}({\zeta_{3}})\mathrm{c}(u_{13})+2\mathrm{s}({\zeta_{1}})\mathrm{s}({\zeta_{3}})\mathrm{c}(u_{23})\right.$
$\displaystyle+\mathrm{s}(2{\zeta_{3}})\mathrm{s}({\zeta_{1}}-{\zeta_{2}})\Bigr{)}\Bigr{]},$
$\displaystyle S^{010}_{010}$
$\displaystyle=\frac{1}{2\mathcal{N}}\Bigl{[}-4\mathrm{s}(\zeta_{2})\left\\{\mathrm{c}(u_{13})+\mathrm{c}(\zeta_{1}-\zeta_{3})\right\\}\left\\{\mathrm{c}(\zeta_{1})\mathrm{c}(u_{23})-\mathrm{c}(\zeta_{3})\mathrm{c}(u_{12})\right\\}+\mathrm{s}(2\zeta_{3})\left\\{\mathrm{c}(2\zeta_{1})-\mathrm{c}(2u_{12})\right\\}$
$\displaystyle-4\mathrm{s}^{2}(\zeta_{2})\mathrm{c}(u_{13})\mathrm{s}(\zeta_{1}-\zeta_{3})+\mathrm{s}(2\zeta_{1})\left\\{\mathrm{c}(2u_{23})\right\\}-\mathrm{c}(2\zeta_{3}))\Bigr{]},$
$\displaystyle S^{010}_{011}$
$\displaystyle=-\frac{\mathrm{i}\,}{2\mathcal{N}}\Bigl{[}2\mathrm{s}(\zeta_{2})\left\\{2\mathrm{c}(\zeta_{1})\left(\mathrm{c}(u_{23})+\mathrm{c}(\zeta_{2}+\zeta_{3})\right)\mathrm{s}(u_{13})-\mathrm{c}(\zeta_{3})\left(\mathrm{s}(u_{12}+u_{13})+\mathrm{c}(2\zeta_{1})\mathrm{s}(u_{23})\right)\right\\}$
$\displaystyle\left.-2\mathrm{s}(\zeta_{3})\left\\{2\mathrm{c}(\zeta_{1})\left(\mathrm{c}(u_{23})+\mathrm{c}(\zeta_{2}+\zeta_{3})\right)\mathrm{s}(u_{12})+\mathrm{c}(\zeta_{2})\left(-\mathrm{s}(u_{12}+u_{13})+\mathrm{c}(2\zeta_{1})\mathrm{s}(u_{23})\right)\right\\}\right.$
$\displaystyle-\mathrm{s}(2u_{13})\mathrm{s}(2\zeta_{2})+\mathrm{s}(2u_{12})\mathrm{s}(2\zeta_{3})\Bigr{]},$
$\displaystyle S^{010}_{100}$
$\displaystyle=\frac{1}{4\mathcal{N}}\Bigl{[}-2\mathrm{c}(2\zeta_{3})\mathrm{s}(2\zeta_{1})+\sin(2(u_{13}-\zeta_{2}))-2\mathrm{s}(u_{12}-\zeta_{1}-\zeta_{2})+\mathrm{s}(u_{13}+u_{23}-\zeta_{1}-\zeta_{2})$
$\displaystyle+\mathrm{s}(u_{13}+u_{23}+\zeta_{1}-\zeta_{2})-\mathrm{s}(2(u_{13}+\zeta_{2}))-\mathrm{s}(u_{13}+u_{23}-\zeta_{1}+\zeta_{2})+2\mathrm{s}(2(\zeta_{1}+\zeta_{2}))$
$\displaystyle+2\mathrm{s}(u_{12}+\zeta_{1}+\zeta_{2})-\mathrm{s}(u_{13}+u_{23}+\zeta_{1}+\zeta_{2})+\mathrm{s}(2(u_{12}-\zeta_{3}))-2\mathrm{s}(u_{12}+u_{13}-\zeta_{2}-\zeta_{3})$
$\displaystyle-8\mathrm{c}(\zeta_{1})\mathrm{s}(u_{12})\mathrm{s}(u_{23})\mathrm{s}(\zeta_{3})-\mathrm{s}(2(u_{12}+\zeta_{3}))-4\mathrm{s}(\zeta_{1})\left\\{\mathrm{c}(u_{12})\mathrm{c}(\zeta_{2}+2\zeta_{3})+2\mathrm{c}(u_{13})\mathrm{s}(\zeta_{2})\mathrm{s}(\zeta_{2}+\zeta_{3})\right\\}$
$\displaystyle+2\mathrm{s}(u_{12}+u_{13}+\zeta_{2}+\zeta_{3})+2\mathrm{c}(2\zeta_{1})\left\\{\mathrm{s}(u_{23}-\zeta_{2}-\zeta_{3})+\mathrm{s}(2\zeta_{3})-\mathrm{s}(u_{23}+\zeta_{2}+\zeta_{3})\right\\}\Bigr{]},$
$\displaystyle S^{010}_{101}$
$\displaystyle=\frac{\mathrm{i}\,}{2\mathcal{N}}\Bigl{[}4\mathrm{c}(\zeta_{2})\left\\{\mathrm{c}(u_{13})+\mathrm{c}(\zeta_{1}-\zeta_{3})\right\\}\left\\{\mathrm{s}(\zeta_{3})\mathrm{s}(u_{12})-\mathrm{s}(\zeta_{1})\mathrm{s}(u_{23})\right\\}-2\mathrm{s}(\zeta_{1}+\zeta_{3})\mathrm{s}(u_{12}-u_{23})$
$\displaystyle-\mathrm{s}(2\zeta_{3})\mathrm{s}(2(u_{12}))+\mathrm{s}(2\zeta_{1})\mathrm{s}(2(u_{23}))+2\mathrm{c}(2\zeta_{2})\mathrm{s}(u_{13})\mathrm{s}(\zeta_{1}-\zeta_{3})\Bigr{]},$
$\displaystyle S^{010}_{110}$
$\displaystyle=\frac{\mathrm{i}\,}{\mathcal{N}}\Bigl{[}8\mathrm{c}(\zeta_{2})\left\\{\mathrm{c}(u_{12})+\mathrm{c}(\zeta_{1}+\zeta_{2})\right\\}\left\\{\mathrm{s}(\zeta_{2})\mathrm{s}(u_{13})-\mathrm{s}(\zeta_{1})\mathrm{s}(u_{23})\right\\}+\mathrm{c}(2(u_{23}-\zeta_{1}))-\mathrm{c}(2(u_{23}+\zeta_{1}))$
$\displaystyle-\mathrm{c}(2(u_{13}-\zeta_{2}))+\mathrm{c}(2(u_{13}+\zeta_{2}))-2\mathrm{c}(u_{13}+u_{23}+\zeta_{1}-\zeta_{2})+2\mathrm{c}(u_{13}+u_{23}-\zeta_{1}+\zeta_{2})$
$\displaystyle-4\mathrm{c}(2\zeta_{3})\mathrm{s}(u_{12})\mathrm{s}(\zeta_{1}+\zeta_{2})\Bigr{]},$
$\displaystyle Y^{111}_{011}$
$\displaystyle=\frac{2\mathrm{i}\,}{\mathcal{N}}\Bigl{[}\left\\{\mathrm{c}(u_{23})+\mathrm{c}(\zeta_{2}+\zeta_{3})\right\\}\left\\{\mathrm{s}(u_{13})\mathrm{s}(\zeta_{1}+\zeta_{2})+\mathrm{s}(u_{12})\mathrm{s}(\zeta_{1}-\zeta_{3})\right\\}\Bigl{]},$
$\displaystyle Y^{111}_{110}$
$\displaystyle=\frac{2\mathrm{i}\,}{\mathcal{N}}\Bigl{[}\left\\{\mathrm{c}(u_{12})+\mathrm{c}(\zeta_{1}+\zeta_{2})\right\\}\left\\{\mathrm{s}(u_{23})\mathrm{s}(\zeta_{1}-\zeta_{3})-\mathrm{s}(u_{13})\mathrm{s}(\zeta_{2}+\zeta_{3})\right\\}\Bigl{]},$
$\displaystyle Y^{111}_{101}$
$\displaystyle=\frac{2\mathrm{i}\,}{\mathcal{N}}\Bigl{[}\left\\{\mathrm{c}(u_{13})+\mathrm{c}(\zeta_{1}-\zeta_{3})\right\\}\left\\{\mathrm{s}(u_{23})\mathrm{s}(\zeta_{1}+\zeta_{2})-\mathrm{s}(u_{12})\mathrm{s}(\zeta_{2}+\zeta_{3})\right\\}\Bigr{]},$
$\displaystyle Y^{111}_{001}$
$\displaystyle=\frac{2}{\mathcal{N}}\Bigl{[}\left\\{\mathrm{c}(u_{13}+\mathrm{c}(\zeta_{1}-\zeta_{3})\right\\}\left\\{\mathrm{c}(u_{23})+\mathrm{c}(\zeta_{2}+\zeta_{3})\right\\}\mathrm{s}(\zeta_{1}+\zeta_{2})\Bigr{]},$
$\displaystyle Y^{111}_{010}$
$\displaystyle=\frac{2}{\mathcal{N}}\Bigl{[}\left\\{\mathrm{c}(u_{12})+\mathrm{c}(\zeta_{1}+\zeta_{2})\right\\}\left\\{\mathrm{c}(u_{23})+\mathrm{c}(\zeta_{2}+\zeta_{3})\right\\}\mathrm{s}(\zeta_{1}-\zeta_{3})\Bigr{]},$
$\displaystyle Y^{111}_{100}$
$\displaystyle=-\frac{2}{\mathcal{N}}\Bigl{[}\left\\{\mathrm{c}(u_{12})+\mathrm{c}(\zeta_{1})+\zeta_{2})\right\\}\left\\{\mathrm{c}(u_{13})+\mathrm{c}(\zeta_{1}-\zeta_{3})\right\\}\mathrm{s}(\zeta_{2}+\zeta_{3})\Bigr{]}.$

In the above formulas, the factor $\mathcal{N}$ is given by the following
expression:

$\displaystyle\mathcal{N}$
$\displaystyle=\left\\{\mathrm{c}(u_{12})-\mathrm{c}(u_{13}+u_{23})\right\\}\mathrm{s}(\zeta_{1}+\zeta_{2})+2\mathrm{s}(u_{12})\mathrm{s}(u_{23})\mathrm{s}(\zeta_{1}-\zeta_{3})$
(C.1)
$\displaystyle-2\mathrm{s}(u_{12})\mathrm{s}(u_{13})\mathrm{s}(\zeta_{2}+\zeta_{3})-2\mathrm{s}(\zeta_{1}+\zeta_{2})\mathrm{s}(\zeta_{1}-\zeta_{3})\mathrm{s}(\zeta_{2}+\zeta_{3}).$

The other coefficients are obtained from the ones above from the relations:
$\displaystyle S^{111}_{001}$ $\displaystyle=S^{010}_{001}+Y^{111}_{001};$
$\displaystyle S^{111}_{011}$ $\displaystyle=S^{010}_{011}+Y^{111}_{011};$
$\displaystyle S^{111}_{100}$ $\displaystyle=S^{010}_{100}+Y^{111}_{100};$
$\displaystyle S^{111}_{101}$ $\displaystyle=S^{010}_{101}+Y^{111}_{101};$
$\displaystyle S^{111}_{110}$ $\displaystyle=S^{010}_{110}+Y^{111}_{110};$
$\displaystyle S^{111}_{010}$ $\displaystyle=S^{010}_{010}+Y^{111}_{010}-1;$
$\displaystyle S^{100}_{010}$ $\displaystyle=S^{001}_{010};$ $\displaystyle
S^{100}_{001}$ $\displaystyle=S^{001}_{001}+1;$ $\displaystyle S^{100}_{011}$
$\displaystyle=S^{001}_{011};$ $\displaystyle S^{100}_{101}$
$\displaystyle=S^{001}_{101};$ $\displaystyle S^{100}_{110}$
$\displaystyle=S^{001}_{110};$ $\displaystyle S^{100}_{100}$
$\displaystyle=S^{001}_{100}-1;$ $\displaystyle S^{001}_{001}$
$\displaystyle=-S^{010}_{001};$ $\displaystyle S^{001}_{010}$
$\displaystyle=1-S^{010}_{010};$ $\displaystyle S^{001}_{100}$
$\displaystyle=1-S^{010}_{100};$ $\displaystyle S^{001}_{011}$
$\displaystyle=-S^{010}_{011};$ $\displaystyle S^{001}_{101}$
$\displaystyle=-S^{010}_{101};$ $\displaystyle S^{001}_{110}$
$\displaystyle=-S^{010}_{110}.$

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