Bethe Vectors for Composite Models with $\mathfrak{gl}(2|1)$ and
$\mathfrak{gl}(1|2)$ Supersymmetry

Bethe Vectors for Composite Models
with $\boldsymbol{\mathfrak{gl}(2|1)}$ and $\boldsymbol{\mathfrak{gl}(1|2)}$
Supersymmetry

Jan FUKSA †‡

J. Fuksa

† Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research,
† Dubna, Russia<EMAIL_ADDRESS>
‡ Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague,
‡ Czech Republic

Received November 18, 2016, in final form March 03, 2017; Published online
March 13, 2017

Supersymmetric composite generalized quantum integrable models solvable by the
algebraic Bethe ansatz are studied. Using a coproduct in the bialgebra of
monodromy matrix elements and their action on Bethe vectors, formulas for
Bethe vectors in the composite models with supersymmetry based on the super-
Yangians $Y[\mathfrak{gl}(2|1)]$ and $Y[\mathfrak{gl}(1|2)]$ are derived.

algebraic Bethe ansatz; composite models

17B37; 81R50; 82B23

##  1 Introduction

The main success of the algebraic Bethe ansatz resides in the general
prescription how to obtain eigenvectors (famous Bethe vectors) for a vast
class of quantum integrable models. Consequently, it enables the calculation
of the correlation functions via the calculation of the form factors. The
Bethe vectors for models related to the algebra $\mathfrak{gl}(2)$ and its
deformations have a very simple form (see [6, 16, 20, 28, 31] and references
therein). However, they become very nontrivial for models based on higher rank
(super)algebras.

What is common for the Bethe vectors in the models with $\mathfrak{gl}(2)$ and
higher rank symmetry (super)algebras is that they belong to the state
(super)space $\mathcal{H}$ with a structure of the Fock space. In other words,
$\mathcal{H}$ contains a cyclic vector $\Omega$ (usually called pseudovacuum)
and the Bethe vectors are generated by the action of certain creation-like
operators on $\Omega$. These creation-like operators belong to the
(super)algebra of matrix elements of the monodromy matrix $T(u)$.

The problem of the higher rank (super)algebras has been addressed by the
nested Bethe ansatz method. Its origin dates back to the works [30, 33] in the
context of the coordinate Bethe ansatz, whereas its algebraic version was
introduced in [21]. The last years have been marked by considerable progress
in finding “user-friendly” forms of Bethe vectors for higher rank
(super)algebras. For $\mathfrak{gl}(3)$ and its quantum deformations, Bethe
vectors were calculated in [3, 4]. Bethe vectors for the superalgebras
$\mathfrak{gl}(2|1)$ and $\mathfrak{gl}(1|2)$ were obtained in [26] and used
to calculate the scalar products [13, 14] and the form factors of the
monodromy matrix elements [11]. Based on this, correlation functions in models
with this type of supersymmetry can be investigated via the form factor
expansion.

The knowledge of form factors of local operators thus becomes an important
condition for the calculation of correlation functions. In their turn, the
form factors of local operators can be computed in two different ways. To
clarify the purpose and content of this paper, we briefly describe these
approaches.

The first method is applied to the models with a known solution of the quantum
inverse scattering problem (see, e.g., [18, 23]). These include the
supersymmetric $t$-$J$ model (in the context of $\mathfrak{gl}(2|1)$) well-
known from the condensed matter physics. The corresponding quantum inverse
scattering problem was solved in [10]. The form factors of local operators in
this model reduce to the form factors of the monodromy matrix elements. Their
calculation, in its turn, reduces to the well-studied problem of calculating
scalar products of Bethe vectors. Thus, in this case, the inverse scattering
problem provides a powerful tool for calculating the form factors of local
operators. However, solutions of the quantum inverse scattering problem are
known for a rather restricted class of integrable models.

The second approach to the calculation of the form factors of local operators
is more general and does not imply knowledge of explicit solutions of the
quantum inverse scattering problem. It is based on the composite model
[16].111The terminology witnessed a long evolution here. The composite model
was originally called the two-site model in [16]. Later the term two-component
model was used [29]. Both these terms can lead to confusion in some
situations. Therefore, the term composite model was proposed recently [24]. We
hold this terminology in our article. There are several integrable models
related to the $\mathfrak{gl}(2|1)$ superalgebra to which the composite model
is applicable. They are, e.g., the impurity version of the $t$-$J$ model [7],
the doped Heisenberg chains [8], or the supersymmetric $U$ model [5] and [2,
27]. Unlike the $t$-$J$ model, they are not based on the fundamental
representation of $\mathfrak{gl}(2|1)$, therefore, the quantum inverse
scattering problem is unsolved for them. However, the method of the composite
model still works in this case, as it is based on the algebraic Bethe ansatz
only.

The main idea of the composite model method is that the interval $[0,L]$, on
which the original model is defined, is divided into two subintervals $[0,x]$
and $]x,L]$. Consequently, the state (super)space $\mathcal{H}$ of the
complete model is divided into two (graded) subspaces $\mathcal{H}^{(1)}$ and
$\mathcal{H}^{(2)}$ corresponding to $[0,x]$ and $]x,L]$, respectively, such
that $\mathcal{H}=\mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}$. Simultaneously,
the monodromy matrix $T(u)$ can be presented as a matrix product of two
partial monodromy matrices $T^{(2)}(u)$ and $T^{(1)}(u)$:

$\displaystyle T(u)=T^{(2)}(u)\cdot T^{(1)}(u).$ (1.1)

The partial monodromy matrices $T^{(\ell)}(u)$, $\ell=1,2$, are to be
understood as particular representations of the complete monodromy matrix
$T(u)$ on the partial state (super)spaces $\mathcal{H}^{(\ell)}$. This shall
be ascribed to a bialgebraic structure on the algebra of matrix elements of
the monodromy matrix. The property (1.1) is equivalent to a comultiplication
in the aforementioned bialgebra. The partial monodromy matrix $T^{(\ell)}(u)$
acts nontrivially only in its corresponding representation (super)space
$\mathcal{H}^{(\ell)}$.

We suppose that the partial state (super)spaces $\mathcal{H}^{(\ell)}$ possess
also the structure of the Fock space with the partial pseudovacua
$\Omega^{(\ell)}$ such that $\Omega=\Omega^{(1)}\otimes\Omega^{(2)}$. A
nontrivial fact is that the Bethe vectors $\mathbb{B}\in\mathcal{H}$ of the
complete model can be represented as bilinear combinations of partial Bethe
vectors $\mathbb{B}^{(1)}\in\mathcal{H}^{(1)}$ and
$\mathbb{B}^{(2)}\in\mathcal{H}^{(2)}$. For the models based on the
$\mathfrak{gl}(2)$ symmetry, this was shown in [16]. The case of
$\mathfrak{gl}(3)$ was recently studied in [24]. The aim of this article is to
find a similar representation for the integrable models based on the
superalgebras $\mathfrak{gl}(2|1)$ and $\mathfrak{gl}(1|2)$. Such a
representation allows one to compute the form factors of the partial monodromy
matrix elements $T_{ij}^{(\ell)}(u)$ in the basis of the Bethe vectors of the
complete model. Based on this, the form factors and correlation functions of
local operators can be investigated.

It is worth mentioning that the composite model is tied with the theory of the
quantized Knizhnik–Zamolodchikov (qKZ) equation. Bethe vectors play a
principal role in finding its integral solutions [1, 32]. This is related to
the fact that the Bethe vectors belong to the same vector space as the
solutions of qKZ. For an $R$-matrix $R(u)$, which at a certain value of the
parameter $u$ becomes a permutation operator, a technique of solving the qKZ
based on the nested Bethe ansatz was developed in [1]. In [32], the so-called
weight functions are constructed as basic building blocks for the integral
solutions of the qKZ. They satisfy a coproduct property. The coproduct
property states that from two weight functions belonging to vector spaces
$V^{(1)}$ and $V^{(2)}$, respectively, a weight function belonging to their
tensor product $V^{(1)}\otimes V^{(2)}$ can be constructed in a unique way.
The weight functions are nothing else but the Bethe vectors for an integrable
model whose monodromy matrix is constructed from the $R$-matrices
corresponding to the qKZ equation. The composite model represents in an
explicit way the coproduct property of the weight functions, i.e., of the
Bethe vectors.

The general strategy of this article is similar to the strategy of the paper
[24] on $\mathfrak{gl}(3)$. However, some technical details appear due to
different commutation relations. We describe these technical differences. We
show how they change calculations for a concrete case, but we do not give all
calculations.

Recall that the superalgebras $\mathfrak{gl}(2|1)$ and $\mathfrak{gl}(1|2)$
are isomorphic. The same holds also for their corresponding super-Yangians
$Y[\mathfrak{gl}(2|1)]$ and $Y[\mathfrak{gl}(1|2)]$. Therefore, we mostly
concern ourselves with the models based on $\mathfrak{gl}(2|1)$ and then
isomorphically map the results to the $\mathfrak{gl}(1|2)$ case. Let us
emphasize that the underlying algebraic structure of the models described in
this paper are the super-Yangians $Y[\mathfrak{gl}(2|1)]$ and
$Y[\mathfrak{gl}(1|2)]$, respectively.

The paper is organized as follows. In Section 2, basic notions of the
$\mathfrak{gl}(2|1)$-invariant quantum integrable models like the monodromy
matrix, the RTT algebra, and the Bethe vectors are introduced, and the
notation used further in the article is described. Section 3 describes the
$\mathfrak{gl}(2|1)$-invariant composite model. The main results are
formulated in two theorems contained in Section 3.1. Section 4 deals with some
technical details from Section 3.1. Section 5 is devoted to the
$\mathfrak{gl}(1|2)$-invariant composite model. The main results are presented
in two theorems in Section 5.1.

##  2 Basic notions of the $\boldsymbol{\mathfrak{gl}(2|1)}$-invariant model

###  2.1 $\boldsymbol{R}$-matrix and RTT algebra

The content of this subsection can be easily generalized to superalgebras of
other ranks.

The $R$-matrix acts in the tensor product of two $\mathbb{Z}_{2}$-graded
auxiliary superspaces $\mathbb{C}^{2|1}$. The basis of the even part of
$\mathbb{C}^{2|1}$ is $\\{e_{1},e_{2}\\}$ and of the odd part is
$\\{e_{3}\\}$. We introduce the parity function $[\
]\colon\\{1,2,3\\}\rightarrow\mathbb{Z}_{2}$ such that $[1]=[2]=0$ and
$[3]=1$. The gradation of the superspace $\mathbb{C}^{2|1}$ is thus described
by this parity function: $\mathop{\rm grad}(e_{i})=[i]$, $i=1,2,3$.

The matrix units $E_{ij}\in\mathrm{End}(\mathbb{C}^{3})$ are introduced in the
standard way $(E_{ij})_{ab}=\delta_{ia}\delta_{jb}$ with the gradation
$\mathop{\rm grad}(E_{ij})=[i]+[j]$. The tensor product is graded in the
following way:

$\displaystyle(E_{ij}\otimes E_{kl})(E_{mn}\otimes
E_{pq})=(-1)^{([k]+[l])([m]+[n])}E_{ij}E_{mn}\otimes E_{kl}E_{pq}.$

The graded permutation (superpermutation) for two auxiliary superspaces [22]
is defined using the matrix units

$\displaystyle P=\sum_{i,j=1}^{3}(-1)^{[j]}E_{ij}\otimes E_{ji}.$

The $\mathfrak{gl}(2|1)$-invariant $R$-matrix has the explicit form

$\displaystyle R(u,v)=\mathbb{I}\otimes\mathbb{I}+g(u,v)P,$ (2.1)

where $\mathbb{I}$ is the unit matrix in $\mathbb{C}^{2|1}$. The function
$g(u,v)$ is antisymmetric and rational

$\displaystyle g(u,v)=\frac{c}{u-v}$

and $c\in\mathbb{C}$ is an auxiliary constant.

The definition of the monodromy matrix is standard
$T(u)=\sum\limits_{i,j=1}^{3}E_{ij}\otimes T_{ij}(u)$. The elements
$T_{ij}(u)$, $i,j=1,2,3$, together with the unit element $\mathbf{1}$ are
generators of the associative superalgebra $\mathcal{A}$. The monodromy matrix
is a globally even matrix due to the fact that $\mathop{\rm
grad}(E_{ij})=\mathop{\rm grad}(T_{ij}(u))$. It satisfies the RTT relation
with the $R$-matrix (2.1)

$\displaystyle R(u,v)(T(u)\otimes\mathbb{I})(\mathbb{I}\otimes
T(v))=(\mathbb{I}\otimes T(v))(T(u)\otimes\mathbb{I})R(u,v).$ (2.2)

The RTT relation is equivalent to the bilinear relations in $\mathcal{A}$
which can be written in two equivalent forms

$\displaystyle[T_{ij}(u),T_{kl}(v)\\}=g(u,v)(-)^{[i][j]+[i][l]+[j][l]}(T_{il}(u)T_{kj}(v)-T_{il}(v)T_{kj}(u))$
$\displaystyle\hphantom{[T_{ij}(u),T_{kl}(v)\\}}{}=-g(u,v)(-)^{[i][k]+[i][l]+[k][l]}(T_{kj}(u)T_{il}(v)-T_{kj}(v)T_{il}(u))$
(2.3)

with the supercommutator $[T_{ij}(u),T_{kl}(v)\\}\equiv
T_{ij}(u)T_{kl}(v)-(-)^{([i]+[j])([k]+[l])}T_{kl}(v)T_{ij}(u)$. The
superalgebra $\mathcal{A}$ is called the RTT algebra.

There is a coalgebraic structure defined on $\mathcal{A}$ with the following
coproduct:

$\displaystyle\Delta(T_{ij}(u))\equiv\sum_{k=1}^{3}T_{kj}(u)\otimes
T_{ik}(u)=\sum_{k=1}^{3}T_{kj}^{(1)}(u)T_{ik}^{(2)}(u).$ (2.4)

The superscripts at $T_{ij}^{(\ell)}(u)$, $\ell=1,2$, are used to distinguish
the two copies of the bialgebra $\mathcal{A}$. The elements of different
copies of $\mathcal{A}$ mutually supercommute
$\big{[}T_{il}^{(1)}(u),T_{jk}^{(2)}(v)\big{\\}}=0$. The partial monodromy
matrices $T^{(\ell)}(u)$, $\ell=1,2$, satisfy the same RTT relation (2.2) as
$\mathcal{A}$ has the structure of bialgebra. It is worth mentioning that the
coproduct (2.4) is equivalent to relation (1.1) for the monodromy matrix of
the composite model.

Let us remind that there is an antimorphism of $\mathcal{A}$ [26]:

$\displaystyle\psi(T_{ij}(u))=(-1)^{[i][j]+[i]}T_{ji}(u),\qquad\psi(AB)=(-1)^{\mathop{\rm
grad}(A)\cdot\mathop{\rm grad}(B)}\psi(B)\psi(A),$ (2.5)

for $A,B\in\mathcal{A}$ of definite gradings. It preserves the supercommutator
$\psi([A,B\\})=-[\psi(A),\psi(B)\\}$ and satisfies the composition rule

$\displaystyle\Delta\circ\psi=(\psi\otimes\psi)\circ\Delta^{\prime}$ (2.6)

with the standard coproduct (2.4) and the opposite coproduct222The name
opposite coproduct is obviously relative. It refers to the fact that it
intertwines the factors in the tensor product in comparison with the
“standard” coproduct (2.4). In the same way, (2.4) is the opposite coproduct
to (2.7). Despite this, we call in this article the coproduct of the type
(2.4) the standard and of (2.7) the opposite.

$\displaystyle\Delta^{\prime}(T_{ij}(u))=\sum_{k}(-1)^{([i]+[k])([k]+[j])}T_{ik}(u)\otimes
T_{kj}(u)=\sum_{k}T^{(2)}_{kj}(u)T^{(1)}_{ik}(u).$ (2.7)

The composition rule (2.6) is the key property for the investigation of dual
Bethe vectors in the composite model, as we will see below.

###  2.2 Notation

The following functions are used throughout the text:

$\displaystyle f(u,v)=1+g(u,v)=\frac{u-v+c}{u-v},\qquad
h(u,v)=\frac{f(u,v)}{g(u,v)}=\frac{u-v+c}{c}.$

We completely follow the notation used, e.g., in [26]. The sets of parameters
are denoted as $\bar{u}$, $\bar{v}$ etc. Their individual elements as $u_{j}$,
$v_{0},$ etc. The notation $\bar{u}_{j}$, $\bar{v}_{0}$ means
$\bar{u}\backslash u_{j}$, $\bar{v}\backslash v_{0}$, etc. To avoid lengthy
and complicated formulas, we use the shorthand notation for products of the
above functions over the sets of parameters. For example,

$\displaystyle f(\bar{u},v)\equiv\prod_{u_{j}\in\bar{u}}f(u_{j},v),\qquad
g(u,\bar{v}_{i})\equiv\prod_{\begin{subarray}{c}v_{j}\in\bar{v}\\\ v_{j}\neq
v_{i}\end{subarray}}g(u,v_{j}),\qquad
h(\bar{u},\bar{v})=\prod_{u_{j}\in\bar{u}}\prod_{v_{k}\in\bar{v}}h(u_{j},v_{k}).$

This notation is preserved also for products of even operators

$\displaystyle
T_{ij}(\bar{u})\equiv\prod_{u_{k}\in\bar{u}}T_{ij}(u_{k}),\qquad[i]+[j]=0.$

For products of odd operators $T_{i3}(u)$ and $T_{3i}(u)$, $i=1,2$, the
symmetrised product is used

$\displaystyle\mathbb{T}_{i3}(\bar{u})\equiv\frac{T_{i3}(u_{1})T_{i3}(u_{2})\cdots
T_{i3}(u_{n})}{\prod\limits_{1\leq j<k\leq
n}h(u_{k},u_{j})},\qquad\mathbb{T}_{3i}(\bar{u})\equiv\frac{T_{3i}(u_{1})T_{3i}(u_{2})\cdots
T_{3i}(u_{n})}{\prod\limits_{1\leq j<k\leq n}h(u_{j},u_{k})}.$

A set of parameters $\bar{u}$ is often divided into its two disjoint subsets
$\bar{u}_{\scriptscriptstyle\rm I}$, $\bar{u}_{\scriptscriptstyle\rm II}$:
$\bar{u}_{\scriptscriptstyle\rm I}\cap\bar{u}_{\scriptscriptstyle\rm
II}=\varnothing$ and $\bar{u}_{\scriptscriptstyle\rm
I}\cup\bar{u}_{\scriptscriptstyle\rm II}=\bar{u}$. We denote it as
$\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}$. Such a partition is always
accompanied by a summation over all partitions of the prescribed type, just
according to the rule: “where a partition is, there is a summation”.

###  2.3 Bethe vectors

The Bethe vectors for the $\mathfrak{gl}(2|1)$-invariant models were
calculated in [26]. They are obtained under the assumption that there is a
cyclic vector $\Omega$ which is an eigenvector of the diagonal elements of the
monodromy matrix $T(u)$ and is annihilated by the lower-triangular elements:

$\displaystyle T_{ii}(u)\Omega=\lambda_{i}(u)\Omega,\qquad
T_{ij}(u)\Omega=0,\quad i>j.$ (2.8)

The eigenfunctions $\lambda_{i}(u)$ vary depending on the model. Leaving them
as arbitrary functions of the parameter $u$, the model is called generalized.
The cyclic vector $\Omega$ is called pseudovacuum. We suppose its gradation to
be vanishing ($\Omega$ is an even supervector). The other Bethe vectors
$\mathbb{B}_{a,b}(\bar{u};\bar{v})$ are generated by the action of the upper-
triangular elements $T_{ij}(u)$, $i<j$, on $\Omega$. They depend on two sets
of spectral parameters $\bar{u}=\\{u_{1},\dots,u_{a}\\}$ and
$\bar{v}=\\{v_{1},\dots,v_{b}\\}$ where $a,b=0,1,2,\dots$ are the
cardinalities $a=\\#\bar{u}$ and $b=\\#\bar{v}$.

The Bethe vectors have several representations [26]. It is convenient to use
the following one for our purposes:

$\displaystyle\mathbb{B}_{a,b}(\bar{u};\bar{v})=\sum
K_{n}(\bar{v}_{\scriptscriptstyle\rm I}|\bar{u}_{\scriptscriptstyle\rm
I})\frac{f(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})g(\bar{v}_{\scriptscriptstyle\rm II},\bar{v}_{\scriptscriptstyle\rm
I})}{\lambda_{2}(\bar{u}_{\scriptscriptstyle\rm
II})\lambda_{2}(\bar{v})f(\bar{v},\bar{u})}\mathbb{T}_{13}(\bar{v}_{\scriptscriptstyle\rm
I})\mathbb{T}_{23}(\bar{v}_{\scriptscriptstyle\rm
II})T_{12}(\bar{u}_{\scriptscriptstyle\rm II})\Omega.$ (2.9)

The sum goes over all partitions
$\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}$ and
$\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm II}\\}$ with the restriction that
$\\#\bar{u}_{\scriptscriptstyle\rm I}=\\#\bar{v}_{\scriptscriptstyle\rm I}=n$,
where $n=0,1,\dots,\min(a,b)$. The function
$K_{n}(\bar{v}_{\scriptscriptstyle\rm I}|\bar{u}_{\scriptscriptstyle\rm I})$
is the partition function of the six-vertex model with the domain wall
boundary condition [19]. It has the following representation [15]:

$\displaystyle
K_{n}(\bar{v}|\bar{u})=\prod_{i<j}^{n}g(v_{i},v_{j})g(u_{j},u_{i})\cdot\frac{f(\bar{v},\bar{u})}{g(\bar{v},\bar{u})}\left.\det\left[\frac{g^{2}(v_{k},u_{l})}{f(v_{k},u_{l})}\right]\right|_{k,l=1,\dots,n}.$

A simple observation states that the Bethe vector
$\mathbb{B}_{a,b}(\bar{u};\bar{v})$ has the gradation $\mathop{\rm
grad}(\mathbb{B}_{a,b})=b$ $(\mathop{\rm mod}2)$.

Similarly, we assume the existence of the dual pseudovacuum $\Omega^{\dagger}$
with the properties

$\displaystyle\Omega^{\dagger}T_{ii}(u)=\lambda_{i}(u)\Omega^{\dagger},\qquad\Omega^{\dagger}T_{ij}(u)=0,\qquad
i<j,$

where the eigenfunctions $\lambda_{i}(u)$ are the same as in (2.8). We suppose
the gradation of $\Omega^{\dagger}$ to be vanishing. The dual Bethe vectors
$\mathbb{C}_{ab}(\bar{u};\bar{v})$ also depend on two sets of spectral
parameters $\bar{u}$, $\bar{v}$ with $a=\\#\bar{u}$, $b=\\#\bar{v}$. Their
explicit form is [26]

$\displaystyle\mathbb{C}_{a,b}(\bar{u};\bar{v})=(-1)^{\frac{b^{2}-b}{2}}\sum
K_{n}(\bar{v}_{\scriptscriptstyle\rm I}|\bar{u}_{\scriptscriptstyle\rm
I})\frac{f(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})g(\bar{v}_{\scriptscriptstyle\rm II},\bar{v}_{\scriptscriptstyle\rm
I})}{\lambda_{2}(\bar{u}_{\scriptscriptstyle\rm
II})\lambda_{2}(\bar{v})f(\bar{v},\bar{u})}\Omega^{\dagger}T_{21}(\bar{u}_{\scriptscriptstyle\rm
II})\mathbb{T}_{32}(\bar{v}_{\scriptscriptstyle\rm
II})\mathbb{T}_{31}(\bar{v}_{\scriptscriptstyle\rm I}),$

where the sum goes over all partitions
$\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}$ and
$\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm II}\\}$ with the restriction that
$\\#\bar{u}_{\scriptscriptstyle\rm I}=\\#\bar{v}_{\scriptscriptstyle\rm I}=n$,
where $n=0,1,\dots,\min(a,b)$. The gradation of the dual Bethe vector
$\mathbb{C}_{a,b}(\bar{u};\bar{v})$ is again $\mathop{\rm
grad}(\mathbb{C}_{a,b})=b$ $(\mathop{\rm mod}2)$.

If the antimorphism (2.5) relates the pseudovacuum to the dual pseudovacuum as
$\psi(\Omega)=\Omega^{\dagger}$, the Bethe vector
$\mathbb{B}_{a,b}(\bar{u};\bar{v})$ is mapped by $\psi$ to the dual Bethe
vector $\mathbb{C}_{a,b}(\bar{u};\bar{v})$

$\displaystyle\psi\left(\mathbb{B}_{a,b}(\bar{u};\bar{v})\right)=\mathbb{C}_{a,b}(\bar{u};\bar{v}).$
(2.10)

##  3 Composite $\boldsymbol{\mathfrak{gl}(2|1)}$-invariant model

The interval $[0,L]$, which the generalized model is defined on, is split into
its two subintervals, as discussed in Introduction. The fact that the
monodromy matrix of the full model $T(u)$ is simply the matrix product of the
partial monodromy matrices $T^{(2)}(u)T^{(1)}(u)$, as expressed in (1.1),
follows from the coproduct (2.4) in the bialgebra $\mathcal{A}$.

This is followed by the split of the original superspace into its two graded
subspaces $\mathcal{H}=\mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}$. We suppose
that the graded subspaces $\mathcal{H}^{(\ell)}$, $\ell=1,2$, contain the
partial pseudovacua $\Omega^{(\ell)}$ with the properties

$\displaystyle
T_{ii}^{(\ell)}(u)\Omega^{(\ell)}=\lambda_{i}^{(\ell)}(u)\Omega^{(\ell)},\qquad
T_{ij}^{(\ell)}(u)\Omega^{(\ell)}=0,\qquad i>j.$

The operators $T_{ij}^{(\ell)}(u)$ are the matrix elements of the partial
monodromy matrices $T^{(\ell)}(u)$ described by (1.1). In other words, the
subspaces $\mathcal{H}^{(\ell)}$ have exactly the same structure of a Fock
space as the full superspace $\mathcal{H}$. The corresponding Bethe vectors
are denoted as $\mathbb{B}^{(\ell)}_{a,b}(\bar{u};\bar{v})$. We suppose that
the partial pseudovacua $\Omega^{(\ell)}$ form the total pseudovacuum
$\Omega=\Omega^{(1)}\Omega^{(2)}$. We omit here the symbol $\otimes$ for the
direct product of the pseudovacua because the superscripts at
$\Omega^{(\ell)}$ indicate that we work in the direct product of superspaces.
This notation is kept also below for direct products of arbitrary Bethe
vectors.

Similarly, we suppose for the partial dual superspaces the existence of the
dual pseudovacua $\Omega^{\dagger(\ell)}$ satisfying

$\displaystyle\Omega^{\dagger(\ell)}T_{ii}^{(\ell)}(u)=\lambda_{i}^{(\ell)}(u)\Omega^{\dagger(\ell)},\qquad\Omega^{\dagger(\ell)}T_{ij}^{(\ell)}(u)=0,\qquad
i<j.$

The corresponding dual Bethe vectors are denoted as
$\mathbb{C}^{(\ell)}_{a,b}(\bar{u};\bar{v})$ and we again suppose that
$\Omega^{\dagger}=\Omega^{\dagger(1)}\Omega^{\dagger(2)}$.

It is useful to introduce two ratio functions $r_{1}(u)$ and $r_{3}(u)$
instead of three independent functions $\lambda_{i}(u)$:

$\displaystyle r_{i}(u)=\frac{\lambda_{i}(u)}{\lambda_{2}(u)},\quad
i=1,3,\qquad
r_{i}^{(\ell)}(u)=\frac{\lambda^{(\ell)}_{i}(u)}{\lambda^{(\ell)}_{2}(u)},\qquad
i=1,3,\quad\ell=1,2.$

This corresponds to the multiplication of the monodromy matrix by
$\lambda_{2}^{-1}(u)$. Obviously
$\lambda_{i}(u)=\lambda_{i}^{(1)}(u)\lambda_{i}^{(2)}(u)$ and
$r_{i}(u)=r_{i}^{(1)}(u)r_{i}^{(2)}(u)$.

###  3.1 Main results

######  Theorem 3.1.

The Bethe vectors of the full model can be expressed as the bilinear
combination of the partial Bethe vectors:

$\displaystyle\mathbb{B}_{a,b}(\bar{u};\bar{v})=\sum
r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})}{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})}\mathbb{B}_{a_{2},b_{2}}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})\mathbb{B}^{(1)}_{a_{1},b_{1}}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I}).$ (3.1)

The summation goes over all partitions
$\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}$ and
$\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm II}\\}$ with no restriction. The
corresponding cardinalities satisfy $a_{1}+a_{2}=a$ and $b_{1}+b_{2}=b$.

The coproduct used in the proof is the standard one (2.4), as can be seen in
Section 4.

###### Idea of the proof.

The proof is based on a recursion relation enjoined by Bethe vectors of
$\mathfrak{gl}(2|1)$-invariant models [26]

$\displaystyle\frac{T_{23}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathbb{B}_{a,b-1}(\bar{u};\bar{v})=f(z,\bar{u})\mathbb{B}_{a,b}(\bar{u};\\{z,\bar{v}\\})$
$\displaystyle\hphantom{\frac{T_{23}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathbb{B}_{a,b-1}(\bar{u};\bar{v})=}{}+\sum_{\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}}g(u_{0},z)f(u_{0},\bar{u}_{0})\frac{T_{13}(z)\mathbb{B}_{a-1,b-1}(\bar{u}_{0};\bar{v})}{\lambda_{2}(z)h(\bar{v},z)}.$
(3.2)

Here, the sum is taken over all partitions
$\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}$ where $\\#u_{0}=1.$

Let us define the following vectors contained in the composite model, i.e., in
$\mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}$:

$\displaystyle\mathcal{B}_{a,b}(\bar{u};\bar{v})=\sum
r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})}{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})}\mathbb{B}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I}).$ (3.3)

The subscripts $a_{1}$, $b_{1}$, $a_{2}$, $b_{2}$ of the partial Bethe vectors
$\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I})$ and
$\mathbb{B}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm II})$ in the definition of
$\mathcal{B}_{a,b}(\bar{u};\bar{v})$ are omitted here and below because they
do not carry any important information. Obviously, if we prove that vector
(3.3) satisfies recursion (3.2) and an initial condition
$\mathcal{B}_{a,0}(\bar{u};\varnothing)=\mathbb{B}_{a,0}(\bar{u};\varnothing)$,
$a=0,1,\dots$, then, using induction over $b$, we immediately obtain that
$\mathcal{B}_{a,b}(\bar{u};\bar{v})=\mathbb{B}_{a,b}(\bar{u};\bar{v})$ for $a$
and $b$ arbitrary.

It is easy to see that $\mathcal{B}_{a,0}(\bar{u};\varnothing)$ coincides with
the known result for the composite $\mathfrak{gl}(2)$-invariant model [16]

$\displaystyle\mathcal{B}_{a,0}(\bar{u};\varnothing)=\sum
r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm I})f(\bar{u}_{\scriptscriptstyle\rm
II},\bar{u}_{\scriptscriptstyle\rm
I})\mathbb{B}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\varnothing)\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\varnothing)=\mathbb{B}_{a,0}(\bar{u};\varnothing).$

Thus, the initial condition is satisfied. It remains to prove that
$\mathcal{B}_{a,b}(\bar{u};\bar{v})$ satisfies the recursion

$\displaystyle\frac{T_{23}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathcal{B}_{a,b-1}(\bar{u};\bar{v})=f(z,\bar{u})\mathcal{B}_{a,b}(\bar{u};\\{z,\bar{v}\\})$
$\displaystyle\hphantom{\frac{T_{23}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathcal{B}_{a,b-1}(\bar{u};\bar{v})=}{}+\sum_{\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}}g(u_{0},z)f(u_{0},\bar{u}_{0})\frac{T_{13}(z)\mathcal{B}_{a-1,b-1}(\bar{u}_{0};\bar{v})}{\lambda_{2}(z)h(\bar{v},z)}.$
(3.4)

Here the notation is the same as in (3.2).

One can easily convince oneself that recursion relation (3.4) is a simple
consequence of the following two relations which we intend to prove in Section
4:

$\displaystyle\frac{T_{13}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathcal{B}_{a-1,b-1}(\bar{u};\bar{v})=\mathcal{B}_{a,b}(\\{z,\bar{u}\\};\\{z,\bar{v}\\}),$
(3.5)
$\displaystyle\frac{T_{23}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathcal{B}_{a,b-1}(\bar{u};\bar{v})=f(z,\bar{u})\mathcal{B}_{a,b}(\bar{u};\\{z,\bar{v}\\})$
$\displaystyle\hphantom{\frac{T_{23}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathcal{B}_{a,b-1}(\bar{u};\bar{v})=}{}+\sum_{\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}}g(u_{0},z)f(u_{0},\bar{u}_{0})\mathcal{B}_{a,b}(\\{z,\bar{u}_{0}\\};\\{z,\bar{v}\\}).$
(3.6)

The sum is performed here over all partitions of the type
$\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}$ where $\\#u_{0}=1$. Thus, the
proof of equations (3.5), (3.6) yields the proof of (3.1). ∎

######  Remark 3.2.

We can commute the factors
$\mathbb{B}^{(1)}_{a_{1},b_{1}}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I})$ and
$\mathbb{B}^{(2)}_{a_{2},b_{2}}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm II})$ in (3.1) according to

$\displaystyle g(\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm
II})\mathbb{B}_{a_{2},b_{2}}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})\mathbb{B}^{(1)}_{a_{1},b_{1}}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I})=g(\bar{v}_{\scriptscriptstyle\rm
II},\bar{v}_{\scriptscriptstyle\rm
I})\mathbb{B}^{(1)}_{a_{1},b_{1}}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm
I})\mathbb{B}_{a_{2},b_{2}}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm II})$

because their gradation is reflected in the product of the antisymmentric
functions $g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})$.

######  Theorem 3.3.

The dual Bethe vectors of the full model can be expressed as the bilinear
combination of the partial dual Bethe vectors:

$\displaystyle\mathbb{C}_{a,b}(\bar{u};\bar{v})=\sum
r_{1}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
II})r_{3}^{(2)}(\bar{v}_{\scriptscriptstyle\rm
I})\frac{f(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})g(\bar{v}_{\scriptscriptstyle\rm II},\bar{v}_{\scriptscriptstyle\rm
I})}{f(\bar{v}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})}\mathbb{C}_{a_{1},b_{1}}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm
I})\mathbb{C}^{(2)}_{a_{2},b_{2}}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm II}).$

The summation goes over all partitions
$\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}$ and
$\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm II}\\}$. The corresponding cardinalities
satisfy $a_{1}+a_{2}=a$ and $b_{1}+b_{2}=b$.

###### Proof.

The Bethe vector $\mathbb{B}_{a,b}(\bar{u};\bar{v})$ can be understood as

$\displaystyle\mathbb{B}_{a,b}(\bar{u};\bar{v})=B_{a,b}(\bar{u};\bar{v})\Omega$

where $B_{a,b}(\bar{u};\bar{v})$ is a polynomial in elements of the bialgebra
$\mathcal{A}$ which acts on the pseudovacuum $\Omega$. In other words, it is
the rest of the Bethe vector (2.9) if we erase the pseudovacuum. Hence,
formula (3.1) for Bethe vectors in the composite model can be written as

$\displaystyle\mathbb{B}_{a,b}(\bar{u};\bar{v})=B_{a,b}(\bar{u};\bar{v})\Omega=\Delta(B_{a,b}(\bar{u};\bar{v}))\Omega^{(1)}\Omega^{(2)}$
$\displaystyle\hphantom{\mathbb{B}_{a,b}(\bar{u};\bar{v})}{}=\sum
r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})}{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})}B^{(2)}_{a_{2},b_{2}}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})B^{(1)}_{a_{1},b_{1}}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I})\Omega^{(1)}\Omega^{(2)}.$

The antimorphism (2.5) relates the Bethe vectors to the dual Bethe vectors,
cf. (2.10),

$\displaystyle\mathbb{C}_{a,b}(\bar{u};\bar{v})=\psi(\mathbb{B}_{a,b}(\bar{u};\bar{v}))=\psi(\Omega)\psi(B_{a,b}(\bar{u};\bar{v}))=\Omega^{\dagger}\psi(B_{a,b}(\bar{u};\bar{v})).$

Due to the composition rule (2.6) for the antimorphism $\psi$ with the
coproducts (2.4) and (2.7), we obtain for the composite model

$\displaystyle\mathbb{C}_{a,b}(\bar{u};\bar{v})=\Omega^{\dagger(1)}\Omega^{\dagger(2)}\left[\Delta\circ\psi(B_{a,b}(\bar{u};\bar{v}))\right]=\Omega^{\dagger(1)}\Omega^{\dagger(2)}\left[(\psi\otimes\psi)\circ\Delta^{\prime}(B_{a,b}(\bar{u};\bar{v}))\right]$
$\displaystyle=\Omega^{\dagger(1)}\Omega^{\dagger(2)}\left[(\psi\otimes\psi)\sum
r_{1}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(2)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})}{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})}B^{(1)}_{a_{2},b_{2}}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})B^{(2)}_{a_{1},b_{1}}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I})\right].$

We stress that the factors
$B_{a_{2},b_{2}}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})B_{a_{1},b_{1}}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I})$ are changed to
$B_{a_{2},b_{2}}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})B_{a_{1},b_{1}}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I})$ in comparison with Theorem 3.1. In the
same way, the functions $r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm II})$ are changed to
$r_{1}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(2)}(\bar{v}_{\scriptscriptstyle\rm II})$. This is due to the use of
the opposite coproduct (2.7) instead of the standard one (2.4). After the
application of $\psi\otimes\psi$

$\displaystyle\mathbb{C}_{a,b}(\bar{u};\bar{v})=\sum
r_{1}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(2)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})}{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})}\mathbb{C}^{(1)}_{a_{2},b_{2}}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})\mathbb{C}^{(2)}_{a_{1},b_{1}}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I}).$

Renaming the sets of variables as $\bar{u}_{\scriptscriptstyle\rm
I}\leftrightarrow\bar{u}_{\scriptscriptstyle\rm II}$,
$\bar{v}_{\scriptscriptstyle\rm
I}\leftrightarrow\bar{v}_{\scriptscriptstyle\rm II}$, we arrive at the
statement of the theorem. ∎

##  4 Action of $\boldsymbol{T_{13}(z)}$ and $\boldsymbol{T_{23}(z)}$ on
$\boldsymbol{\mathcal{B}_{a,b}(\bar{u};\bar{v})}$

We aim to prove that the supervectors (3.3) satisfy

$\displaystyle\frac{T_{13}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathcal{B}_{a-1,b-1}(\bar{u};\bar{v})=\mathcal{B}_{a,b}(\bar{\eta};\bar{\xi}),$
(4.1)

where we introduce new sets of spectral parameters
$\bar{\eta}=\\{z,\bar{u}\\}$ and $\bar{\xi}=\\{z,\bar{v}\\}$. Equation (4.1)
is just one of the properties satisfied by the Bethe vector
$\mathbb{B}_{a-1,b-1}(\bar{u};\bar{v})$, as remarked in Appendix A. The
strategy of the proof is simple. We investigate both sides of (4.1) separately
and then show that they coincide. To this end, we use the known formulas for
the action of the monodromy matrix elements on the Bethe vectors listed in
Appendix A.

The right-hand side of (4.1) has the form

$\displaystyle\mathcal{B}_{a,b}(\bar{\eta};\bar{\xi})=\sum
r_{1}^{(2)}(\bar{\eta}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(\bar{\xi}_{\scriptscriptstyle\rm
II})\frac{f(\bar{\eta}_{\scriptscriptstyle\rm
II},\bar{\eta}_{\scriptscriptstyle\rm I})g(\bar{\xi}_{\scriptscriptstyle\rm
I},\bar{\xi}_{\scriptscriptstyle\rm II})}{f(\bar{\xi}_{\scriptscriptstyle\rm
II},\bar{\eta}_{\scriptscriptstyle\rm
I})}\mathbb{B}^{(2)}(\bar{\eta}_{\scriptscriptstyle\rm
II};\bar{\xi}_{\scriptscriptstyle\rm
II})\mathbb{B}^{(1)}(\bar{\eta}_{\scriptscriptstyle\rm
I};\bar{\xi}_{\scriptscriptstyle\rm I}),$

where we just used definition (3.3). From the analysis how the parameter $z$
can enter the subsets $\bar{\eta}_{\scriptscriptstyle\rm I}$,
$\bar{\xi}_{\scriptscriptstyle\rm I}$, $\bar{\eta}_{\scriptscriptstyle\rm
II}$, $\bar{\xi}_{\scriptscriptstyle\rm II}$ we obtain three cases:

$\displaystyle(i)\quad$ $\displaystyle\bar{\eta}_{\scriptscriptstyle\rm
I}=\\{z,\bar{u}_{\scriptscriptstyle\rm I}\\},\qquad$
$\displaystyle\bar{\xi}_{\scriptscriptstyle\rm
I}=\\{z,\bar{v}_{\scriptscriptstyle\rm I}\\},\qquad$
$\displaystyle\bar{\eta}_{\scriptscriptstyle\rm
II}=\bar{u}_{\scriptscriptstyle\rm II},\qquad$
$\displaystyle\bar{\xi}_{\scriptscriptstyle\rm
II}=\bar{v}_{\scriptscriptstyle\rm II},$ $\displaystyle(ii)\quad$
$\displaystyle\bar{\eta}_{\scriptscriptstyle\rm
I}=\bar{u}_{\scriptscriptstyle\rm I},\qquad$
$\displaystyle\bar{\xi}_{\scriptscriptstyle\rm
I}=\bar{v}_{\scriptscriptstyle\rm I},\qquad$
$\displaystyle\bar{\eta}_{\scriptscriptstyle\rm
II}=\\{z,\bar{u}_{\scriptscriptstyle\rm II}\\},\qquad$
$\displaystyle\bar{\xi}_{\scriptscriptstyle\rm
II}=\\{z,\bar{v}_{\scriptscriptstyle\rm II}\\},$ $\displaystyle(iii)\quad$
$\displaystyle\bar{\eta}_{\scriptscriptstyle\rm
I}=\bar{u}_{\scriptscriptstyle\rm I},\qquad$
$\displaystyle\bar{\xi}_{\scriptscriptstyle\rm
I}=\\{z,\bar{v}_{\scriptscriptstyle\rm I}\\},\qquad$
$\displaystyle\bar{\eta}_{\scriptscriptstyle\rm
II}=\\{z,\bar{u}_{\scriptscriptstyle\rm II}\\},\qquad$
$\displaystyle\bar{\xi}_{\scriptscriptstyle\rm
II}=\bar{v}_{\scriptscriptstyle\rm II}.$

The case, where $z\in\bar{\xi}_{\scriptscriptstyle\rm II}$ and
$z\in\bar{\eta}_{\scriptscriptstyle\rm I}$, gives a vanishing contribution
because of the function $f(\bar{\xi}_{\scriptscriptstyle\rm
II},\bar{\eta}_{\scriptscriptstyle\rm I})$ in the denominator. The vector
$\mathcal{B}_{a,b}(\bar{\eta};\bar{\xi})$ is thus composed of three parts with
different structure

$\displaystyle\mathcal{B}_{a,b}(\bar{\eta};\bar{\xi})=A_{1}+A_{2}+A_{3},$

where

$\displaystyle A_{1}=\sum
r_{1}^{(2)}(z)r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(\bar{u}_{\scriptscriptstyle\rm
II},z)f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})g(z,\bar{v}_{\scriptscriptstyle\rm II})}{f(\bar{v}_{\scriptscriptstyle\rm
II},\bar{u}_{\scriptscriptstyle\rm I})f(\bar{v}_{\scriptscriptstyle\rm
II},z)}$
$\displaystyle\hphantom{A_{1}=}{}\times\mathbb{B}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})\mathbb{B}^{(1)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
I}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm I}\\}),$ $\displaystyle A_{2}=\sum
r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(z)r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},z)g(\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm II})}{f(\bar{v}_{\scriptscriptstyle\rm
II},\bar{u}_{\scriptscriptstyle\rm
I})}\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
II}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm
II}\\})\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I}),$ $\displaystyle A_{3}=\sum
r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(z,\bar{u}_{\scriptscriptstyle\rm
I})f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(z,\bar{v}_{\scriptscriptstyle\rm II})g(\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm II})}{f(\bar{v}_{\scriptscriptstyle\rm
II},\bar{u}_{\scriptscriptstyle\rm
I})}\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
II}\\};\bar{v}_{\scriptscriptstyle\rm
II})\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\\{z,\bar{v}_{\scriptscriptstyle\rm I}\\}).$

As the supervector $\mathcal{B}_{a-1,b-1}(\bar{u};\bar{v})$ belongs to
$\mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}$, the action of $T_{13}(z)$ on it
is defined via a coproduct. We use the standard one (2.4)

$\displaystyle\Delta(T_{13}(z))=T_{13}^{(1)}(z)T_{11}^{(2)}(z)+T_{23}^{(1)}(z)T_{12}^{(2)}(z)+T_{33}^{(1)}(z)T_{13}^{(2)}(z).$

Hence, the left-hand side of (4.1) decomposes into three parts

$\displaystyle\frac{T_{13}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathcal{B}_{a-1,b-1}(\bar{u};\bar{v})=C_{1}+C_{2}+C_{3}$
$\displaystyle\qquad{}=\left(\frac{T_{13}^{(1)}(z)T_{11}^{(2)}(z)}{\lambda_{2}(z)h(\bar{v},z)}+\frac{T_{23}^{(1)}(z)T_{12}^{(2)}(z)}{\lambda_{2}(z)h(\bar{v},z)}+\frac{T_{33}^{(1)}(z)T_{13}^{(2)}(z)}{\lambda_{2}(z)h(\bar{v},z)}\right)\mathcal{B}_{a-1,b-1}(\bar{u};\bar{v}).$
(4.2)

The specific parts $C_{k}$, $k=1,2,3$, can be written as

$\displaystyle C_{k}=\sum(-1)^{(1+[k])\cdot
b_{2}}r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})}{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})}$
$\displaystyle\hphantom{C_{k}=}{}\times\frac{T_{1k}^{(2)}(z)}{\lambda_{2}^{(2)}(z)h(\bar{v}_{\scriptscriptstyle\rm
II},z)}\mathbb{B}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})\frac{T_{k3}^{(1)}(z)}{\lambda_{2}^{(1)}(z)h(\bar{v}_{\scriptscriptstyle\rm
I},z)}\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I}),$

where $b_{2}=\\#\bar{v}_{\scriptscriptstyle\rm II}$. The sign factor in
$C_{k}$ appears because of the oddness of the monodromy matrix element
$T_{i3}(z)$, $i=1,2$. It is absorbed during the calculations by the
antisymmetric functions $g(u,v)$.

Using formulas for the action of the monodromy matrix elements on the Bethe
vectors listed in Appendix A, we obtain the explicit forms of $C_{k}$:

$\displaystyle
C_{1}=\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}\\\
\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm
II}\\}\end{subarray}}r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})}{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})}\Bigg{\\{}r_{1}^{(2)}(z)\frac{f(\bar{u}_{\scriptscriptstyle\rm
II},z)g(z,\bar{v}_{\scriptscriptstyle\rm
II})}{f(\bar{v}_{\scriptscriptstyle\rm
II},z)}\mathbb{B}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm II})$
$\displaystyle\hphantom{C_{1}=}{}+\sum_{\bar{u}_{\scriptscriptstyle\rm
II}\Rightarrow\\{u_{\scriptscriptstyle\rm i},\bar{u}_{\scriptscriptstyle\rm
ii}\\}}r_{1}^{(2)}(u_{\scriptscriptstyle\rm
i})\frac{f(\bar{u}_{\scriptscriptstyle\rm ii},u_{\scriptscriptstyle\rm
i})g(z,u_{\scriptscriptstyle\rm i})g(z,\bar{v}_{\scriptscriptstyle\rm
II})}{f(\bar{v}_{\scriptscriptstyle\rm II},u_{\scriptscriptstyle\rm
i})}\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
ii}\\};\bar{v}_{\scriptscriptstyle\rm II})$
$\displaystyle\hphantom{C_{1}=}{}+\\!\sum_{\begin{subarray}{c}\bar{u}_{\scriptscriptstyle\rm
II}\Rightarrow\\{u_{\scriptscriptstyle\rm i},\bar{u}_{\scriptscriptstyle\rm
ii}\\}\\\ \bar{v}_{\scriptscriptstyle\rm
II}\Rightarrow\\{v_{\scriptscriptstyle\rm i},\bar{v}_{\scriptscriptstyle\rm
ii}\\}\end{subarray}}\\!r_{1}^{(2)}(u_{\scriptscriptstyle\rm
i})\frac{f(\bar{u}_{\scriptscriptstyle\rm ii},u_{\scriptscriptstyle\rm
i})g(v_{\scriptscriptstyle\rm i},z)g(v_{\scriptscriptstyle\rm
i},\bar{v}_{\scriptscriptstyle\rm ii})}{f(\bar{v}_{\scriptscriptstyle\rm
ii},u_{\scriptscriptstyle\rm i})h(v_{\scriptscriptstyle\rm
i},z)h(v_{\scriptscriptstyle\rm i},u_{\scriptscriptstyle\rm
i})}\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
ii}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm
ii}\\})\Bigg{\\}}\mathbb{B}^{(1)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
I}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm I}\\}),\\!$ $\displaystyle
C_{2}=\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}\\\
\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm
II}\\}\end{subarray}}r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})}{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})}$
$\displaystyle\hphantom{C_{2}=}{}\times\Bigg{\\{}g(z,\bar{v}_{\scriptscriptstyle\rm
II})\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
II}\\};\bar{v}_{\scriptscriptstyle\rm
II})+\sum_{\bar{v}_{\scriptscriptstyle\rm
II}\Rightarrow\\{v_{\scriptscriptstyle\rm i},\bar{v}_{\scriptscriptstyle\rm
ii}\\}}\frac{g(v_{\scriptscriptstyle\rm i},z)g(v_{\scriptscriptstyle\rm
i},\bar{v}_{\scriptscriptstyle\rm ii})}{h(v_{\scriptscriptstyle\rm
i},z)}\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
II}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm ii}\\})\Bigg{\\}}$
$\displaystyle\hphantom{C_{2}=}{}\times\Bigg{\\{}f(z,\bar{u}_{\scriptscriptstyle\rm
I})\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\\{z,\bar{v}_{\scriptscriptstyle\rm
I}\\})+\sum_{\bar{u}_{\scriptscriptstyle\rm
I}\Rightarrow\\{u_{\scriptscriptstyle\rm i},\bar{u}_{\scriptscriptstyle\rm
ii}\\}}g(u_{\scriptscriptstyle\rm i},z)f(u_{\scriptscriptstyle\rm
i},\bar{u}_{\scriptscriptstyle\rm
ii})\mathbb{B}^{(1)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
ii}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm I}\\})\Bigg{\\}},$ $\displaystyle
C_{3}=\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}\\\
\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm
II}\\}\end{subarray}}r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})}{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})}\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
II}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm II}\\})$
$\displaystyle\hphantom{C_{3}=}{}\times\Bigg{\\{}r_{3}^{(1)}(z)g(\bar{v}_{\scriptscriptstyle\rm
I},z)\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I})+\\!\sum_{\bar{v}_{\scriptscriptstyle\rm
I}\Rightarrow\\{v_{\scriptscriptstyle\rm i},\bar{v}_{\scriptscriptstyle\rm
ii}\\}}\\!r_{3}^{(1)}(v_{\scriptscriptstyle\rm
i})\frac{f(z,\bar{u}_{\scriptscriptstyle\rm I})g(z,v_{\scriptscriptstyle\rm
i})g(\bar{v}_{\scriptscriptstyle\rm ii},v_{\scriptscriptstyle\rm
i})}{h(v_{\scriptscriptstyle\rm i},z)f(v_{\scriptscriptstyle\rm
i},\bar{u}_{\scriptscriptstyle\rm
I})}\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\\{z,\bar{v}_{\scriptscriptstyle\rm ii}\\})$
$\displaystyle\hphantom{C_{3}=}{}+\sum_{\begin{subarray}{c}\bar{u}_{\scriptscriptstyle\rm
I}\Rightarrow\\{u_{\scriptscriptstyle\rm i},\bar{u}_{\scriptscriptstyle\rm
ii}\\}\\\ \bar{v}_{\scriptscriptstyle\rm
I}\Rightarrow\\{v_{\scriptscriptstyle\rm i},\bar{v}_{\scriptscriptstyle\rm
ii}\\}\end{subarray}}r_{3}^{(1)}(v_{\scriptscriptstyle\rm
i})\frac{g(u_{\scriptscriptstyle\rm i},z)f(u_{\scriptscriptstyle\rm
i},\bar{u}_{\scriptscriptstyle\rm ii})g(z,v_{\scriptscriptstyle\rm
i})g(\bar{v}_{\scriptscriptstyle\rm ii},v_{\scriptscriptstyle\rm
i})}{h(v_{\scriptscriptstyle\rm i},u_{\scriptscriptstyle\rm
i})f(v_{\scriptscriptstyle\rm i},z)f(v_{\scriptscriptstyle\rm
i},\bar{u}_{\scriptscriptstyle\rm
ii})}\mathbb{B}^{(1)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
ii}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm ii}\\})\Bigg{\\}}.$

The summations are performed over all possible partitions of the sets
$\bar{u}$, $\bar{v}$ of the original Bethe parameters into their subsets
$\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}$,
$\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm II}\\}$. Some of these subsets are divided
into additional subsets, e.g., $\bar{u}_{\scriptscriptstyle\rm
II}\Rightarrow\\{u_{\scriptscriptstyle\rm i},\bar{u}_{\scriptscriptstyle\rm
ii}\\}$ where $\\#u_{\scriptscriptstyle\rm i}=1$, and the summation is
performed again over all such partitions. The same for the other additional
divisions of $\bar{v}_{\scriptscriptstyle\rm II}$,
$\bar{u}_{\scriptscriptstyle\rm I}$, $\bar{v}_{\scriptscriptstyle\rm I}$.

We can moreover see that the sum over partitions in $C_{1}$ involving the
product of Bethe vectors $\mathbb{B}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})\mathbb{B}^{(1)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
I}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm I}\\})$ coincides with the term
$A_{1}$. The sum over partitions involving
$\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
II}\\};\bar{v}_{\scriptscriptstyle\rm
II})\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\\{z,\bar{v}_{\scriptscriptstyle\rm I}\\})$ in $C_{2}$ coincides with
$A_{3}$. Similarly, the sum over partitions in $C_{3}$ containing
$\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
II}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm
II}\\})\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I})$ coincides with $A_{2}$. The remaining
terms of $C_{1}$, $C_{2}$, $C_{3}$ cancel mutually, as we show below.

There are two remaining terms containing the product of Bethe vectors of this
type

$\displaystyle\mathbb{B}^{(2)}(\\{z,\bar{u}^{\prime}\\};\\{z,\bar{v}^{\prime}\\})\mathbb{B}^{(1)}(\bar{u}^{\prime\prime};\\{z,\bar{v}^{\prime\prime}\\}),$

where primes mean any subset of $\bar{u}$ or $\bar{v}$. Namely, the first term
comes from $C_{2}$

$\displaystyle
C_{2,3}=\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}\\\
\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},v_{\scriptscriptstyle\rm i},\bar{v}_{\scriptscriptstyle\rm
ii}\\}\end{subarray}}r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(v_{\scriptscriptstyle\rm
i})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
ii})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
ii})g(\bar{v}_{\scriptscriptstyle\rm I},v_{\scriptscriptstyle\rm
i})g(v_{\scriptscriptstyle\rm i},\bar{v}_{\scriptscriptstyle\rm
ii})f(z,\bar{u}_{\scriptscriptstyle\rm I})}{f(\bar{v}_{\scriptscriptstyle\rm
ii},\bar{u}_{\scriptscriptstyle\rm I})f(v_{\scriptscriptstyle\rm
i},\bar{u}_{\scriptscriptstyle\rm I})h(v_{\scriptscriptstyle\rm i},z)}$
$\displaystyle\hphantom{C_{2,3}=}{}\times g(v_{\scriptscriptstyle\rm
i},z)\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
II}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm
ii}\\})\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\\{z,\bar{v}_{\scriptscriptstyle\rm I}\\})$

and the second term from $C_{3}$

$\displaystyle
C_{3,2}=\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}\\\
\bar{v}\Rightarrow\\{v_{i},\bar{v}_{\scriptscriptstyle\rm
ii},\bar{v}_{\scriptscriptstyle\rm
II}\\}\end{subarray}}r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(v_{\scriptscriptstyle\rm
i})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm ii},\bar{v}_{\scriptscriptstyle\rm
II})g(v_{\scriptscriptstyle\rm i},\bar{v}_{\scriptscriptstyle\rm
II})g(\bar{v}_{\scriptscriptstyle\rm ii},v_{\scriptscriptstyle\rm
i})f(z,\bar{u}_{\scriptscriptstyle\rm I})}{f(\bar{v}_{\scriptscriptstyle\rm
II},\bar{u}_{\scriptscriptstyle\rm I})f(v_{\scriptscriptstyle\rm
i},\bar{u}_{\scriptscriptstyle\rm I})h(v_{\scriptscriptstyle\rm i},z)}$
$\displaystyle\hphantom{C_{3,2}=}{}\times g(z,v_{\scriptscriptstyle\rm
i})\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
II}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm
II}\\})\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\\{z,\bar{v}_{\scriptscriptstyle\rm ii}\\}).$ (4.3)

Renaming the sets $\bar{v}_{\scriptscriptstyle\rm
II}\rightarrow\bar{v}_{\scriptscriptstyle\rm ii}$ and
$\bar{v}_{\scriptscriptstyle\rm ii}\rightarrow\bar{v}_{\scriptscriptstyle\rm
I}$ in (4.3), we obtain

$\displaystyle
C_{3,2}=\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}\\\
\bar{v}\Rightarrow\\{v_{\scriptscriptstyle\rm
i},\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
ii}\\}\end{subarray}}r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{3}^{(1)}(v_{\scriptscriptstyle\rm
i})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
ii})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})g(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
ii})g(v_{\scriptscriptstyle\rm i},\bar{v}_{\scriptscriptstyle\rm
ii})g(\bar{v}_{\scriptscriptstyle\rm I},v_{\scriptscriptstyle\rm
i})f(z,\bar{u}_{\scriptscriptstyle\rm I})}{f(\bar{v}_{\scriptscriptstyle\rm
ii},\bar{u}_{\scriptscriptstyle\rm I})f(v_{\scriptscriptstyle\rm
i},\bar{u}_{\scriptscriptstyle\rm I})h(v_{\scriptscriptstyle\rm i},z)}$
$\displaystyle\hphantom{C_{3,2}=}{}\times g(z,v_{\scriptscriptstyle\rm
i})\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
II}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm
ii}\\})\mathbb{B}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I};\\{z,\bar{v}_{\scriptscriptstyle\rm I}\\}).$

Due to the antisymmetry of the function $g(z,v_{\scriptscriptstyle\rm i})$, we
see that $C_{2,3}+C_{3,2}=0$.

There are two terms containing the product of Bethe vectors of the type

$\displaystyle\mathbb{B}^{(2)}(\\{z,\bar{u}^{\prime}\\};\bar{v}^{\prime})\mathbb{B}^{(1)}(\\{z,\bar{u}^{\prime\prime}\\};\\{z,\bar{v}^{\prime\prime}\\}).$

One such term is contained in $C_{1}$ and one in $C_{2}$. One can prove that
their sum vanishes by similar argumentation as above.

The remaining three terms contain Bethe vectors of the type

$\displaystyle\mathbb{B}^{(2)}(\\{z,\bar{u}^{\prime}\\};\\{z,\bar{v}^{\prime}\\})\mathbb{B}^{(1)}(\\{z,\bar{u}^{\prime\prime}\\};\\{z,\bar{v}^{\prime\prime}\\}).$

They are the following:

$\displaystyle
C_{1,3}=\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},u_{\scriptscriptstyle\rm i},\bar{u}_{\scriptscriptstyle\rm ii}\\}\\\
\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},v_{\scriptscriptstyle\rm i},\bar{v}_{\scriptscriptstyle\rm
ii}\\}\end{subarray}}r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})r_{1}^{(2)}(u_{\scriptscriptstyle\rm
i})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
ii})r_{3}^{(1)}(v_{\scriptscriptstyle\rm i})\frac{f(u_{\scriptscriptstyle\rm
i},\bar{u}_{\scriptscriptstyle\rm I})f(\bar{u}_{\scriptscriptstyle\rm
ii},\bar{u}_{\scriptscriptstyle\rm I})f(\bar{u}_{\scriptscriptstyle\rm
ii},u_{\scriptscriptstyle\rm i})}{f(\bar{v}_{\scriptscriptstyle\rm
ii},\bar{u}_{\scriptscriptstyle\rm I})f(v_{\scriptscriptstyle\rm
i},\bar{u}_{\scriptscriptstyle\rm I})f(\bar{v}_{\scriptscriptstyle\rm
ii},u_{\scriptscriptstyle\rm i})f(v_{\scriptscriptstyle\rm
i},u_{\scriptscriptstyle\rm i})}$ (4.4)
$\displaystyle\hphantom{C_{1,3}=}{}\times\frac{g(\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm ii})g(\bar{v}_{\scriptscriptstyle\rm
I},v_{\scriptscriptstyle\rm i})g(v_{\scriptscriptstyle\rm
i},\bar{v}_{\scriptscriptstyle\rm ii})}{h(v_{\scriptscriptstyle\rm
i},z)}g(v_{\scriptscriptstyle\rm i},z)g(v_{\scriptscriptstyle\rm
i},u_{\scriptscriptstyle\rm
i})\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
ii}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm
ii}\\})\mathbb{B}^{(1)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
I}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm I}\\}),$ $\displaystyle
C_{2,4}=\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{u_{\scriptscriptstyle\rm
i},\bar{u}_{\scriptscriptstyle\rm ii},\bar{u}_{\scriptscriptstyle\rm II}\\}\\\
\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},v_{\scriptscriptstyle\rm i},\bar{v}_{\scriptscriptstyle\rm
ii}\\}\end{subarray}}r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
ii})r_{1}^{(2)}(u_{\scriptscriptstyle\rm
i})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
ii})r_{3}^{(1)}(v_{\scriptscriptstyle\rm
i})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
ii})f(\bar{u}_{\scriptscriptstyle\rm II},u_{\scriptscriptstyle\rm
i})f(u_{\scriptscriptstyle\rm i},\bar{u}_{\scriptscriptstyle\rm
ii})}{f(\bar{v}_{\scriptscriptstyle\rm ii},\bar{u}_{\scriptscriptstyle\rm
ii})f(\bar{v}_{\scriptscriptstyle\rm ii},u_{\scriptscriptstyle\rm
i})f(v_{\scriptscriptstyle\rm i},\bar{u}_{\scriptscriptstyle\rm
ii})f(v_{\scriptscriptstyle\rm i},u_{\scriptscriptstyle\rm i})}$ (4.5)
$\displaystyle\hphantom{C_{2,4}=}{}\times\frac{g(\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm ii})g(\bar{v}_{\scriptscriptstyle\rm
I},v_{\scriptscriptstyle\rm i})g(v_{\scriptscriptstyle\rm
i},\bar{v}_{\scriptscriptstyle\rm ii})}{h(v_{\scriptscriptstyle\rm
i},z)}g(v_{\scriptscriptstyle\rm i},z)g(u_{\scriptscriptstyle\rm
i},z)\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
II}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm
ii}\\})\mathbb{B}^{(1)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
ii}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm I}\\}),$ $\displaystyle
C_{3,3}=\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{u_{\scriptscriptstyle\rm
i},\bar{u}_{\scriptscriptstyle\rm ii},\bar{u}_{\scriptscriptstyle\rm II}\\}\\\
\bar{v}\Rightarrow\\{v_{\scriptscriptstyle\rm
i},\bar{v}_{\scriptscriptstyle\rm ii},\bar{v}_{\scriptscriptstyle\rm
II}\\}\end{subarray}}r_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
ii})r_{1}^{(2)}(u_{\scriptscriptstyle\rm
i})r_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})r_{3}^{(1)}(v_{\scriptscriptstyle\rm
i})\frac{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
ii})f(\bar{u}_{\scriptscriptstyle\rm II},u_{\scriptscriptstyle\rm
i})f(u_{\scriptscriptstyle\rm i},\bar{u}_{\scriptscriptstyle\rm
ii})}{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
ii})f(\bar{v}_{\scriptscriptstyle\rm II},u_{\scriptscriptstyle\rm
i})f(v_{\scriptscriptstyle\rm i},\bar{u}_{\scriptscriptstyle\rm
ii})f(v_{\scriptscriptstyle\rm i},u_{\scriptscriptstyle\rm i})}$ (4.6)
$\displaystyle\hphantom{C_{3,3}=}{}\times\frac{g(\bar{v}_{\scriptscriptstyle\rm
ii},\bar{v}_{\scriptscriptstyle\rm II})g(v_{\scriptscriptstyle\rm
i},\bar{v}_{\scriptscriptstyle\rm II})g(\bar{v}_{\scriptscriptstyle\rm
ii},v_{\scriptscriptstyle\rm i})}{h(v_{\scriptscriptstyle\rm
i},z)}g(z,u_{\scriptscriptstyle\rm i})g(v_{\scriptscriptstyle\rm
i},u_{\scriptscriptstyle\rm
i})\mathbb{B}^{(2)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
II}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm
II}\\})\mathbb{B}^{(1)}(\\{z,\bar{u}_{\scriptscriptstyle\rm
ii}\\};\\{z,\bar{v}_{\scriptscriptstyle\rm ii}\\}).$

We rename the sets as $\bar{u}_{\scriptscriptstyle\rm
II}\rightarrow\bar{u}_{\scriptscriptstyle\rm ii}$ and
$\bar{u}_{\scriptscriptstyle\rm ii}\rightarrow\bar{u}_{\scriptscriptstyle\rm
I}$ in (4.5). In (4.6), we rename the sets as $\bar{u}_{\scriptscriptstyle\rm
II},\bar{v}_{\scriptscriptstyle\rm
II}\rightarrow\bar{u}_{\scriptscriptstyle\rm
ii},\bar{v}_{\scriptscriptstyle\rm ii}$ and $\bar{u}_{\scriptscriptstyle\rm
ii},\bar{v}_{\scriptscriptstyle\rm
ii}\rightarrow\bar{u}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
I}$. Thus, we obtain all the Bethe vectors in (4.4)–(4.6) with the same
arguments. Due to the identity

$\displaystyle g(v_{\scriptscriptstyle\rm i},z)g(v_{\scriptscriptstyle\rm
i},u_{\scriptscriptstyle\rm i})+g(v_{\scriptscriptstyle\rm
i},z)g(u_{\scriptscriptstyle\rm i},z)+g(z,u_{\scriptscriptstyle\rm
i})g(v_{\scriptscriptstyle\rm i},u_{\scriptscriptstyle\rm i})=0,$

we see that $C_{1,3}+C_{2,4}+C_{3,3}=0$. Equality (4.1) is thus proved.

The action of $T_{23}(z)$ on $\mathcal{B}_{a,b-1}(\bar{u};\bar{v})$ described
in (3.6) can be proved in a similar manner. We again use the coproduct (2.4)
to define the action of $T_{23}(z)$ on the direct product of two partial Bethe
vectors $\mathbb{B}^{(2)}_{a_{2},b_{2}}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})\mathbb{B}^{(1)}_{a_{1},b_{1}}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I})$ and formulas from Appendix A for the
action of the monodromy matrix elements on Bethe vectors. The case of
$T_{23}(z)$ involves more lengthy calculations than that of $T_{13}(z)$. As
the reasoning is rather similar, we do not provide the details here.

##  5 Composite $\boldsymbol{\mathfrak{gl}(1|2)}$-invariant model

Let us denote the RTT algebra corresponding to $\mathfrak{gl}(1|2)$ as
$\widetilde{\mathcal{A}}$ and its elements as $\widetilde{T}_{ij}(u)$ to
distinguish them from their $\mathfrak{gl}(2|1)$ equivalents. The gradation in
the $\mathfrak{gl}(1|2)$ case is described by the parity function
$\widetilde{[\ ]}$, where $\widetilde{[1]}=0$ and
$\widetilde{[2]}=\widetilde{[3]}=1$. The algebraic structure is governed by
the bilinear relation (2.3) provided that all relevant objects are marked by
tildas.

For the $\mathfrak{gl}(1|2)$-invariant models the pseudovacuum is denoted as
$\widetilde{\Omega}$ and the corresponding eigenvalues of
$\widetilde{T}_{jj}(u)$, $j=1,2,3$, are $\widetilde{\lambda}_{j}(u)$. The dual
pseudovacuum is denoted as $\widetilde{\Omega}^{\dagger}$.

There is a relation between gradations on $\mathcal{A}$ and
$\widetilde{\mathcal{A}}$: $[i]=\widetilde{[4-i]}+1$ $(\mathop{\rm{mod}}2)$,
$i=1,2,3$.

The superalgebras $\mathcal{A}$ and $\widetilde{\mathcal{A}}$ are isomorphic,
as was shown in [26]. This is due to the map

$\displaystyle\varphi\colon\
\begin{cases}\mathcal{A}\rightarrow\widetilde{\mathcal{A}},\\\
T_{ij}(u)\rightarrow(-1)^{[i][j]+[j]+1}\widetilde{T}_{4-j,4-i}(u),\\\
\lambda_{j}(u)\rightarrow-\widetilde{\lambda}_{4-j}(u)=\lambda_{j}(u).\end{cases}$

The ratio function $r_{i}(u)$, $i=1,3$, is mapped to
$\widetilde{r}_{4-i}(u)=\widetilde{\lambda}_{4-i}(u)/\widetilde{\lambda}_{2}(u)$.
Moreover $\varphi$ is a homomorphism $\varphi(AB)=\varphi(A)\varphi(B)$. The
map

$\displaystyle\widetilde{\varphi}\colon\
\begin{cases}\widetilde{\mathcal{A}}\rightarrow\mathcal{A},\\\
\widetilde{T}_{ij}(u)\rightarrow(-1)^{\widetilde{[i]}\widetilde{[j]}+\widetilde{[j]}+1}T_{4-j,4-i}(u),\\\
\widetilde{\lambda_{j}}(u)\rightarrow-\lambda_{4-j}(u)=\widetilde{\lambda}_{j}(u)\end{cases}$

is inverse to $\varphi$, i.e.,
$\widetilde{\varphi}\circ\varphi=\mathop{\mathrm{id}}$ and
$\varphi\circ\widetilde{\varphi}=\widetilde{\mathop{\mathrm{id}}}$, where
$\mathop{\mathrm{id}}$ is the identity map on $\mathcal{A}$ and
$\widetilde{\mathop{\mathrm{id}}}$ is the identity map on
$\widetilde{\mathcal{A}}$.

The (dual) Bethe vectors for the $\mathfrak{gl}(1|2)$-invariant models were
constructed in [26]:

$\displaystyle\widetilde{\mathbb{B}}_{a,b}(\bar{u};\bar{v})=(-1)^{a}\sum\frac{g(\bar{u}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm I})f(\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm II})g(\bar{u}_{\scriptscriptstyle\rm
II},\bar{u}_{\scriptscriptstyle\rm I})h(\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm
I})}{\widetilde{\lambda}_{2}(\bar{u}_{\scriptscriptstyle\rm
II})\widetilde{\lambda}_{2}(\bar{v})f(\bar{u},\bar{v})}\widetilde{\mathbb{T}}_{13}(\bar{v}_{\scriptscriptstyle\rm
I})\widetilde{T}_{23}(\bar{v}_{\scriptscriptstyle\rm
II})\widetilde{\mathbb{T}}_{12}(\bar{u}_{\scriptscriptstyle\rm
II})\widetilde{\Omega},$
$\displaystyle\widetilde{\mathbb{C}}_{a,b}(\bar{u};\bar{v})=(-1)^{\frac{a(a-1)}{2}}\sum\frac{g(\bar{u}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm I})f(\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm II})g(\bar{u}_{\scriptscriptstyle\rm
II},\bar{u}_{\scriptscriptstyle\rm I})h(\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm
I})}{\widetilde{\lambda}_{2}(\bar{u}_{\scriptscriptstyle\rm
II})\widetilde{\lambda}_{2}(\bar{v})f(\bar{u},\bar{v})}\widetilde{\Omega}^{\dagger}\widetilde{\mathbb{T}}_{21}(\bar{u}_{\scriptscriptstyle\rm
II})\widetilde{T}_{32}(\bar{v}_{\scriptscriptstyle\rm
II})\widetilde{\mathbb{T}}_{31}(\bar{v}_{\scriptscriptstyle\rm I}),$

where the sums go over all partitions
$\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}$ and
$\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm II}\\}$ under the constraint
$\\#\bar{u}_{\scriptscriptstyle\rm I}=\\#\bar{v}_{\scriptscriptstyle\rm I}$.
We remind that the products of the odd operators $\widetilde{T}_{1i}(u)$ and
$\widetilde{T}_{i1}(u)$, $i=2,3$, are symmetrised

$\displaystyle\widetilde{\mathbb{T}}_{1i}(\bar{u})\equiv\frac{\widetilde{T}_{1i}(u_{1})\widetilde{T}_{1i}(u_{2})\cdots\widetilde{T}_{1i}(u_{n})}{\prod\limits_{1\leq
j<k\leq
n}h(u_{k},u_{j})},\qquad\widetilde{\mathbb{T}}_{i1}(\bar{u})\equiv\frac{\widetilde{T}_{i1}(u_{1})\widetilde{T}_{i1}(u_{2})\cdots\widetilde{T}_{i1}(u_{n})}{\prod\limits_{1\leq
j<k\leq n}h(u_{j},u_{k})}.$

If we assume that $\varphi(\Omega)=\widetilde{\Omega}$ and
$\varphi(\Omega^{\dagger})=\widetilde{\Omega}^{\dagger}$, it can be shown that

$\displaystyle\varphi(\mathbb{B}_{a,b}(\bar{u};\bar{v}))=\widetilde{\mathbb{B}}_{b,a}(\bar{v};\bar{u}),\qquad\varphi(\mathbb{C}_{a,b}(\bar{u};\bar{v}))=\widetilde{\mathbb{C}}_{b,a}(\bar{v};\bar{u}).$

The coalgebraic structure on $\widetilde{\mathcal{A}}$ and $\mathcal{A}$ is
related by the isomorphism $\varphi$ in the following way:

$\displaystyle\widetilde{\Delta}\circ\varphi=(\varphi\otimes\varphi)\circ\Delta^{\prime},$
(5.1)

where $\Delta^{\prime}$ is the opposite coproduct (2.7) on $\mathcal{A}$ and
$\widetilde{\Delta}$ is the standard coproduct on $\widetilde{\mathcal{A}}$

$\displaystyle\widetilde{\Delta}(\widetilde{T}_{ij}(u))=-\widetilde{T}_{kj}(u)\otimes\widetilde{T}_{ik}(u).$

It seems useful to incorporate the minus in the definition of
$\widetilde{\Delta}$. This has a consequence for the pseudovacuum eigenvalues
in the composite model:
$\widetilde{\lambda}_{j}(u)=-\widetilde{\lambda}^{(1)}_{j}(u)\widetilde{\lambda}^{(2)}_{j}(u)$.
On the other hand, the ratio functions satisfy the same relation as for
$\mathfrak{gl}(2|1)$:
$\widetilde{r}_{j}(u)=\widetilde{r}^{(1)}_{j}(u)\widetilde{r}^{(2)}_{j}(u)$.

###  5.1 Main results

From the above remarks and results of Section 3.1, we can conclude about the
form of Bethe vectors in the $\mathfrak{gl}(1|2)$-invariant composite model.
Similarly to the case of Theorem 3.3, the proof of the following two theorems
is based on the composition rule for the isomorphism $\varphi$ with the
coproducts $\widetilde{\Delta}$ and $\Delta^{\prime}$.

######  Theorem 5.1.

The Bethe vectors of the full model can be expressed as the bilinear
combination of the partial Bethe vectors:

$\displaystyle\widetilde{\mathbb{B}}_{a,b}(\bar{u};\bar{v})=\sum\widetilde{r}_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II})\widetilde{r}_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I})\frac{f(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})g(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})}{f(\bar{u}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})}\widetilde{\mathbb{B}}^{(1)}_{a_{1},b_{1}}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm
I})\widetilde{\mathbb{B}}_{a_{2},b_{2}}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm II}).$

The summation goes over all partitions
$\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}$ and
$\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm II}\\}$ with no restriction. The
corresponding cardinalities satisfy $a_{1}+a_{2}=a$ and $b_{1}+b_{2}=b$.

###### Proof.

Let $\widetilde{B}_{ab}(\bar{u};\bar{v})$ denote the polynomial in elements of
$\widetilde{\mathcal{A}}$ which acts on the pseudovacuum, i.e.,
$\widetilde{\mathbb{B}}_{ab}(\bar{u};\bar{v})=\widetilde{B}_{ab}(\bar{u};\bar{v})\widetilde{\Omega}$.
We use the composition rule (5.1) and the results of Theorem 3.1

$\displaystyle\widetilde{\mathbb{B}}_{a,b}(\bar{u};\bar{v})=\widetilde{B}_{a,b}(\bar{u};\bar{v})\widetilde{\Omega}=\widetilde{\Delta}(\widetilde{B}_{a,b}(\bar{u};\bar{v}))\widetilde{\Omega}^{(1)}\widetilde{\Omega}^{(2)}=\big{[}\widetilde{\Delta}\circ\varphi(B_{b,a}(\bar{v};\bar{u}))\big{]}\widetilde{\Omega}^{(1)}\widetilde{\Omega}^{(2)}$
$\displaystyle=\big{[}(\varphi\otimes\varphi)\circ\Delta^{\prime}(B_{b,a}(\bar{v};\bar{u}))\big{]}\widetilde{\Omega}^{(1)}\widetilde{\Omega}^{(2)}$
$\displaystyle=\left[(\varphi\otimes\varphi)\sum
r_{1}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
I})r_{3}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II})\frac{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{v}_{\scriptscriptstyle\rm
I})g(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})}{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{v}_{\scriptscriptstyle\rm
I})}B^{(1)}_{b_{2},a_{2}}(\bar{v}_{\scriptscriptstyle\rm
II};\bar{u}_{\scriptscriptstyle\rm
II})B^{(2)}_{b_{1},a_{1}}(\bar{v}_{\scriptscriptstyle\rm
I};\bar{u}_{\scriptscriptstyle\rm
I})\right]\widetilde{\Omega}^{(1)}\widetilde{\Omega}^{(2)}.$

We stress that there was again used the opposite coproduct (2.7) in contrast
to Theorem 3.1. The application of the map $\varphi\otimes\varphi$ maps not
only the polynomials $B^{(1)}_{b_{2},a_{2}}(\bar{v}_{\scriptscriptstyle\rm
II};\bar{u}_{\scriptscriptstyle\rm
II})B^{(2)}_{b_{1},a_{1}}(\bar{v}_{\scriptscriptstyle\rm
I};\bar{u}_{\scriptscriptstyle\rm I})$ to
$\widetilde{B}_{a_{2},b_{2}}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})\widetilde{B}^{(2)}_{a_{1},b_{1}}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I})$ but also
$r_{1}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
I})r_{3}^{(2)}(\bar{u}_{\scriptscriptstyle\rm II})$ to
$\widetilde{r}_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
I})\widetilde{r}_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm II})$. Hence,

$\displaystyle\widetilde{\mathbb{B}}_{a,b}(\bar{u};\bar{v})=\sum\widetilde{r}_{3}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
I})\widetilde{r}_{1}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II})\frac{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{v}_{\scriptscriptstyle\rm
I})g(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})}{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{v}_{\scriptscriptstyle\rm
I})}\widetilde{\mathbb{B}}_{a_{2},b_{2}}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})\widetilde{\mathbb{B}}^{(2)}_{a_{1},b_{1}}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I}).$

After renaming the sets of variables as $\bar{u}_{\scriptscriptstyle\rm
I}\leftrightarrow\bar{u}_{\scriptscriptstyle\rm II}$ and
$\bar{v}_{\scriptscriptstyle\rm
I}\leftrightarrow\bar{v}_{\scriptscriptstyle\rm II}$, we arrive at the
statement of the theorem. ∎

######  Theorem 5.2.

The dual Bethe vectors of the full model can be expressed as the bilinear
combination of the partial Bethe vectors:

$\displaystyle\widetilde{\mathbb{C}}_{a,b}(\bar{u};\bar{v})=\sum\widetilde{r}_{3}^{(2)}(\bar{v}_{\scriptscriptstyle\rm
I})\widetilde{r}_{1}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
II})\frac{f(\bar{v}_{\scriptscriptstyle\rm II},\bar{v}_{\scriptscriptstyle\rm
I})g(\bar{u}_{\scriptscriptstyle\rm I},\bar{u}_{\scriptscriptstyle\rm
II})}{f(\bar{u}_{\scriptscriptstyle\rm II},\bar{v}_{\scriptscriptstyle\rm
I})}\widetilde{\mathbb{C}}_{a_{2},b_{2}}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm
II})\widetilde{\mathbb{C}}^{(1)}_{a_{1},b_{1}}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm I}).$

The summation goes over all partitions
$\bar{u}\Rightarrow\\{\bar{u}_{\scriptscriptstyle\rm
I},\bar{u}_{\scriptscriptstyle\rm II}\\}$ and
$\bar{v}\Rightarrow\\{\bar{v}_{\scriptscriptstyle\rm
I},\bar{v}_{\scriptscriptstyle\rm II}\\}$ with no restriction. The
corresponding cardinalities satisfy $a_{1}+a_{2}=a$ and $b_{1}+b_{2}=b$.

###### Proof.

We use the results of Theorem 3.3 and the composition rule (5.1). Everything
is analogous to the proof of the previous theorem

$\displaystyle\widetilde{\mathbb{C}}_{a,b}(\bar{u};\bar{v})=\varphi(\mathbb{C}_{b,a}(\bar{v};\bar{u}))=\widetilde{\Omega}^{\dagger}\varphi(C_{b,a}(\bar{v};\bar{u}))=\widetilde{\Omega}^{\dagger(1)}\widetilde{\Omega}^{\dagger(2)}\big{[}\widetilde{\Delta}\circ\varphi(C_{b,a}(\bar{v};\bar{u}))\big{]}$
$\displaystyle=\widetilde{\Omega}^{\dagger(1)}\widetilde{\Omega}^{\dagger(2)}\big{[}(\varphi\otimes\varphi)\circ\Delta^{\prime}(C_{b,a}(\bar{v};\bar{u}))\big{]}$
$\displaystyle=\widetilde{\Omega}^{\dagger(1)}\widetilde{\Omega}^{\dagger(2)}\left[(\varphi\otimes\varphi)\sum
r_{1}^{(2)}(\bar{v}_{\scriptscriptstyle\rm
II})r_{3}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I})\frac{f(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})g(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})}{f(\bar{u}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})}C_{b_{1},a_{1}}^{(2)}(\bar{v}_{\scriptscriptstyle\rm
I};\bar{u}_{\scriptscriptstyle\rm
I})C_{b_{2},a_{2}}^{(1)}(\bar{v}_{\scriptscriptstyle\rm
II};\bar{u}_{\scriptscriptstyle\rm II})\right]$
$\displaystyle=\sum\widetilde{r}_{3}^{(2)}(\bar{v}_{\scriptscriptstyle\rm
II})\widetilde{r}_{1}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
I})\frac{f(\bar{v}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})g(\bar{u}_{\scriptscriptstyle\rm II},\bar{u}_{\scriptscriptstyle\rm
I})}{f(\bar{u}_{\scriptscriptstyle\rm I},\bar{v}_{\scriptscriptstyle\rm
II})}\mathbb{C}_{a_{1},b_{1}}^{(2)}(\bar{u}_{\scriptscriptstyle\rm
I};\bar{v}_{\scriptscriptstyle\rm
I})\mathbb{C}_{a_{2},b_{2}}^{(1)}(\bar{u}_{\scriptscriptstyle\rm
II};\bar{v}_{\scriptscriptstyle\rm II}).$

Renaming the sets of variables as $\bar{u}_{\scriptscriptstyle\rm
I}\leftrightarrow\bar{u}_{\scriptscriptstyle\rm II}$ and
$\bar{v}_{\scriptscriptstyle\rm
I}\leftrightarrow\bar{v}_{\scriptscriptstyle\rm II}$ leads to the statement of
the theorem. ∎

##  6 Conclusion

We have obtained explicit formulas for Bethe vectors for the composite
$\mathfrak{gl}(2|1)$\- and $\mathfrak{gl}(1|2)$-invariant generalized quantum
integrable models. The method of calculation was straightforward. We used the
known action of the monodromy matrix elements on the Bethe vectors [12]. Since
the RTT algebra $\mathcal{A}$ has the structure of a bialgebra, we expressed
the action of the monodromy matrix elements of the complete model on the
tensor product of the superspaces of the partial models using the coproduct in
$\mathcal{A}$. The corresponding dual Bethe vectors were obtained using a
certain antimorphism of $\mathcal{A}$. Similarly, the (dual) Bethe vectors for
the $\mathfrak{gl}(1|2)$-invariant model were obtained with the help of
isomorphism of the RTT algebras $\mathcal{A}$ and $\widetilde{\mathcal{A}}$.

The authors of [24] used apart from this approach also the coproduct property
of the weight functions [17].

We are now prepared to calculate the form factors of the partial monodromy
elements $T_{ij}^{(\ell)}(u)$ in the basis of the Bethe vectors of the
complete model. They allow one to calculate the form factors of local
operators depending on an internal point of the original interval $[0,L]$. The
correlation functions of the local operators can be consequently investigated.

Our subsequent publication [9] is devoted to the investigation of the form
factors of the partial monodromy matrix elements and local operators using the
method of zero modes [25].

##  Appendix A Action of monodromy matrix elements on Bethe vectors

We list here some useful formulas. The summation is usually performed over all
partitions of the type $\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}$ and
$\bar{v}\Rightarrow\\{v_{0},\bar{v}_{0}\\}$ where $\\#u_{0}=\\#v_{0}=1$. It
also happens that the summation goes over all partitions of the type
$\bar{u}\Rightarrow\\{u_{0},u_{1},\bar{u}_{2}\\}$ with the condition
$\\#u_{0}=\\#u_{1}=1$. All formulas listed in this appendix are special cases
of the results obtained in [12].

  * •

Action of the diagonal elements:

$\displaystyle\frac{T_{11}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathbb{B}_{a,b}(\bar{u};\bar{v})=r_{1}(z)\frac{f(\bar{u},z)}{h(\bar{v},z)}\mathbb{B}_{a,b}(\bar{u};\bar{v})$
$\displaystyle\qquad{}+\sum_{\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}}r_{1}(u_{0})\frac{g(z,u_{0})f(\bar{u}_{0},u_{0})g(\bar{v},z)}{f(\bar{v},u_{0})}\mathbb{B}_{a,b}(\\{z,\bar{u}_{0}\\};\bar{v})$
$\displaystyle\qquad{}+\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}\\\
\bar{v}\Rightarrow\\{v_{0},\bar{v}_{0}\\}\end{subarray}}r_{1}(u_{0})\frac{f(\bar{u}_{0},u_{0})g(z,v_{0})g(\bar{v}_{0},v_{0})}{f(\bar{v}_{0},u_{0})h(v_{0},z)h(v_{0},u_{0})}\mathbb{B}_{a,b}(\\{z,\bar{u}_{0}\\};\\{z,\bar{v}_{0}\\}),$
$\displaystyle\frac{T_{22}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathbb{B}_{a,b}(\bar{u};\bar{v})=f(z,\bar{u})g(\bar{v},z)\mathbb{B}_{a,b}(\bar{u};\bar{v})$
$\displaystyle\qquad{}+\sum_{\bar{v}\Rightarrow\\{v_{0},\bar{v}_{0}\\}}\frac{f(z,\bar{u})g(z,v_{0})g(\bar{v}_{0},v_{0})}{h(v_{0},z)}\mathbb{B}_{a,b}(\bar{u};\\{z,\bar{v}_{0}\\})$
$\displaystyle\qquad{}+\sum_{\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}}g(u_{0},z)f(u_{0},\bar{u}_{0})g(\bar{v},z)\mathbb{B}_{a,b}(\\{z,\bar{u}_{0}\\};\bar{v})$
$\displaystyle\qquad{}+\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}\\\
\bar{v}\Rightarrow\\{v_{0},\bar{v}_{0}\\}\end{subarray}}\frac{g(u_{0},z)f(u_{0},\bar{u}_{0})g(z,v_{0})g(\bar{v}_{0},v_{0})}{h(v_{0},z)}\mathbb{B}_{a,b}(\\{z,\bar{u}_{0}\\};\\{z,\bar{v}_{0}\\}),$
$\displaystyle\frac{T_{33}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathbb{B}_{a,b}(\bar{u};\bar{v})=r_{3}(z)g(\bar{v},z)\mathbb{B}_{a,b}(\bar{u};\bar{v})$
$\displaystyle\qquad{}+\sum_{\bar{v}\Rightarrow\\{v_{0},\bar{v}_{0}\\}}r_{3}(v_{0})\frac{f(z,\bar{u})g(z,v_{0})g(\bar{v}_{0},v_{0})}{h(v_{0},z)f(v_{0},\bar{u})}\mathbb{B}_{a,b}(\bar{u};\\{z,\bar{v}_{0}\\})$
$\displaystyle\qquad{}+\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}\\\
\bar{v}\Rightarrow\\{v_{0},\bar{v}_{0}\\}\end{subarray}}r_{3}(v_{0})\frac{g(u_{0},z)f(u_{0},\bar{u}_{0})g(z,v_{0})g(\bar{v}_{0},v_{0})}{h(v_{0},u_{0})f(v_{0},z)f(v_{0},\bar{u}_{0})}\mathbb{B}_{a,b}(\\{z,\bar{u}_{0}\\};\\{z,\bar{v}_{0}\\}).$

  * •

Action of the upper-triangular elements:

$\displaystyle\frac{T_{13}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathbb{B}_{a,b}(\bar{u};\bar{v})=\mathbb{B}_{a+1,b+1}(\\{z,\bar{u}\\};\\{z,\bar{v}\\}),$
$\displaystyle\frac{T_{23}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathbb{B}_{a,b}(\bar{u};\bar{v})$
$\displaystyle\qquad{}=f(z,\bar{u})\mathbb{B}_{a,b+1}(\bar{u};\\{z,\bar{v}\\})+\sum_{\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}}g(u_{0},z)f(u_{0},\bar{u}_{0})\mathbb{B}_{a,b+1}(\\{z,\bar{u}_{0}\\},\\{z,\bar{v}\\}),$
$\displaystyle\frac{T_{12}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathbb{B}_{a,b}(\bar{u};\bar{v})$
$\displaystyle\qquad{}=g(\bar{v},z)\mathbb{B}_{a+1,b}(\\{z,\bar{u}\\};\bar{v})+\sum_{\bar{v}\Rightarrow\\{v_{0},\bar{v}_{0}\\}}\frac{g(z,v_{0})g(\bar{v}_{0},v_{0})}{h(v_{0},z)}\mathbb{B}_{a+1,b}(\\{z,\bar{u}\\};\\{z,\bar{v}_{0}\\}).$

  * •

Action of the lower-triangular elements:

$\displaystyle\frac{T_{21}(z)}{\lambda_{2}(z)h(\bar{v},z)}\mathbb{B}_{a,b}(\bar{u};\bar{v})=\sum_{\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}}r_{1}(z)\frac{f(u_{0},z)f(u_{0},\bar{u}_{0})f(\bar{u}_{0},z)g(\bar{v},z)}{f(\bar{v},z)h(u_{0},z)}\mathbb{B}_{a-1,b}(\bar{u}_{0};\bar{v})$
$\displaystyle\qquad{}+\sum_{\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}}r_{1}(u_{0})\frac{f(z,u_{0})f(z,\bar{u}_{0})f(\bar{u}_{0},u_{0})g(\bar{v},z)}{f(\bar{v},u_{0})h(z,u_{0})}\mathbb{B}_{a-1,b}(\bar{u}_{0};\bar{v})$
$\displaystyle\qquad{}+\sum_{\bar{u}\Rightarrow\\{u_{0},u_{1},\bar{u}_{2}\\}}r_{1}(u_{0})\frac{f(u_{1},u_{0})f(u_{1},\bar{u}_{2})f(u_{1},z)f(\bar{u}_{2},u_{0})f(z,u_{0})g(\bar{v},z)}{f(\bar{v},u_{0})h(u_{1},z)h(z,u_{0})}$
$\displaystyle\qquad{}\times\mathbb{B}_{a-1,b}(\\{z,\bar{u}_{2}\\};\bar{v})$
$\displaystyle\qquad{}+\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{u_{0},\bar{u}_{0}\\}\\\
\bar{v}\Rightarrow\\{v_{0},\bar{v}_{0}\\}\end{subarray}}r_{1}(u_{0})\frac{f(z,\bar{u}_{0})f(\bar{u}_{0},u_{0})g(z,v_{0})g(\bar{v}_{0},v_{0})}{h(v_{0},z)f(\bar{v}_{0},u_{0})h(v_{0},u_{0})}\mathbb{B}_{a-1,b}(\bar{u}_{0};\\{z,\bar{v}_{0}\\})$
$\displaystyle\qquad{}+\sum_{\begin{subarray}{c}\bar{u}\Rightarrow\\{u_{0},u_{1},\bar{u}_{2}\\}\\\
\bar{v}\Rightarrow\\{v_{0},\bar{v}_{0}\\}\end{subarray}}r_{1}(u_{0})\frac{f(u_{1},u_{0})g(u_{1},z)f(u_{1},\bar{u}_{2})f(\bar{u}_{2},u_{0})g(z,v_{0})g(\bar{v}_{0},v_{0})}{h(v_{0},z)f(\bar{v}_{0},u_{0})h(v_{0},u_{0})}$
$\displaystyle\qquad{}\times\mathbb{B}_{a-1,b}(\\{z,\bar{u}_{2}\\};\\{z,\bar{v}_{0}\\}).$

### Acknowledgements

The author wants to express his gratitude to N.A. Slavnov for the proposal to
investigate this topic and discussions. He thanks also to S. Pakuliak for
discussions and to A.P. Isaev and Č. Burdík for their support. The work of the
author has been supported by the Grant Agency of the Czech Technical
University in Prague, grant No. SGS15/215/OHK4/3T/14, and by the Grant of the
Plenipotentiary of the Czech Republic at JINR, Dubna.

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