Boundary Algebraic Bethe Ansatz for a nineteen vertex model with $U_{q}[\mathrm{osp}(2|2)^{(2)}]$ symmetry

TL;DR

This paper develops a boundary algebraic Bethe Ansatz for a supersymmetric nineteen vertex model with $U_{q}[osp(2|2)^{(2)}]$ symmetry, deriving eigenvalues, eigenvectors, and Bethe equations.

Contribution

It introduces a novel boundary algebraic Bethe Ansatz approach for a specific supersymmetric vertex model with twisted quantum affine Lie superalgebra symmetry.

Findings

Eigenvalues and eigenvectors of the transfer matrix are explicitly calculated.

Bethe Ansatz equations for the model are derived.

The method applies to models with diagonal reflection K-matrices.

Abstract

The boundary algebraic Bethe Ansatz for a supersymmetric nineteen vertex-model constructed from a three-dimensional representation of the twisted quantum affine Lie superalgebra $U_{q}[\mathrm{osp}(2|2)^{(2)}]$ is presented. The eigenvalues and eigenvectors of Sklyanin's transfer matrix, with diagonal reflection $K$-matrices, are calculated and the corresponding Bethe Ansatz equations are obtained.

Equations

$$A{\otimes }^{g}B=\sum _{\left\{i,j,k,l\right\}=1}^n{\left(-1\right)}^{p(i)p(k)+p(j)p(k)}{A}_{ij}{B}_{kl}\left(e_{ij}\otimes e_{kl}\right),$$

$$P^g=\sum _{\left\{i,j\right\}=1}^n{\left(-1\right)}^{p(i)p(j)}\left(e_{ij}\otimes e_{ji}\right).$$

$$A^{t^g}=\sum _{\left\{i,j\right\}=1}^n{\left(-1\right)}^{p(i)p(j)+p(i)}{A}_{ji}{e}_{ij},A^{{\tau }^g}=\sum _{\left\{i,j\right\}=1}^n{\left(-1\right)}^{p(i)p(j)+p(j)}{A}_{ji}{e}_{ij},$$

$${\mathrm{tr}}^g(A)=\sum _{i=1}^n{\left(-1\right)}^{p(i)}{A}_{ii}{e}_{ii}.$$

$$R_{12}(x)R_{13}(xy)R_{23}(y)=R_{23}(y)R_{13}(xy)R_{12}(x),$$

$$R_{12}(x/y)K_1^{-}(x)R_{21}(xy)K_2^{-}(y)=K_2^{-}(y)R_{12}(xy)K_1^{-}(x)R_{21}(x/y),$$

$$R(x)=\left(\begin{array}[]{ccccccccc}r_{1}(x)&0&0&0&0&0&0&0&0\\ 0&r_{2}(x)&0&r_{5}(x)&0&0&0&0&0\\ 0&0&r_{3}(x)&0&r_{6}(x)&0&r_{7}(x)&0&0\\ 0&s_{5}(x)&0&r_{2}(x)&0&0&0&0&0\\ 0&0&s_{6}(x)&0&r_{4}(x)&0&r_{6}(x)&0&0\\ 0&0&0&0&0&r_{2}(x)&0&r_{5}(x)&0\\ 0&0&s_{7}(x)&0&s_{6}(x)&0&r_{3}(x)&0&0\\ 0&0&0&0&0&s_{5}(x)&0&r_{2}(x)&0\\ 0&0&0&0&0&0&0&0&r_{1}(x)\end{array}\right),$$

$$r_1\left(x\right)$$

$$r_2\left(x\right)$$

$$r_3\left(x\right)$$

$$r_4\left(x\right)$$

$$r_5\left(x\right)$$

$$r_6\left(x\right)$$

$$r_7\left(x\right)$$

$$s_5\left(x\right)$$

$$s_6\left(x\right)$$

$$s_7\left(x\right)$$

$$R_{12}\left(1\right)=f{\left(1\right)}^{1/2}{P}_{12}^g,$$

$$R_{12}\left(x\right)=f\left(x\right)R_{21}^{-1}\left(x^{-1}\right),$$

$$R_{12}\left(x\right)=R_{21}^{t_1^g{\tau }_2^g}\left(x\right),$$

$$R_{12}\left(x\right)=g\left(x\right)\left[V_1{R}_{12}^{t_2^g}\left({\eta }^{-1}{x}^{-1}\right)V_1^{-1}\right],$$

$$f\left(x\right)=r_1(x)r_1\left(\frac{1}{x}\right)=\left(q^{2}x-1\right)\left(\frac{q^2}{x}-1\right),g(x)=-\frac{qx\left(x-1\right)}{\left(qx+1\right)}.$$

$$M=V^{t^g}V=\mathrm{diag}\left(1/q,1,q\right)$$

$$K^{-}(x)=\mathrm{diag}\left(k_{1,1}^{-}(x),k_{2,2}^{-}(x),k_{3,3}^{-}(x)\right),$$

$$k_{1,1}^{-}(x)$$

$$k_{2,2}^{-}(x)$$

$$k_{3,3}^{-}(x)$$

$$R_{12}\left(\frac{y}{x}\right)K_1^{+}\left(x\right)M_1^{-1}{R}_{21}\left(\frac{\eta }{xy}\right)M_1{K}_2^{+}\left(y\right)=M_1{K}_2^{+}\left(y\right)R_{12}\left(\frac{\eta }{xy}\right)M_1^{-1}{K}_1^{+}\left(x\right)R_{12}\left(\frac{y}{x}\right),$$

$$K^{+}(x)=K^{-}({\eta }^{-1}{x}^{-1})M,$$

$$t(x)={\mathrm{tr}}_a^g\left[K^{+}(x)U(x)\right],$$

$$U(x)=\left[K^{+}(x)T(x)K^{-}(x)T^{-1}(x^{-1})\right],T(x)=R_{aN}(x)R_{aN-1}(x)\cdots R_{a1}(x),$$

$$\left[t(x),t(y)\right]=0,\forall \left\{x,y\right\}\in \mathbb{C}.$$

$$H=\sum _{i=1}^{N-1}{H}_{i,i+1}+\frac{1}{2}{\frac{\mathrm{d}{K}_1^{-}(x)}{\mathrm{d}x}|}_{x=1}+\frac{{\mathrm{tr}}_a^g\left[K_a^{+}(1)H_{N,a}\right]}{{\mathrm{tr}}_a^g\left[K_a^{+}(1)\right]},$$

$$H_{i,i+1}={\frac{\mathrm{d}}{\mathrm{d}x}\left[P_{i,i+1}{R}_{i,i+1}(x)\right]|}_{x=1},1\le i\le N-1.$$

$$U(x)=\left(\begin{array}[]{ccc}A_{1}(x)&B_{1}(x)&B_{2}(x)\\ C_{1}(x)&A_{2}(x)&B_{3}(x)\\ C_{2}(x)&C_{3}(x)&A_{3}(x)\end{array}\right).$$

$$t(x)=k_{1,1}^{+}(x)A_1(x)-k_{2,2}^{+}(x)A_2(x)+k_{3,3}^{+}(x)A_3(x),$$

$${\Psi }_0={\left(1,0,0\right)}^t$$

Full text

Keywords: Algebraic Bethe Ansatz, Open boundary conditions, Lie superalgebras, twisted Lie algebras, nineteen vertex-models.

1 The model

Over the last decades, great interest has been aroused in the study of supersymmetric integrable systems. In fact, supersymmetry is now present in several fields of contemporary mathematics and physics, ranging from condensed matter physics to superstring theory. We can cite, for instance, the graded generalizations of Hubbard and t-J models [1, 2, 3], which play an important role in condensed matter physics, and also the search for solutions of the graded Yang-Baxter equation [4, 5, 6, 7, 8, 9], which gave origin to important algebraic construction as the supersymmetric Hopf algebras and quantum groups [10]. More recently, the integrability of supersymmetric models also proved to be important in superstring theory, more specifically in the AdS/CFT correspondence [11, 12, 13, 14].

The most powerful and beautiful method to analyze these integrable quantum systems is probably the *Algebraic Bethe Ansatz *(aba) [15, 16, 17]. This technique allows to diagonalize the transfer matrix of a given integrable quantum system in an analytical way and the commutability of the transfer matrix (which is guaranteed by the Yang-Baxter equation) ensures the existence of several conserved quantities in evolution, the Hamiltonian being one of them. A complete analytical answer for the problem, however, require the solution of the Bethe Ansatz equations, a system of non-linear equations which have not been completely solved so far [18]. The aba was originally applied to systems with periodic boundary conditions but after the work of Sklyanin [19], integrable models with non-periodic boundaries could also be handled. Further generalizations [20, 21] showed that the aba can be applied to several classes of integrable models, described by different Lie algebras and superalgebras, with both periodic as non-periodic boundary conditions. In the case of the periodic aba, the fundamental object is a $R$-matrix, solution of the Yang-Baxter equation, while in the case of the boundary aba, other fundamental ingredient is necessary: the $K$-matrices (or reflection matrices), which are solutions of the boundary Yang-Baxter equations (a.k.a. reflection equations) [19, 20, 21].

The aba was successfully applied to nineteen vertex models. In fact, after Tarasov [22] used this technique to solve the Izergin-Korepin model [23] with periodic boundary conditions, the Zamolodchikov-Fateev vertex-model [24] was also solved by Lima-Santos [25]. The boundary aba for these vertex models were performed in [26, 27] together with the supersymmetric $\mathrm{sl}\left(2|1\right)$ and $\mathrm{osp}\left(2|1\right)$ vertex models. Several other important results were obtained for nineteen vertex-models $-$ see, for instance, [28] and references therein.

In this work we will study another graded three-state nineteen-vertex model with reflecting boundary conditions. We derive the boundary aba for a supersymmetric nineteen-vertex model that was presented by Yang and Zhang in [29]. The $R$-matrix associated with this model is constructed from a three-dimensional free boson representation $V$ of the twisted quantum affine Lie superalgebra $U_{q}[\mathrm{osp}(2|2)^{(2)}]$ and the periodic aba for this model was presented in [30]. We would like to emphasize that vertex-models described by Lie superalgebras $-$ and, in particular, by twisted Lie superalgebras $-$ are usually the most complex ones, which of course is due to the high complexity of such Lie superalgebras [31, 32, 33, 34, 35, 36]. In fact, even the reflection $K$-matrices of those models were not yet completely classified, although a great advance in this direction has been obtained in the last years [37, 38, 39, 40, 41, 42]. In the recent work [28], we derived the reflection $K$-matrices of the Yang-Zhang model which allowed us to implement now the boundary aba for this model.

Since we shall deal here with a supersymmetric system, it will be helpful to remember first the basics of the graded Lie algebras [43]. Let $W=V\oplus U$ be a $\mathbb{Z}_{2}$-graded vector space where $V$ and $U$ denote their even and odd parts, respectively. In a $\mathbb{Z}_{2}$-graded vector space we associate a graduation $p(i)$ to each element $\epsilon_{i}$ of a given basis of $V$. In the present case, we shall consider only a three-dimensional representation of the twisted quantum affine Lie superalgebra $U_{q}[\mathrm{osp}(2|2)^{(2)}]$ with a basis $E=\{\epsilon_{1},\epsilon_{2},\epsilon_{3}\}$ and the grading $p(1)=0$, $p(2)=1$ and $p(3)=0$. Multiplication rules in the graded vector space $W$ differ from the ordinary ones by the appearance of additional signs. For example, the graded tensor product of two homogeneous even elements $A\in\mathrm{End}(V)$ and $B\in\mathrm{End}(V)$ turns out to be defined by the formula,

[TABLE]

where $n=3$ is the dimension of the vector space $V$ and $e_{ij}$ are the Weyl matrices ($e_{ij}$ is a matrix in which all elements are null, except that element on the $[i,j]$ position, which equals $1$). In the same fashion, the graded permutation operator $P^{g}$ is defined by

[TABLE]

and the graded transposition $A^{t^{g}}$ of a matrix $A\in\mathrm{End}(V)$ as well as its inverse graded transposition, $A^{\tau^{g}}$, are defined, respectively, by

[TABLE]

so that $A^{t^{g}\tau^{g}}=A^{\tau^{g}t^{g}}=A$. Finally, the graded trace of a matrix $A\in\mathrm{End}(V)$ is given by

[TABLE]

In the graded case, both the periodic YB equation [4, 5, 6, 7, 8, 9],

[TABLE]

as well as the boundary YB equation [19, 20, 21],

[TABLE]

can be written in the same way as in in the non-graded case: it is only necessary to employ graded operations instead of the usual operations [21].

The $R$-matrix, solution of the graded YB equation (5), associated with the Yang-Zhang vertex-model [29] can be written, up to a normalizing factor and employing a different notation, as follows:

[TABLE]

where the amplitudes $r_{i}\left(x\right)$ and $s_{i}\left(x\right)$ are given respectively by

[TABLE]

This $R$-matrix has the following properties or symmetries [28]:

[TABLE]

where

[TABLE]

Here, $t_{1}^{g}$ and $t_{2}^{2}$ mean graded partial transpositions in the first and second vector spaces, respectively, and $\tau_{1}^{2}$ and $\tau_{2}^{2}$ the corresponding inverse operations. Besides, $\eta=-q$ is the crossing parameter while

[TABLE]

is the crossing matrix.

Solutions of the boundary YB equation (6) for this vertex model were presented recently in [28]. The most general regular diagonal reflection $K$-matrix,

[TABLE]

of the Yang-Zhang vertex model $-$ which is of interest in the present work $-$ has the following entries [28]:

[TABLE]

where $\beta_{1,1}$ and $\beta_{2,2}$ are the boundary free-parameters of the solutions. Notice that the properties (18), (19), (20), and (21), enjoyed by the $R$-matrix (7), ensure the existence of the* dual reflection equation*,

[TABLE]

as described by Bracken* et Al.* [21]. Besides, the dual reflection matrices $K^{+}(x)$ which are solutions of the dual reflection equation (28) can determined by the following isomorphism [21, 28]

[TABLE]

with a new set of boundary free-parameters (say, with $\alpha_{i,j}$ replacing $\beta_{i,j}$). It is to be noticed that special values for the boundary free-parameters lead to particular reflection matrices, for instance, the quantum group invariant solutions $K^{-}(x)=I$ and $K^{+}(x)=M$.

2 The Boundary Algebraic Bethe Ansatz

The aba for quantum integrable systems containing diagonal boundaries was developed by Sklyanin [19] for integrable systms described by symmetric $R$-matrices. Menisezcu and Nepomechie extended Sklyanin’s formalism to get account of non-periodic $R$-matrices [20] and the graded case was developed further by Bazhanov and Shadrikov [9] and also Bracken et Al. [21].

The fundamental ingredient of the boundary aba is the Sklyanin transfer matrix,

[TABLE]

where $K^{-}(x)$ and $K^{+}(x)$ are any pair of reflection $K$-matrices satisfying the reflection equations (6) and (28), respectively, and

[TABLE]

are, respectively, the Sklyanin monodromy and the usual periodic monodromy. Notice that the operators $R_{aq}(x)$ appearing in the expressions above act in $\mathrm{End}\left(V_{a}\otimes V_{q}\right)$, where $V_{a}$ denotes the auxiliary space and $V_{q}$, for $1\leq q\leq N$, are in the quantum spaces associated to a lattice of $N$ sites (the graded trace should be taken in the auxiliary space only). In performing the aba, we consider any pair $\left\{K^{-}(x),K^{+}(x)\right\}$ of reflection $K$-matrices which do not necessarily need to be related by the isomorphism (29).

As commented already in the introduction, he mean feature of the boundary aba is that the transfer matrix (30) commutes with itself for any values of the spectral parameters $x$ and $y$, that is,

[TABLE]

This means that $t(x)$ can be thought as the generator of infinitely many conserved quantities in evolution, which justifies the name of *integrable *to systems that can be solved by the (boundary) aba. The commutative property of $t(x)$ can be proved from the unitarity and crossing symmetries of the $R$-matrix plus the algebra provided by the reflection equations [19, 20, 21]; in particular, this implies the integrability of open quantum spin chains whose Hamiltonian is given by

[TABLE]

where

[TABLE]

We remark, that the free boson realization of the $U_{q}[\mathrm{osp}(2|2)^{(2)}]$ quantum Lie superalgebra considered by Yang-Zhang does not have a classical limit as $q\rightarrow 1$ [29]. This particularity prevent us to study the Gaudin magnets through the off-shell aba for this model. Nevertheless, other realizations as, for instance, that one presented in [34], could provide other vertex models with $U_{q}[\mathrm{osp}(2|2)^{(2)}]$ symmetry that do have a classical limit.

2.1 The reference state

In order to find the eigenvalues and eigenvectors of the transfer matrix through the Bethe Ansatz method, it is necessary to know at least one of its eigenvectors that is simple enough so that the corresponding eigenvalue can be directly computed [19]. This simple eigenvector is called reference state.

To find the reference state is useful to rewrite the transfer matrix in the Lax representation, that is, as an three-by-three operator valued matrix, say,

[TABLE]

In this representation, the diagonal Sklyanin’s transfer matrix (30), becomes,

[TABLE]

and we can easily verify that the following state

[TABLE]

is an eigenvector of the transfer matrix and, hence, it is the reference state we were looking for.

In fact, the action of the monodromy elements over $\Psi_{0}$ can be evaluated following [26, 27], and reads,

[TABLE]

where $\xi$ can be any complex number and

[TABLE]

where,

[TABLE]

Therefore, the action of the transfer matrix on $\Psi_{0}$ reads:

[TABLE]

It will be more convenient, however, to introduce a new set of diagonal operators, namely,

[TABLE]

so that their action on the reference state $\Psi_{0}$ simplifies to

[TABLE]

with

[TABLE]

Similarly, the transfer matrix can be rewritten as

[TABLE]

with

[TABLE]

so that their action on the reference state reads now,

[TABLE]

2.2 The 1-particle state

Once the action of the transfer matrix over the reference state is determined, we can ask about their excited states. In the framework of the boundary aba, these excited states are constructed from the action of the creators operators $B_{1}$ and $B_{2}$ (it can be verified, however, that the excited states can be constructed without use the operator $B_{3}$ [22, 26, 27]) over the reference state $\Psi_{0}$ . Further, we can verify that the action of $B_{2}$ on $\Psi_{0}$ is proportional to a double action of $B_{1}$ on $\Psi_{0}$ (in the sense that $B_{2}$ rises the magnon number of the system twice when compared to $B_{1}$). From this follows that the first excited state, named here the 1-particle state, should be defined as,

[TABLE]

The new variable $x_{1}$ – called rapidity – must be determined in order to $\Psi_{1}\left(x_{1}\right)$ be indeed an eigenvector of the transfer matrix; we shall see below that this requirement is provided by the Bethe Ansatz equation of the 1-particle state.

From (58) and (59) we can write down the action of $t(x)$ over $\Psi_{1}\left(x_{1}\right)$:

[TABLE]

Notice that we shall need to know the commutation relations between the diagonal operators $D_{1}(x)$, $D_{2}(x)$ and $D_{3}(x)$ with $B_{1}\left(x_{1}\right)$ in order to evaluate the action of $t(x)$ on $\Psi_{1}\left(x_{1}\right)$. These commutation relations are provided by the* fundamental relation* of the boundary aba (102) and they are presented in the Appendix. Making use of the commutation relations (103), (104) and (105) and simplifying the results, we can realize that the action of $t(x)$ on $\Psi_{1}\left(x_{1}\right)$ can be written as follows:

[TABLE]

where,

[TABLE]

and,

[TABLE]

[TABLE]

Now, if $\Psi_{1}(x_{1})$ is an eigenstate of $t(x)$, then we must have $\beta_{1}^{1}\left(x_{1}\right)=\beta_{1}^{2}\left(x_{1}\right)=0$. This provides the Bethe Ansatz equation of the 1-particle state that implicitly fix the rapidity $x_{1}$:

[TABLE]

After simplify we can verify that both $\beta_{1}^{1}\left(x_{1}\right)$ as $\beta_{1}^{2}\left(x_{1}\right)$ vanish and also that the right-hand-side of (65) does not actually depend on the spectral parameter $x$, as it should. The conclusion in that $\Psi_{1}\left(x_{1}\right)$ is an eigenvector of the transfer matrix with eigenvalue $\tau_{1}\left(x|x_{1}\right)$ given by (62).

2.3 The 2-Particle state

In the construction of the next excited state $-$ the 2-particle state $-$ both the operators $B_{1}$ and $B_{2}$ should be taken into account. This is necessary because both $B_{2}\Psi_{0}$ as $B_{1}B_{1}\Psi_{0}$ are in the same sector (i.e., both states have the same magnon number). Therefore, the most general 2-particle state should be constructed through a linear combination of these operators and we can verify a posteriori that the adequate linear combination is as follows:

[TABLE]

The coefficient $\lambda\left(x_{1},x_{2}\right)$ of this linear combination can be fixed by the condition that

[TABLE]

for some phase function $\omega\left(x_{1},x_{2}\right)$ [22, 26, 27]. Making use of the commutation relation (112) between $B_{1}(x_{1})$ and $B_{1}(x_{2})$, we get that

[TABLE]

from which follows that

[TABLE]

where we made use of the following properties:

[TABLE]

Once $\lambda\left(x_{1},x_{2}\right)$ is determined, we can find the action of the transfer matrix on the 2-particle state. To this end, will be necessary to use several other commutation relations besides the previous one, namely, the commutation relations provided by (106), (107), (108), (121) and (123). Although this computation maybe somewhat extensive, we can verify that the action of $t(x)$ over $\Psi_{2}\left(x_{1},x_{2}\right)$ can be written as,

[TABLE]

where,

[TABLE]

and

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Now, in order to $\Psi_{2}\left(x_{1},x_{2}\right)$ given at (66) be an eigenstate of the transfer matrix (58), all terms on (71) but the first one must vanish. This is indeed true, provided that the Bethe Ansatz equations of the 2-particle state,

[TABLE]

are satisfied. Moreover, we can realize again that all dependence of the Bethe Ansatz equations on the spectral parameter $x$ is only apparent, as they should. Therefore, provided that the Bethe Ansatz equations (82) and (83) are satisfied, $\Psi_{2}\left(x_{1},x_{2}\right)$ as given by (66) will be an eigenvector of the transfer matrix (58) with eigenvalue $\tau_{2}\left(x|x_{1},x_{2}\right)$ given by (72).

2.4 The $n$-particle state

From the previous cases we can figure out what should be the appropriated $n$-particle state of the transfer matrix (58). It follows that $\Psi_{n}(x_{1},\ldots,x_{n})$ can be defined through a recurrence relation of the form,

[TABLE]

where $\Psi_{0}$ is given by (37) and $\Psi_{1}(x_{1})$ is given by (59). We have also introduced the notation,

[TABLE]

The functions $\lambda_{i}(x_{1},\ldots,x_{n})$ appearing in (84) can be determined imposing the following exchange conditions,

[TABLE]

and using the commutations relations between the creator operators in order to put them in a well-ordered form (see Appendix). This lead us to the expressions,

[TABLE]

and

[TABLE]

The action of $t(x)$ on $\Psi_{n}(x_{1},\ldots,x_{n})$ can be computed using many others commutation relations presented in the Appendix. It follows that this action can be written as,

[TABLE]

where,

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

The requirement that $\Psi_{n}(x_{1},\ldots,x_{n})$ be an eigenstate of the transfer matrix means that all terms in (89) that are not proportional to $\Psi_{n}(x_{1},\ldots,x_{n})$ itself must vanish. This lead us to the Bethe Ansatz equations of the general $n$-particle state:

[TABLE]

3 Explicit results

Making use of the amplitudes of the $R$-matrix (7), the expressions of the elements of the diagonal reflection $K$-matrix (24) and the coefficients of the commutation relations presented in the appendix, we can explicitly write down the results of the boundary aba for the Yang-Zhang model. It follows that the $n$-state eigenvector of the transfer matrix (58) is given by (84) where

[TABLE]

and

[TABLE]

The eigenvalues of the Sklyanin transfer matrix (58) are given by

[TABLE]

where

[TABLE]

Finally, the Bethe Ansatz equations are:

[TABLE]

4 Conclusion

In this work we derived the boundary aba for the supersymetric nineteen vertex model constructed from a three-dimensional boson free representation $V$ of the twisted quantum affine Lie superalgebra $U_{q}[\mathrm{osp}(2|2)^{(2)}]$. The $R$-matrix of this model was introduced by Yang and Zhang in [29] and the correponding reflection $K$-matrices were derived by us recently in [28]. The eigenvalues and eigenvectors of Sklyanin’s transfer matrix with diagonal reflection $K$-matrices were determined, as well as the corresponding Bethe Ansatz equations. Explicit results were also presented.

This work was supported in part by Brazilian Research Council (CNPq), grant #310625/2013-0 and FAPESP, grant #2011/18729-1.

Appendix A The fundamental commutation relations

To perform the boundary aba, we need to know how the diagonal operators $D_{1}$, $D_{2}$ and $D_{3}$ pass through the creators operators $B_{1}$, $B_{2}$ and $B_{3}$ (as an intermediate step, we shall also need known how the $C$’s operators pass through the $B$’s). These exchange rules are provided by the commutation relations that can be derived from the so-called fundamental relation of the boundary aba:

[TABLE]

In fact, writing $U(x)$ in the Lax representation as (35), and using the relations (42), (43), (44) and (45), the following commutation relations ca be obtained (for details about how these expressions are obtained, please see [26, 27]):

[TABLE]

The commutation relation among the creators operators $B_{1}$, $B_{2}$ and $B_{3}$ themselves are:

[TABLE]

Notice that we have chosen an appropriated order for the creators operators, namely, that $B_{i}(x_{k})\prec B_{j}(x_{l})$ if, and only if, $x_{k}<x_{l}$, for the indexes $\{i,j,k,l\}$ running from $1$ to $3$. This order is important to the implementation of the boundary aba, since we should use these commutation relations until all operators be well ordered $-$ i.e., so that all diagonal operators $D_{i}$, and all annihilator operators $C_{i}$, be at right of the creator operators $B_{j}$ and, further, that among the creator operators themselves, we always have $B_{i}\prec B_{j}$.

Finally, the commutation relations between the $C$’s and the $B$’s are:

[TABLE]

The fundamental relation (102) also provides the commutation relations between the operators $C_{i}$ and $D_{j}$, which can be used to decrease a bit the number of terms of some commutation relations. However, these supplementary commutation relations are not necessary in the implementation of the boundary aba and they will not be presented (the reader can consult references [26, 27] for this purpose).

Next we will write down the coefficients of the commutation relations above. We restrict ourselves only to the coefficients that appear explicitly in the text; the others expressions (which are usually very cumbersome and, as a mater of fact, not necessary) can also be found following the lines of [26, 27].

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]