Reflection K-Matrices of the 19-Vertex Model and XXZ Spin-1 Chain with General Boundary Terms

We derive and classify all regular solutions of the boundary Yang-Baxter equation for 19-vertex models known as Zamolodchikov-Fateev or $A_{1}^{(1)}$ model, Izergin-Korepin or $A_{2}^{(2)}$ model, sl(2|1) model and osp(2|1) model. We find that there is a general solution for $A_{1}^{(1)}$ and sl(2|1) models. In both models it is a complete K-matrix with three free parameters. For the $A_{2}^{(2)}$ and osp(2|1) models we find three general solutions, being two complete reflect...

We derive the solutions of the boundary Yang-Baxter equation associated with a supersymmetric nineteen vertex model constructed from the three-dimensional representation of the twisted quantum affine Lie superalgebra $U_{q}[\mathrm{osp}\left(2|2\right)^{\left(2\right)}]\simeq U_{q}[C\left(2\right)^{\left(2\right)}]$. We found three classes of solutions. The type I solution is characterized by three boundary free-parameters and all elements of the corresponding reflection $K$-...

In the algebraic approach to Baxter's Q-operators for the closed Heisenberg XXZ spin chain, certain infinite-dimensional 'prefundamental' representations of the q-deformed Borel subalgebras play a central role. To extend this formalism to open spin chains, one needs a factorization identity for particular solutions of the reflection equation associated to these representations. In the case of quantum affine $\mathfrak{sl}_2$, we derive such an identity using the recent theory...

This work concerns to the studies of boundary integrability of the vertex models from representations of the Temperley-Lieb algebra associated with the quantum group ${\cal U}_{q}[X_{n}]$ for the affine Lie algebras $X_{n}$ = $A_{1}^{(1)}$, $B_{n}^{(1)}$, $C_{n}^{(1)}$ and $D_{n}^{(1)}$. A systematic computation method is used to constructed solutions of the boundary Yang-Baxter equations. We find a $2n^{2}+1$ free parameter solution for $A_{1}^{(1)} $ spin-$(n-1/2)$ and ...

We investigate the possible regular solutions of the boundary Yang-Baxter equation for the fundamental $U_q[G_2]$ vertex model. We find four distinct classes of reflection matrices such that half of them are diagonal while the other half are non-diagonal. The latter are parameterized by two continuous parameters but only one solution has all entries non-null. The non-diagonal solutions do not reduce to diagonal ones at any special limit of the free-parameters.

We investigate the possible regular solutions of the boundary Yang-Baxter equation for the vertex models associated with the graded version of the $A_{n-1}^{(1)}$ affine Lie algebra, the $U_{q}[sl(m|n)^{(1)}]$ vertex model, also known as Perk-Schultz model.

In this paper we consider families of multiparametric $R$-matrices to make a systematic study of the boundary Yang-Baxter equations in order to discuss the corresponding families of multiparametric $K$-matrices. Our results are indeed non-trivial generalization of the $K$-matrix solutions of the {\cal {U}}_{q}[D_{n+1}^{(2)}] vertex model when distinct reflections and extra free-parameters are admissible.

This work concerns the boundary integrability of the spin-s ${\cal{U}}_{q}[sl(2)]$ Temperley-Lieb model. A systematic computation method is used to constructed the solutions of the boundary Yang-Baxter equations. For $s$ half-integer, a general $2s(s+1)+3/2$ free parameter solution is presented. It turns that for $s$ integer, the general solution has $2s(s+1)+1$ free parameters. Moreover, some particular solutions are discussed.

We investigate the regular solutions of the boundary Yang-Baxter equation for the vertex models associated with the $B_{n}^{(1)}$ and $A_{2n}^{(2)}$ affine Lie algebras. In both class of models we find two general solutions with $n+1$ free parameters. In addition, we have find $2n-1$ diagonal solutions for $B_{n}^{(1)}$ models and $2n+1$ diagonal solutions for $% A_{2n}^{(2)}$ models. It turns out that for each $B_{n}^{(1)}$ model there exist a diagonal K-matrix with one free...

The general solutions for the factorization equations of the reflection matrices $K^{\pm}(\theta)$ for the eight vertex and six vertex models (XYZ, XXZ and XXX chains) are found. The associated integrable magnetic Hamiltonians are explicitly derived, finding families dependig on several continuous as well as discrete parameters.