On the multiparametric ${\mathcal{U}}_{q}[{D}_{n+1}^{(2)}]$ vertex model

Vieira, R. S.; Lima-Santos, A.

doi:10.1088/1742-5468/2013/02/p02011

Cited by 6 publications

(5 citation statements)

References 24 publications

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“…We remark that the first solutions of the boundary YB equation associated to the U q [o (2) (2n + 2)] vertex-model were the diagonal and block-diagonal solutions found by Martins and Guan in [44]; soon after Lima-Santos deduced the general K-matrices of this vertex-model in [45]. The corresponding reflection K-matrices for the multiparametric U q [o (2) (2n + 2)] vertex-model were deduced and classified by Vieira and Lima-Santos in [86] and a new family of solutions for Jimbo's U q [o (2) (2n + 2)] vertex-model was also derived in [86].…”

Section: Solutions Of the Boundary Yb Equationmentioning

confidence: 81%

Reflection matrices withU_q[osp⁽²⁾(2|2m)] symmetry

Vieira¹,

Lima-Santos²

2017

J. Phys. A: Math. Theor.

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We propose a classification of the reflection K-matrices (solutions of the boundary Yang-Baxter equation) for the Uq[ospmodel. We have found four families of solutions, namely, the complete solutions, in which no elements of the reflection K-matrix is null, the block-diagonal solutions, the X -shape solutions and the diagonal solutions. We highlight that these diagonal Kmatrices also hold for the Uq[osp

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Section: Solutions Of the Boundary Yb Equationmentioning

confidence: 81%

Reflection matrices withU_q[osp⁽²⁾(2|2m)] symmetry

Vieira¹,

Lima-Santos²

2017

J. Phys. A: Math. Theor.

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“…Notice, moreover, that although Abel's method requires the solutions to be differential (there can be non-differential solutions of some functional equations), this restriction is not a problem when dealing with the YBE, as its solutions are always assumed to be differentiable because of the connection between the R matrix and the corresponding local Hamiltonian. Concerning the theory of integrable systems, the differential method is perhaps most known in connection with the boundary YBE [40][41][42].…”

Section: The Differential Yang-baxter Equationsmentioning

confidence: 99%

Solutions of the Yang–Baxter equation for (n + 1) (2n + 1)-vertex models using a differential approach

Vieira¹,

Lima-Santos²

2021

J. Stat. Mech.

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The formal derivatives of the Yang–Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of differential equations. The derivatives of the R matrix elements, however, can be regarded as independent variables and eliminated from the systems, after which, two systems of polynomial equations are obtained in their place. In general, these polynomial systems have a non-zero Hilbert dimension, which means that not all elements of the R matrix can be fixed through them. Nevertheless, the remaining unknowns can be found by solving a few simple differential equations that arise as consistency conditions of the method. The branches of the solutions can also be easily analyzed by this method, which ensures the uniqueness and generality of the solutions. In this work, we consider the Yang–Baxter equation for (n + 1) (2n + 1)-vertex models with a generalization based on the A n−1 symmetry. This differential approach allows us to solve the Yang–Baxter equation in a systematic way.

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“…Notice moreover that although Abel's method requires the solutions to be differentiable (there can be non-differentiable solutions of some functional equations), this restriction is not a problem when dealing with the YBE, as its solutions are always assumed to be differentiable because of the connection between the R matrix and the corresponding local Hamiltonian. Concerning the theory of integrable systems, the differential method is perhaps most known in connection with boundary YBE [34,35,36]. Now we can look for the matrices (3) what are solutions of (1).…”

Section: The Differential Yang-baxter Equationmentioning

confidence: 99%

The solutions of the Yang-Baxter equation for the $(n+1)(2n+1)$-vertex models through a differential approach

Vieira,

Lima-Santos

2020

Preprint

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The formal derivatives of the Yang-Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of dierential equations. The derivatives of the R matrix elements, however, can be regarded as independent variables and eliminated from the systems, after which two systems of polynomial equations are obtained in place. In general, these polynomial systems have a non-zero Hilbert dimension, which means that not all elements of the R matrix can be fixed through them. Nevertheless, the remaining unknowns can be found by solving a few number of simple differential equations that arise as consistency conditions of the method. The branches of the solutions can also be easily analyzed by this method, which ensures the uniqueness and generality of the solutions. In this work we considered the Yang-Baxter equation for the (n + 1)(2n + 1)-vertex models with a generalization based on the An−1 symmetry. This differential approach allowed us to solve the Yang-Baxter equation in a systematic way. .

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On the multiparametric ${\mathcal{U}}_{q}[{D}_{n+1}^{(2)}]$ vertex model

Cited by 6 publications

References 24 publications

Reflection matrices withU_q[osp⁽²⁾(2|2m)] symmetry

Reflection matrices withU_q[osp⁽²⁾(2|2m)] symmetry

Solutions of the Yang–Baxter equation for (n + 1) (2n + 1)-vertex models using a differential approach

The solutions of the Yang-Baxter equation for the $(n+1)(2n+1)$-vertex models through a differential approach

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On the multiparametric ${\mathcal{U}}_{q}[{D}_{n+1}^{(2)}]$ vertex model

Cited by 6 publications

References 24 publications

Reflection matrices withUq[osp(2)(2|2m)] symmetry

Reflection matrices withUq[osp(2)(2|2m)] symmetry

Solutions of the Yang–Baxter equation for (n + 1) (2n + 1)-vertex models using a differential approach

The solutions of the Yang-Baxter equation for the $(n+1)(2n+1)$-vertex models through a differential approach

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Product

Resources

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Reflection matrices withU_q[osp⁽²⁾(2|2m)] symmetry

Reflection matrices withU_q[osp⁽²⁾(2|2m)] symmetry