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Lima-Santos, A.; Galleas, W.

doi:10.1016/j.nuclphysb.2010.02.009

Cited by 9 publications

(13 citation statements)

References 42 publications

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“…This particularity makes the differential method as simple as powerful to study the reflection equation (7). In fact, this method was extensively used by Lima-Santos and collaborators in a series of papers, where solutions of (7) associated with non-exceptional Lie algebras and superalgebras were found and classified [35,36,37,38,39,40,41,42]. Differential methods were also employed to solve the periodic ybe (3).…”

Section: The Differential Yang-baxter Equationsmentioning

confidence: 99%

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Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems

Vieira

2018

J. High Energ. Phys.

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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 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. Nonetheless, 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 two-state systems, up to the eight-vertex model. This differential approach allowed us to solve the Yang-Baxter equation in a systematic way and also to completely classify its regular solutions.

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Section: The Differential Yang-baxter Equationsmentioning

confidence: 99%

“…For the remaining cases of the unusual six-vertex R matrices given by (39), the functional equations are quite simple, reason for which we shall report only the final results here. They are:…”

Section: The Remaining Casesmentioning

confidence: 99%

Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems

Vieira

2018

J. High Energ. Phys.

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“…Uq[spo (2n|2m)] and Uq[sl (1) (m|n)] vertex-models were obtained in a period of almost one year [53][54][55][56], the corresponding reflection K-matrices of the Uq[osp (2) (2|2m)] vertex-model were derived only now, approximately eight years after. Indeed, the general K-matrices for the Uq[osp (2) (2n + 2|2m)] vertex-model (except for the diagonal ones which we report in this work) are yet unknown.…”

Section: Solutions Of the Boundary Yb Equationmentioning

confidence: 99%

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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“…In fact, the classification of the reflection K-matrices for models associated with Lie superalgebras is not yet complete. A great advance in this direction was obtained in the last years by Lima-Santos and collaborators, where a classification of the reflection K-matrices associated with the U q [sl (r|2m) (2) ], U q [osp (r|2m) (1) ], U q [spo (2n|2m)] and U q [sl (m|n) (1) ] vertex models were given [71][72][73][74]. More recently, Vieira and Lima-Santos derived the reflection K-matrices associated with (2m + 2)-dimensional representations of the U q [osp (2|2m) (2) ] U q [C (m + 1) (2) ] quantum affine Lie superalgebra [75].…”

Section: (Although Its Hamiltonianmentioning

confidence: 99%

Reflection K-matrices for a nineteen vertex model with Uq[osp(2|2)
Vieira
¹
,
Santos
²

2017
Physics Letters A
2
0
3
0
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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 [osp (2|2) (2) ]U q [C (2) (2) ]. 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-matrix are different from zero. In the type II solution, the reflection K-matrix is even (every element of the K-matrix with an odd parity is null) and it has only one boundary free-parameter. Finally, the type III solution corresponds to a diagonal reflection K-matrix with two boundary free-parameters. The modelThis letter concerns with the reflection K-matrices of a supersymmetric nineteen vertex model introduced by Yang and Zhang in [1] (see also [2][3][4][5]). The Yang-Zhang R-matrix is constructed from a three-dimensional representation V of the twisted quantum affine Lie superalgebra(2) ], and the periodic algebraic Bethe Ansatz of this vertex model was performed in [6].Vertex models with underlying symmetries corresponding to Lie superalgebras are important in several fields of physics and mathematics. For instance, Z 2 -graded Lie superalgebras appear in the study of lattice models of strongly correlated electrons, such as the supersymmetric generalizations of the t-J model [7][8][9][10][11] and the Hubbard model [12,13], among others. The integrability of supersymmetric two-dimensional quantum chains had proved to be important as well in the AdS/CFT correspondence, either in the N = 4 super Yang-Mills side of the duality [14][15][16], or in the AdS 5 × S 5 string theory side [17]. Furthermore, these mathematical structures also are important in the construction of supersymmetric Hopf algebras and quantum groups [18][19][20][21] [30]. Considering now nonperiodic boundary conditions, the diagonal K-matrices for the ZFvm were first derived in [31] and the general Kmatrices were deduced in [32] and [33]; for the IKvm, the diagonal K-matrices were found in [34] and the general K-matrices were derived in [33] and [35]; besides, the coordinate Bethe Ansätze for both models were presented in [36], while their algebraic versions were performed in [37] (although the ZFvm had been considered before in [31] with the help of the fusion procedure [26]). Among the ZFvm and IKvm, other nineteen vertex models were discovered since then, for instance, the supersymmetric vertex models associated with the sl (2|1) and osp (2|1) Lie superalgebras, whose R-matrix was presented in [38] and the corresponding reflection K-matrices were obtained in [33]; for the Bethe Ansätze of these models, see [36,39]. Finally, other more complex nineteen vertex models were also discovered in the last decade − see, for instance, [40][41][42][43][44][45].The supersymmetric nineteen vertex model which we consider here is not included in the list above. The Rmatrix associated with this model was constructed in [1] and it can be wri...

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Reflection matrices for the Uq[sl(r|2m)(2

Abstract: We investigate the possible regular solutions of the boundary Yang-Baxter equation for the vertex models associated with the graded version of the A (1) n−1 affine Lie algebra, the U q [sl(m|n) (1) ] vertex model, also known as Perk-Schultz model.

Cited by 9 publications

References 42 publications

Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems

Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems

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

Reflection K-matrices for a nineteen vertex model with Uq[osp(2|2)
Vieira
¹
,
Santos
²

2017
Physics Letters A
2
0
3
0
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Reflection matrices for the Uq[sl(r|2m)(2

Abstract: We investigate the possible regular solutions of the boundary Yang-Baxter equation for the vertex models associated with the graded version of the A (1) n−1 affine Lie algebra, the U q [sl(m|n) (1) ] vertex model, also known as Perk-Schultz model.

Cited by 9 publications

References 42 publications

Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems

Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems

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

Reflection K-matrices for a nineteen vertex model with Uq[osp(2|2)Vieira1, Santos2 2017Physics Letters A2030View full textAdd to dashboardCite

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

Reflection K-matrices for a nineteen vertex model with Uq[osp(2|2)
Vieira
¹
,
Santos
²

2017
Physics Letters A
2
0
3
0
View full text Add to dashboard Cite