The hyperbolic modular double and the Yang-Baxter equation

Chicherin, Dmitry; Spiridonov, V. P.

doi:10.2969/aspm/07610095

Cited by 7 publications

(9 citation statements)

References 49 publications

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“…A generalization of the Faddeev-Volkov model still associated with the γ(z)-function was found in [38]. It leads to a more general R-operator than we consider here [11], which can be obtained as a limit from the most complicated known R-operator derived in [16].…”

Section: Representations Of the Quantum Algebramentioning

confidence: 73%

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From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation

Chicherin

Derkachov

Spiridonov

2016

SIGMA

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Abstract. We start from known solutions of the Yang-Baxter equation with a spectral parameter defined on the tensor product of two infinite-dimensional principal series representations of the group SL(2, C) or Faddeev's modular double. Then we describe its restriction to an irreducible finite-dimensional representation in one or both spaces. In this way we obtain very simple explicit formulas embracing rational and trigonometric finite-dimensional solutions of the Yang-Baxter equation. Finally, we construct these finite-dimensional solutions by means of the fusion procedure and find a nice agreement between two approaches.

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Section: Representations Of the Quantum Algebramentioning

confidence: 73%

“…In [7] new compact factorization formulae were derived for finite-dimensional R-matrices in several cases (for different forms of factorizations, see [27] and references therein). Reduction of the integral R-operator for the generalized Faddeev-Volkov model of [38] is considered in [11].…”

Section: Introductionmentioning

confidence: 99%

From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation

Chicherin

Derkachov

Spiridonov

2016

SIGMA

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“…Indeed, as found in [23][24][25][26], the 4D superconformal quiver gauge theories are closely related to previously known 2D lattice models [12,27] and also lead to rather non-trivial new ones [28][29][30][31]. More generally, new advances were achieved in understanding the algebraic structure of solutions of the Yang-Baxter equation, see, e.g., [32][33][34][35][36][37][38][39][40][41][42][43][44]. Moreover, the 3D approach [45,46] to quantum R-matrices resulted in an extremely concise expression [47,48] of the higher-spin R-matrix of the XXZ model in terms of q-Hahn polynomials.…”

Section: Introductionmentioning

confidence: 77%

Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs

Bazhanov¹,

Kels²,

Сергеев³

2016

J. Phys. A: Math. Theor.

109

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In this paper we give an overview of exactly solved edge-interaction models, where the spins are placed on sites of a planar lattice and interact through edges connecting the sites. We only consider the case of a single spin degree of freedom at each site of the lattice. The Yang-Baxter equation for such models takes a particular simple form called the star-triangle relation. Interestigly all known solutions of this relation can be obtained as particular cases of a single "master solution", which is expressed through the elliptic gamma function and have continuous spins taking values on the circle. We show that in the low-temperature (or quasi-classical) limit these lattice models reproduce classical discrete integrable systems on planar graphs previously obtained and classified by Adler, Bobenko and Suris through the consistency-around-a-cube approach. We also discuss inversion relations, the physicical meaning of Baxter's rapidity-independent parameter in the startriangle relations and the invariance of the action of the classical systems under the star-triangle (or cube-flip) transformation of the lattice, which is a direct consequence of Baxter's Z-invariance in the associated lattice models.

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“…A new basic hypergeometric sum/integral identity conjectured in this note needs to be proven (the univariate integrals were considered in [21][22][23][24][25] ). Another possible future direction is to study the aspects of this solution in the framework of quantum algebras along the lines of [26,27]. This paper is organized as follows.…”

Section: Introduction and Conclusionmentioning

confidence: 99%

Comments on the multi-spin solution to the Yang–Baxter equation and basic hypergeometric sum/integral identity

Gahramanov

Jafarzade

2019

Mod. Phys. Lett. A

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We present a multi-spin solution to the Yang-Baxter equation. The solution corresponds to the integrable lattice spin model of statistical mechanics with positive Boltzmann weights and parameterized in terms of the basic hypergeometric functions. We obtain this solution from a non-trivial basic hypergeometric sum-integral identity which originates from the equality of supersymmetric indices for certain three-dimensional N = 2 Seiberg dual theories.

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The hyperbolic modular double and the Yang-Baxter equation

Cited by 7 publications

References 49 publications

From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation

From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation

Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs

Comments on the multi-spin solution to the Yang–Baxter equation and basic hypergeometric sum/integral identity

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