From the tetrahedron equation to universal 𝑅-matrices

Kashaev, R. M.; Volkov, А. Yu.

doi:10.1090/trans2/201/06

Cited by 20 publications

(31 citation statements)

References 23 publications

(30 reference statements)

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“…is an easy consequence of the pentagon relation, as was shown by R. Kashaev and A. Volkov [11]. They also have obtained corresponding construction for the SL q (N) case.…”

Section: The Explicite Constructionsupporting

confidence: 52%

Modular double of a Quantum Group

Фаддеев¹

2016

Fifty Years of Mathematical Physics

102

132

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“…is an easy consequence of the pentagon relation, as was shown by R. Kashaev and A. Volkov [11]. They also have obtained corresponding construction for the SL q (N) case.…”

Section: The Explicite Constructionsupporting

confidence: 52%

Modular double of a Quantum Group

Фаддеев¹

2016

Fifty Years of Mathematical Physics

102

132

View full text Add to dashboard Cite

“…. , ∞ and each vertex is assigned with Boltzmann weights, given by the matrix R abc , (29), so that its indices are identified with the spin variables on six edges surrounding the vertex. Namely, the three pairs of indices (α, α ′ ), (β, β ′ ) and (γ, γ ′ ) are associated with the edges oriented along three different lattice directions "a", "b" and "c" respectively.…”

Section: A Solution Of the Tetrahedron Equationmentioning

confidence: 99%

Zamolodchikov's tetrahedron equation and hidden structure of quantum groups

Bazhanov

Sergeev

2006

J. Phys. A: Math. Gen.

110

348

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The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finitedimensional highest weight representations for all quantum affine algebra U q ( sl(n)), where the rank n coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the sl(n)-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2 + 1 dimensions.

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“…The case α 1 = 0, α 3 = −1 was considered in [16], the case α 1 = α 3 = −1 in [9], the case α 3 = −1 − α 1 in [1].…”

Section: Theorem 2 (I) Letmentioning

confidence: 99%

Tetrahedron Equation, Weyl Group, and Quantum Dilogarithm

Bytsko

Volkov

2014

Lett Math Phys

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We derive a family of solutions to the tetrahedron equation using the RTT presentation of a two parametric quantized algebra of regular functions on an upper triangular subgroup of G L(n). The key ingredients of the construction are the longest element of the Weyl group, the quantum dilogarithm function, and central elements of the quantized division algebra of rational functions on the subgroup in question.Mathematics Subject Classification. 20G42, 16K20, 16T25.

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From the tetrahedron equation to universal 𝑅-matrices

Cited by 20 publications

References 23 publications

Modular double of a Quantum Group

Modular double of a Quantum Group

Zamolodchikov's tetrahedron equation and hidden structure of quantum groups

Tetrahedron Equation, Weyl Group, and Quantum Dilogarithm

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