Tetrahedral Zamolodchikov algebras corresponding to Baxter'sL-operators

Korepanov, I. G.

doi:10.1007/bf02096833

Cited by 63 publications

(119 citation statements)

References 8 publications

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“…The relation (3.5) is equivalent to quantum Korepanov equation, it can be also seen as the tetrahedral Zamolodchikov algebra/local Yang-Baxter equation for the adjoint action of R. See the long story of [12,13,14,15,4,11] for details. We fix the normalization of R by…”

Section: Theorem 31 There Is a Unique (Up To A Constant Multiple) Imentioning

confidence: 99%

Tetrahedron Equation and Quantum R Matrices for Spin Representations of $${B^{(1)}_n}$$ B n ( 1 ) , $${D^{(1)}_n}$$ D n ( 1 ) and $${D^{(2)}_{n+1}}$$ D n + 1 ( 2 )

Kuniba

Sergeev

2013

Commun. Math. Phys.

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(2) n+1 ATSUO KUNIBA AND SERGEY SERGEEV Dedicated to Professor Vladimir Bazhanov on the occasion of his sixtieth birthday. AbstractIt is known that a solution of the tetrahedron equation generates infinitely many solutions of the Yang-Baxter equation via suitable reductions. In this paper this scheme is applied to an oscillator solution of the tetrahedron equation involving bosons and fermions by using special 3d boundary conditions. The resulting solutions of the Yang-Baxter equation are identified with the quantum R matrices for the spin representations of B

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Section: Theorem 31 There Is a Unique (Up To A Constant Multiple) Imentioning

confidence: 99%

Tetrahedron Equation and Quantum R Matrices for Spin Representations of $${B^{(1)}_n}$$ B n ( 1 ) , $${D^{(1)}_n}$$ D n ( 1 ) and $${D^{(2)}_{n+1}}$$ D n + 1 ( 2 )

Kuniba

Sergeev

2013

Commun. Math. Phys.

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“…Then, following the arguments of [19], one can show that the map (3) satisfies the functional tetrahedron equation [20] …”

Section: Consider Four Pointsmentioning

confidence: 99%

“…The analog of the Yang-Baxter equation for integrable quantum systems in 3D is called the tetrahedron equation. It was introduced by Zamolodchikov in [13,14] (see also [15][16][17][18][19][20][21] for further results in this field, used in this paper). Similarly to the Yang-Baxter equation the tetrahedron equation provides local integrability conditions which are not related to the size of the lattice.…”

Section: Introductionmentioning

confidence: 99%

Quantum geometry of three-dimensional lattices

Bazhanov¹,

Mangazeev²,

Sergeev³

2008

J. Stat. Mech.

100

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We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable "ultra-local" Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation). These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasi-classical limit.

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“…This static limit appears to be the solution of the Tetrahedron Equation proposed by Korepanov [4]. Moreover, in the planar limit when β 2 = 0 the vertex weight (5.1) coincides with the N = 2 solution by Hietarinta [5,6].…”

Section: The Case N =mentioning

confidence: 53%

“…Recently two new solutions of the vertex type Tetrahedron equation [1][2][3] for the number of spin states N = 2 [4,5] were obtained. In our previous paper we have tried to generalize these solutions for N > 2 and for general spectral parameters.…”

Section: Introductionmentioning

confidence: 99%

The vertex formulation of the Bazhanov-Baxter model

1996

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In this paper we formulate an integrable model on the simple cubic lattice. The N -valued spin variables of the model belong to edges of the lattice. The Boltzmann weights of the model obey the vertex type Tetrahedron Equation. In the thermodynamic limit our model is equivalent to the Bazhanov -Baxter Model. In the case when N = 2 we reproduce the Korepanov's and Hietarinta's solutions of the Tetrahedron equation as some special cases.

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Tetrahedral Zamolodchikov algebras corresponding to Baxter'sL-operators

Cited by 63 publications

References 8 publications

Tetrahedron Equation and Quantum R Matrices for Spin Representations of $${B^{(1)}_n}$$ B n ( 1 ) , $${D^{(1)}_n}$$ D n ( 1 ) and $${D^{(2)}_{n+1}}$$ D n + 1 ( 2 )

Tetrahedron Equation and Quantum R Matrices for Spin Representations of $${B^{(1)}_n}$$ B n ( 1 ) , $${D^{(1)}_n}$$ D n ( 1 ) and $${D^{(2)}_{n+1}}$$ D n + 1 ( 2 )

Quantum geometry of three-dimensional lattices

The vertex formulation of the Bazhanov-Baxter model

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