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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… (More)

The Faddeev-Volkov solution of the star-triangle relation is connected with the modular double of the quantum group Uq(sl2). It defines an Ising-type lattice model with positive Boltzmann weights where the spin variables take continuous values on the real line. The free energy of the model is exactly calculated in the thermodynamic limit. The model… (More)

A quantum evolution model in 2+1 discrete space – time, connected with 3D fundamental map R, is investigated. Map R is derived as a map providing a zero curvature of a two dimensional lattice system called “the current system”. In a special case of the local Weyl algebra for dynamical variables the map appears to be canonical one and it corresponds to known… (More)

- R . M . Kashaev, Sergey M. Sergeev
- 1996

The tetrahedron equation in a special substitution is reduced to a pair of pentagon and one ten-term equations. Various examples of solutions are found. O-doubles of Novikov, which generalize the Heisenberg double of a Hopf algebra, provide a particular algebraic solution to the problem.

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… (More)

- I. G. Korepanov, Sergey M. Sergeev
- 1998

We describe a scheme of constructing classical integrable models in 2+1-dimensional discrete space-time, based on the functional tetrahedron equation—equation that makes manifest the symmetries of a model in local form. We construct a very general “block-matrix model” together with its algebro-geometric solutions, study its various particular cases, and… (More)

We investigate some classical evolution model in the discrete 2+1 space-time. A map, giving an one-step time evolution, may be derived as the compatibility condition for some systems of linear equations for a set of auxiliary linear variables. Dynamical variables for the evolution model are the coefficients of these systems of linear equations. Determinant… (More)

Quantum 3D R – matrix in the classical (i. e. functional) limit gives a symplectic map of dynamical variables. The corresponding 3D evolution model is considered. An auxiliary problem for it is a system of linear equations playing the role of the monodromy matrix in 2D models. A generating function for the integrals of motion is constructed as a determinant… (More)

- Sergey M. Sergeev
- Int. J. Math. Mathematical Sciences
- 2006

A lattice model of interacting q-oscillators, proposed by V. Bazhanov and S. Sergeev in 2005 is the quantum-mechanical integrable model in 2 + 1 dimensional space-time. Its layer-to-layer transfer matrix is a polynomial of two spectral parameters, it may be regarded in terms of quantum groups both as a sum of sl(N) transfer matrices of a chain of length M… (More)

A sort of two dimensional linear auxiliary problem for the complex of 3D R – operators associated with the Zamolodchikov – Bazhanov – Baxter statistical model is proposed. This problem resembles the problem of the local Yang – Baxter equation but does not coincide with it. The formulation of the auxiliary problem admits a notion of a “fusion”, and usual… (More)