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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… (More)
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)
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 Tetra-hedron Equation. In the thermodynamic limit our model is equivalent to the Bazhanov – Baxter Model. In the case when N = 2 we reproduce the… (More)
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… (More)
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)
The Faddeev-Volkov solution of the star-triangle relation is connected with the modular double of the quantum group U q (sl 2). 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)
In this paper we formulate a new N-state spin integrable model on a three-dimensional lattice with spins interacting round each elementary cube of the lattice. This model can be also reformulated as a vertex type model. Weight functions of the model satisfy tetrahedron equations.
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 give an exact solution of a classical model of a discrete integrable evolution whose auxiliary system is a complete linear system. Generating function for the integrals of motion defines an algebraic curve of a high genus. The curve is determined by the boundary conditions of the dynamical system. We parametrize the dynamical variables in terms of… (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)