arXiv subjects
Publications and source records attributed to S. M. Sergeev.
Simple periodic 3d->2d compactification of the tetrahedron equations gives the Yang-Baxter equations for various evaluation representations of U_q(sl_n). In this paper we construct an example of fixed non-periodic 3d boundary conditions producing a set of Yang-Baxter equations for U_q(D_n^1). These boundary conditions resemble a fusion in hidden direction.
In this paper we give a detailed classification scheme for three-dimensional quantum zero curvature representation and tetrahedron equations. This scheme includes both even and odd parity components, the resulting algebras of observables are either Bose q-oscillators or Fermi oscillators. Three-dimensional $R$-matrices intertwining variously oriented tensor products of Bose and Fermi oscillators and satisfying tetrahedron and super-tetrahedron equations are derived. The 3d->2d compactification reproduces U_q(gl(n|m)) super-algebras and their representation theory.
A relation between q-oscillator R-matrix of the tetrahedron equation and decompositions of Poinkare-Birkhoff-Witt type bases for nilpotent subalgebras of U_q(sl_n) is observed.
Discrete-time evolution operators in integrable quantum lattice models are sometimes more fundamental objects then Hamiltonians. In this paper we study an evolution operator for the one-dimensional integrable q-deformed Bose gas with XXZ-type impurities and find its spectrum. Evolution operators give a new interpretation of known integrable systems, for instance our system describes apparently a simplest laser with a clear resonance peak in the spectrum.
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 operator-valued R-matrix. The current system is a kind of the linear problem for 2+1 evolution model. A generating function for the integrals of motion for the evolution is derived with a help of the current system. The subject of the paper is rather new, and so the perspectives of further investigations are widely discussed.
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 of the auxiliary system.
In this paper we reformulate free field theory models defined on the rectangular $D+1$ dimensional lattices as $D+1$ evolution models. This evolution is in part a simple linear evolution on free (``creation'' and ``annihilation'') operators. Formal eigenvectors of this linear evolution can be directly constructed, and them play the role of the ``physical'' creation and annihilation operators. These operators being completed by a ``physical'' vacuum vector give the spectrum of the evolution operator, as well as the trace of the evolution operator give a correct expression for the partition function. As an example, Bazhanov -- Baxter's free bosonic model is considered.
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 also present a remarkably simple scheme of quantization for one of those cases.
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 local Yang -- Baxter equation appears among other results of this ``fusion''.
Local (or modified) Yang -- Baxter equation (LYBE) gives the functional map from the parameters of the weights in the left hand side to the parameters of the correspondent weights in the right hand side of LYBE. Such maps solve the functional tetrahedron equation. In this paper all the maps associated with LYBE of the ferro-electric type with single parameter in each weight matrix are classified.
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.
This paper has been temporarily withdrawn for corrections.
In this paper we apply the method of psi-vectors to three dimensional statistical models. This method gives the correspondence between the Bazhanov -- Baxter model and its vertex formulation. Considering psi-vectors for the Planar model, we obtain its self-duality.
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.
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.
Using a modified version of the tetrahedron equations we construct a new family of $N$-state three-dimensional integrable models with commuting two-layer transfer-matrices. We investigate a particular class of solutions to these equations and parameterize them in terms of elliptic functions. The corresponding models contain one free parameter $k$ -- an elliptic modulus.
A transfer matrix method relating the process of refinement of a fractal measure to thermodynamic formalism of an appropriate Ising model is applied to the analysis of intermittency in hadron collisions revealing that underlying dynamics is that of period doublings.
In this paper we present a new series of 3-dimensional integrable lattice models with $N$ colors. The case $N=2$ generalizes the elliptic model of our previous paper. The weight functions of the models satisfy modified tetrahedron equations with $N$ states and give a commuting family of two-layer transfer-matrices. The dependence on the spectral parameters corresponds to the static limit of the modified tetrahedron equations and weights are parameterized in terms of elliptic functions. The models contain two free parameters: elliptic modulus and additional parameter $η$. Also we briefly discuss symmetry properties of weight functions of the models.