Sergey M. Sergeev

Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry

Abstract 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 describes quantum fluctuations of circle patterns and the associated discrete conformal transformations connected with the Thurstons discrete analogue of the Riemann mappings theorem. In particular, in the quasi-classical limit the model precisely describe the geometry of integrable circle patterns with prescribed intersection angles.

Functional Tetrahedron Equation

We describe a method for constructing classical integrable models in a (2+1)-dimensional discrete spacetime based on the functional tetrahedron equation, an equation that makes the symmetries of a model obvious in a local form. We construct a very general “block-matrix model,” find its algebraic-geometric solutions, and study its various particular cases. We also present a remarkably simple quantization scheme for one of those cases.

Elliptic gamma-function and multi-spin solutions of the Yang–Baxter equation

Abstract We present a generalization of the master solution to the quantum Yang–Baxter equation (obtained recently in arXiv:1006.0651 ) to the case of multi-component continuous spin variables taking values on a circle. The Boltzmann weights are expressed in terms of the elliptic gamma-function. The associated solvable lattice model admits various equivalent descriptions, including an interaction-round-a-face formulation with positive Boltzmann weights. In the quasi-classical limit the model leads to a new series of classical discrete integrable equations on planar graphs.

Tetrahedron equation and generalized quantum groups

We construct

Exact solution of the Faddeev-Volkov model

2^n

Comment on star-star relations in statistical mechanics and elliptic gamma-function identities

-families of solutions of the Yang-Baxter equation from

Supertetrahedra and superalgebras

n

An integrable 3D lattice model with positive Boltzmann weights

-products of three-dimensional

The pentagon relation and incidence geometry

R

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

and