On the parametrization of solutions of the Yang--Baxter equations

We study all five-, six-, and one eight-vertex type two-state solutions of the Yang-Baxter equations in the form $A_{12} B_{13} C_{23} = C_{23} B_{13} A_{12}$, and analyze the interplay of the `gauge' and `inversion' symmetries of these solution. Starting with algebraic solutions, whose parameters have no specific interpretation, and then using these symmetries we can construct a parametrization where we can identify global, color and spectral parameters. We show in particular how the distribution of these parameters may be changed by a change of gauge.Comment: 19 pages in LaTe

Integrable lattice equations with vertex and bond variables

We present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables. In some of the models the vertex dynamics is independent of the evolution of the bond variables, and one can write the equations as non-autonomous "Yang-Baxter maps". We also present a model in which the vertex and bond variables are fully coupled. Integrability is tested with algebraic entropy as well as multidimensional consistencyComment: 15 pages, remarks added, other minor change

A comment on free-fermion conditions for lattice models in two and more dimensions

We analyze free-fermion conditions on vertex models. We show --by examining examples of vertex models on square, triangular, and cubic lattices-- how they amount to degeneration conditions for known symmetries of the Boltzmann weights, and propose a general scheme for such a process in two and more dimensions.Comment: 12 pages, plain Late

Integrability of Difference Equations Through Algebraic Entropy and Generalized Symmetries

Given an equation arising from some application or theoretical consideration one of the first questions one might ask is: What is its behavior? It is integrable? In these lectures we will introduce two different ways for establishing (and in some sense also defining) integrability for difference equations: Algebraic Entropy and Generalized Symmetries. Algebraic Entropy deals with the degrees of growth of the solution of any kind of discrete equation (ordinary, partial or even differential-difference) and usually provides a quick test to establish if an equation is or not integrable. The approach based on Generalized Symmetries also provides tools for investigating integrable equations and to find particular solutions by symmetry reductions. The main focus of the lectures will be on the computational tools that allow us to calculate Generalized Symmetries and extract the value of the Algebraic Entropy from a finite number of iterations of the map