Yang-Baxter maps associated to elliptic curves

AbstractWe present Yang–Baxter maps associated to elliptic curves. They are related to discrete ver-sions of the Krichever-Novikov and the Landau-Lifshits equations. A lifting of scalar integrablequad–graph equations to two–field equations is also shown. 1 Introduction The importance of the Yang-Baxter (YB) equation to a variety of branches in physics and mathe-matics is well known. Its solutions are intimately related to exactly solvable statistical mechanicalmodels, link polynomials in knot theory, quantum and classical integrable models, conformal fieldtheories, representations of groups and algebras, quantum groups and many others. More interest-ingly, YB equation provides various connections among the aforementioned disciplines.Historically, YB equation has its roots in the theory of exactly solvable models in statisticalmechanics [35, 8] and the quantum inverse scattering method [31]. For an extensive account of earlywork on the YB equation see [14]. In its original form the quantum YB equation is the relationR