Two-component Yang-Baxter maps and star-triangle relations

Abstract

It is shown how Yang-Baxter maps may be directly obtained from classical counterparts of the star-triangle relations and quantum Yang-Baxter equations. This is based on reinterpreting the latter equation and its solutions which are given in terms of special functions, as a set-theoretical form of the Yang-Baxter equation whose solutions are given by quadrirational Yang-Baxter maps. The Yang-Baxter maps obtained through this approach are found to satisfy two different types of Yang-Baxter equations, one that is the usual equation involving a single map, and another equation that involves a pair of maps, which is a case of what is also known as an entwining Yang-Baxter equation. Apart from the elliptic case, each of these Yang-Baxter maps are quadrirational, but only maps that solve the former type of Yang-Baxter equation are reversible. The Yang-Baxter maps are expressed in terms of two-component variables, and two-component parameters, and have a natural QRT-like composition of separate maps for each component. Through this approach, sixteen different Yang-Baxter maps are derived from known solutions of the classical star-triangle relations.