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.