Yang–Baxter equation

In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their momentum while changing their quantum internal states. It states that a matrix , acting on two out of three objects, satisfies In one-dimensional quantum systems, is the scattering matrix and if it satisfies the Yang–Baxter equation then the system is integrable. The Yang–Baxter equation also shows up when discussing knot theory and the braid groups where corresponds to swapping two strands. Since one can swap three strands two different ways, the Yang–Baxter equation enforces that both paths are the same. It takes its name from independent work of C. N. Yang from 1968, and R. J. Baxter from 1971. Let be a unital associative algebra. In its most general form, the parameter-dependent Yang–Baxter equation is an equation for , a parameter-dependent element of the tensor product (here, and are the parameters, which usually range over the real numbers R in the case of an additive parameter, or over positive real numbers R+ in the case of a multiplicative parameter). Let for , with algebra homomorphisms determined by The general form of the Yang–Baxter equation is for all values of , and . Let be a unital associative algebra. The parameter-independent Yang–Baxter equation is an equation for , an invertible element of the tensor product . The Yang–Baxter equation is where , , and . Often the unital associative algebra is the algebra of endomorphisms of a vector space over a field , that is, . With respect to a basis of , the components of the matrices are written , which is the component associated to the map . Omitting parameter dependence, the component of the Yang–Baxter equation associated to the map reads Let be a module of , and . Let be the linear map satisfying for all . The Yang–Baxter equation then has the following alternate form in terms of on .