Yang–Baxter equation


In physics, the Yang–Baxter equation 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.

General form of the parameter-dependent Yang–Baxter equation

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 .
Let for, with algebra homomorphisms determined by
The general form of the Yang–Baxter equation is
for all values of , and .

Parameter-independent form

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.

Alternate form and representations of the braid group

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.
Alternatively, we can express it in the same notation as above, defining , in which case the alternate form is
In the parameter-independent special case where does not depend on parameters, the equation reduces to
and a representation of the braid group,, can be constructed on by for. This representation can be used to determine quasi-invariants of braids, knots and links.

Parametrizations and example solutions

A common ansatz for computing solutions is the difference property, , where R depends only on a single parameter. Equivalently, taking logarithms, we may choose the parametrization , in which case R is said to depend on a multiplicative parameter. In those cases, we may reduce the YBE to two free parameters in a form that facilitates computations:
for all values of and. For a multiplicative parameter, the Yang–Baxter equation is
for all values of and.
The braided forms read as:
In some cases, the determinant of can vanish at specific values of the spectral parameter. Some matrices turn into a one dimensional projector at
. In this case a quantum determinant can be defined.

Example solutions of the parameter-dependent YBE