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.
General form of the parameter-dependent Yang–Baxter equation
Let be a unitalassociativealgebra. 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
A common ansatz for computing solutions is the difference property, , where R depends only on a single (additive) 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[clarification needed].
Example solutions of the parameter-dependent YBE
A particularly simple class of parameter-dependent solutions can be obtained from solutions of the parameter-independent YBE satisfying , where the corresponding braid group representation is a permutation group representation. In this case, (equivalently, ) is a solution of the (additive) parameter-dependent YBE. In the case where and , this gives the scattering matrix of the Heisenberg XXX spin chain.
H.-D. Doebner, J.-D. Hennig, eds, Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Clausthal, FRG, 1989, Springer-Verlag Berlin, ISBN3-540-53503-9.