Yang–Baxter equation

It depends on the idea that in some scattering situations, particles may preserve their momentum while changing their quantum internal states.

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

Since one can swap three strands in two different ways, the Yang–Baxter equation enforces that both paths are the same.

According to Jimbo,[1] the Yang–Baxter equation (YBE) manifested itself in the works of J.

They considered a quantum mechanical many-body problem on a line having

Using Bethe's Ansatz techniques, they found that the scattering matrix factorized to that of the two-body problem, and determined it exactly.

Here YBE arises as the consistency condition for the factorization.

In statistical mechanics, the source of YBE probably goes back to Onsager's star-triangle relation, briefly mentioned in the introduction to his solution of the Ising model[4] in 1944.

The hunt for solvable lattice models has been actively pursued since then, culminating in Baxter's solution of the eight vertex model[5] in 1972.

The YBE has also manifested itself in a study of Young operators in the group algebra

determined by The general form of the Yang–Baxter equation is for all values of

Omitting parameter dependence, the component of the Yang–Baxter equation associated to the map

This representation can be used to determine quasi-invariants of braids, knots and links.

matrix to be invariant under the action of a Lie group

The Yang–Baxter equation is homogeneous in parameter dependence in the sense that if one defines

For example translation invariance enforces that the dependence on the arguments

A common ansatz for computing solutions is the difference property,

Equivalently, taking logarithms, we may choose the parametrization

In those cases, we may reduce the YBE to two free parameters in a form that facilitates computations: for all values of

The braided forms read as: In some cases, the determinant of

can vanish at specific values of the spectral parameter

In this case a quantum determinant can be defined [clarification needed].

Then the parametrized Yang-Baxter equation (in braided form) with the multiplicative parameter is satisfied: There are broadly speaking three classes of solutions: rational, trigonometric and elliptic.

These are related to quantum groups known as the Yangian, affine quantum groups and elliptic algebras respectively.

The set-theoretic Yang–Baxter equation is then defined using the 'twisted' alternate form above, asserting

Solutions to the classical YBE were studied and to some extent classified by Belavin and Drinfeld.

-matrix, unlike the usual quantum YBE which is cubic in

This equation emerges from so called quasi-classical solutions to the quantum YBE, in which the

coefficient of the quantum YBE (and the equation trivially holds at orders