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
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
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
Solutions to the Yang–Baxter equation are often constrained by requiring the matrix to be invariant under the action of a Lie group. For example, in the case and , the only -invariant maps in are the identity and the permutation map . The general form of the -matrix is then for scalar functions .
The Yang–Baxter equation is homogeneous in parameter dependence in the sense that if one defines , where is a scalar function, then also satisfies the Yang–Baxter equation.
The argument space itself may have symmetry. For example translation invariance enforces that the dependence on the arguments must be dependence only on the translation-invariant difference , while scale invariance enforces that is a function of the scale-invariant ratio .
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.
The -matrices of the evaluation modules of the quantum group are given explicitly by the matrix
Then the parametrized Yang-Baxter equation with the multiplicative parameter is satisfied:
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.