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Wikipedia / 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, acting on two out of three objects, satisfies

is the scattering matrix and if it satisfies the Yang–Baxter equation then the system is integrable. In quantum integrable models, the Yang–Baxter equation ensures that multi-particle scattering processes can be factorized into a sequence of two-body interactions, preserving exact solvability. The Yang–Baxter equation also shows up when discussing knot theory and the braid groups corresponds to swapping two strands. Since one can swap three strands in two different ways, the Yang–Baxter equation ensures that both paths are the same.

History

According to Michio Jimbo, the Yang–Baxter equation (YBE) manifested itself in the works of J. B. McGuire in 1964 and C. N. Yang in 1967. They considered a quantum mechanical many-body problem as the potential. Using the Bethe ansatz techniques, they found that the scattering matrix factorized to that of the two-body problem, and determined it exactly. This result established the Yang–Baxter equation as the defining criterion for quantum integrability, linking factorized scattering in one dimension to exactly solvable models. Here the YBE arises as the consistency condition for the factorization.

In statistical mechanics, the source of the YBE probably goes back to Onsager's star-triangle relation, briefly mentioned in the introduction to his solution of the Ising model in 1944. The hunt for solvable lattice models has been actively pursued since then, culminating in Rodney Baxter's solution of the eight vertex model in 1972.

Another line of development was the theory of factorized S-matrix in two dimensional quantum field theory. Alexander B. Zamolodchikov pointed out that the algebraic mechanics working here is the same as that in the Baxter's and others' works.

The YBE has also manifested itself in a study of Young operators in the group algebra of the symmetric group in the work of A. A. Jucys in 1966.

General form of the parameter-dependent Yang–Baxter equation

be a unital associative algebra, a parameter-dependent element of the tensor product 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).

, with algebra homomorphisms determined by

The general form of the Yang–Baxter equation is

.

Parameter-independent form

. The Yang–Baxter equation is

.

With respect to a basis

Often the unital associative algebra is the algebra of endomorphisms of a vector space over a field. With respect to a basis reads

Alternate form and representations of the braid group

be a module be the linear map.

, in which case the alternate form is

does not depend on parameters, the equation reduces to

is invertible) a representation of the braid group. This representation can be used to determine quasi-invariants of braids, knots and links.

Symmetry

matrix to be invariant under the action of a Lie group.

also satisfies the Yang–Baxter equation.

, while scale invariance.

Parametrizations and example solutions

, 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 a multiplicative parameter, the Yang–Baxter equation is

.

The braided forms read as:

. In this case a quantum determinant can be defined .

Example solutions of the parameter-dependent YBE

Classification of solutions

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.

Set-theoretic Yang–Baxter equation

Set-theoretic solutions were studied by Drinfeld.. The equation can then be considered purely as an equation in the category of sets.

Examples

  • .

  • , the transposition map.

  • is a (right) shelf is a set-theoretic solution to the YBE.

Classical Yang–Baxter equation

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

-matrix admits an asymptotic expansion).

See also

References

  • 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, .

  • Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge .

  • Jacques H.H. Perk and Helen Au-Yang, "YangBaxter Equations", (2006), .