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physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
. 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 R, acting on two out of three objects, satisfies :(\check\otimes \mathbf)(\mathbf\otimes \check)(\check\otimes \mathbf) =(\mathbf\otimes \check)(\check \otimes \mathbf)(\mathbf\otimes \check), where \check is R followed by a swap of the two objects. In one-dimensional quantum systems, R is the scattering matrix and if it satisfies the Yang–Baxter equation then the system is
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
. The Yang–Baxter equation also shows up when discussing
knot theory In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...
and the
braid group In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of ...
s where R corresponds to swapping two strands. Since one can swap three strands in two different ways, the Yang–Baxter equation enforces that both paths are the same.


History

According to
Michio Jimbo is a Japanese mathematician working in mathematical physics and is a professor of mathematics at Rikkyo University. He is a grandson of the linguist . Career After graduating from the University of Tokyo in 1974, he studied under Mikio Sato a ...
, 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 on a line having c\sum_\delta(x_i - x_j) as the potential. Using the
Bethe ansatz In physics, the Bethe ansatz is an ansatz for finding the exact wavefunctions of certain quantum many-body models, most commonly for one-dimensional lattice models. It was first used by Hans Bethe in 1931 to find the exact eigenvalues and eigenv ...
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 In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, 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 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 \mathbb _n of the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
in the work of A. A. Jucys in 1966.


General form of the parameter-dependent Yang–Baxter equation

Let A be a unital
associative algebra In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
. In its most general form, the parameter-dependent Yang–Baxter equation is an equation for R(u,u'), a parameter-dependent element of the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
A \otimes A (here, u and u' are the parameters, which usually range over the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s ℝ in the case of an additive parameter, or over
positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
+ in the case of a multiplicative parameter). Let R_(u,u') = \phi_(R(u,u')) for 1\le i< j \le3, with algebra
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s \phi_ : A \otimes A \to A \otimes A \otimes A determined by :\phi_(a \otimes b) = a \otimes b \otimes 1, :\phi_(a \otimes b) = a \otimes 1 \otimes b, :\phi_(a \otimes b) = 1 \otimes a \otimes b. The general form of the Yang–Baxter equation is :R_(u_1,u_2) \ R_(u_1,u_3) \ R_(u_2,u_3) = R_(u_2,u_3) \ R_(u_1,u_3) \ R_(u_1,u_2), for all values of u_1, u_2 and u_3.


Parameter-independent form

Let A be a unital associative algebra. The parameter-independent Yang–Baxter equation is an equation for R, an invertible element of the tensor product A \otimes A. The Yang–Baxter equation is :R_ \ R_ \ R_ = R_ \ R_ \ R_, where R_ = \phi_(R), R_ = \phi_(R), and R_ = \phi_(R).


With respect to a basis

Often the unital associative algebra is the algebra of endomorphisms of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
V over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
k, that is, A = \text(V). With respect to a
basis Basis is a term used in mathematics, finance, science, and other contexts to refer to foundational concepts, valuation measures, or organizational names; here, it may refer to: Finance and accounting * Adjusted basis, the net cost of an asse ...
\ of V, the components of the matrices R\in \text(V)\otimes\text(V) \cong \text(V\otimes V) are written R_^, which is the component associated to the map e_i\otimes e_j \mapsto e_k\otimes e_l. Omitting parameter dependence, the component of the Yang–Baxter equation associated to the map e_a\otimes e_b\otimes e_c \mapsto e_d \otimes e_e \otimes e_f reads :(R_)_^(R_)_^(R_)_^ = (R_)_^ (R_)_^ (R_)_^.


Alternate form and representations of the braid group

Let V be a module of A, and P_ = \phi_(P) . Let P : V \otimes V \to V \otimes V be the linear map satisfying P(x \otimes y) = y \otimes x for all x, y \in V. The Yang–Baxter equation then has the following alternate form in terms of \check(u,u') = P \circ R(u,u') on V \otimes V. :(\mathbf \otimes \check(u_1,u_2)) (\check(u_1,u_3) \otimes \mathbf)(\mathbf\otimes \check(u_2,u_3)) = (\check(u_2,u_3) \otimes \mathbf) ( \mathbf\otimes \check(u_1,u_3) ) ( \check(u_1,u_2) \otimes \mathbf). Alternatively, we can express it in the same notation as above, defining \check_(u,u') = \phi_(\check(u,u')) , in which case the alternate form is :\check_(u_1,u_2) \ \check_(u_1,u_3) \ \check_(u_2,u_3) = \check_(u_2,u_3) \ \check_(u_1,u_3) \ \check_(u_1,u_2). In the parameter-independent special case where \check does not depend on parameters, the equation reduces to :(\mathbf\otimes \check)(\check \otimes \mathbf)(\mathbf\otimes \check) = (\check\otimes \mathbf)(\mathbf\otimes \check)(\check\otimes \mathbf), and (if R is invertible) a
representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
of the
braid group In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of ...
, B_n, can be constructed on V^ by \sigma_i = 1^ \otimes \check \otimes 1^ for i = 1,\dots,n-1. This representation can be used to determine quasi-invariants of
braids A braid (also referred to as a plait; ) is a complex structure or pattern formed by interlacing three or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strand ...
,
knots A knot is a fastening in rope or interwoven lines. Knot or knots may also refer to: Other common meanings * Knot (unit), of speed * Knot (wood), a timber imperfection Arts, entertainment, and media Films * ''Knots'' (film), a 2004 film * ''Kn ...
and links.


Symmetry

Solutions to the Yang–Baxter equation are often constrained by requiring the R matrix to be invariant under the action of a
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
G. For example, in the case G = GL(V) and R(u,u')\in \text(V\otimes V), the only G-invariant maps in \text(V\otimes V) are the identity I and the permutation map P. The general form of the R-matrix is then R(u, u') = A(u,u')I + B(u, u')P for scalar functions A, B. The Yang–Baxter equation is homogeneous in parameter dependence in the sense that if one defines R'(u_i, u_j) = f(u_i, u_j)R(u_i,u_j), where f is a scalar function, then R' also satisfies the Yang–Baxter equation. The argument space itself may have symmetry. For example translation invariance enforces that the dependence on the arguments (u, u') must be dependent only on the translation-invariant difference u-u', while scale invariance enforces that R is a function of the scale-invariant ratio u/u'.


Parametrizations and example solutions

A common ansatz for computing solutions is the difference property, R(u,u') = R(u - u') , where R depends only on a single (additive) parameter. Equivalently, taking logarithms, we may choose the parametrization R(u,u') = R(u/u') , 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: :R_(u) \ R_(u+v) \ R_(v) = R_(v) \ R_(u+v) \ R_(u), for all values of u and v. For a multiplicative parameter, the Yang–Baxter equation is :R_(u) \ R_(uv) \ R_(v) = R_(v) \ R_(uv) \ R_(u), for all values of u and v. The braided forms read as: : (\mathbf\otimes \check(u)) (\check(u + v) \otimes \mathbf) (\mathbf\otimes \check(v)) = (\check(v) \otimes \mathbf) (\mathbf\otimes \check(u + v))(\check(u) \otimes \mathbf ) : (\mathbf\otimes \check(u)) (\check(uv) \otimes \mathbf) (\mathbf\otimes \check(v)) = (\check(v) \otimes \mathbf) (\mathbf\otimes \check(uv))(\check(u) \otimes \mathbf ) In some cases, the determinant of R (u) can vanish at specific values of the spectral parameter u=u_ . Some R matrices turn into a one dimensional projector at u=u_ . In this case a quantum determinant can be defined .


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 \check^2 = \mathbf , where the corresponding braid group representation is a permutation group representation. In this case, \check(u) = \mathbf + u \check (equivalently, R(u) = P + u P \circ \check ) is a solution of the (additive) parameter-dependent YBE. In the case where \check = P and R(u) = P + u \mathbf , this gives the scattering matrix of the Heisenberg XXX spin chain. * The R-matrices of the evaluation modules of the quantum group U_q(\widehat(2)) are given explicitly by the matrix : \check(z) = \begin q z - q^z^ & & & \\ & q-q^ & z-z^ &\\ & z-z^ & q-q^ &\\ & & & q z - q^z^ \end. Then the parametrized Yang-Baxter equation (in braided form) with the multiplicative parameter is satisfied: : (\mathbf\otimes\check(z)) (\check(zz') \otimes \mathbf) (\mathbf\otimes \check(z')) = (\check(z') \otimes \mathbf) (\mathbf\otimes \check(zz'))(\check(z) \otimes \mathbf )


Classification of solutions

There are broadly speaking three classes of solutions: rational, trigonometric and elliptic. These are related to
quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...
s known as the Yangian, affine quantum groups and elliptic algebras respectively.


Set-theoretic Yang–Baxter equation

Set-theoretic solutions were studied by Drinfeld. In this case, there is an R-matrix invariant basis X for the vector space V in the sense that the R-matrix maps the induced basis of V\otimes V to itself. This then induces a map r: X\times X \rightarrow X\times X given by restriction of the R-matrix to the basis. The set-theoretic Yang–Baxter equation is then defined using the 'twisted' alternate form above, asserting (id\times r)(r\times id)(id\times r) = (r\times id)(id\times r)(r \times id) as maps on X\times X\times X. The equation can then be considered purely as an equation in the
category of sets In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...
.


Examples

* R = id. * R = \tau where \tau(u\otimes v) = v\otimes u, the transposition map. * If (X, \triangleleft) is a (right) shelf, then r(x,y) = (y, x \triangleleft y) 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. Given a 'classical r-matrix' r: V\otimes V \rightarrow V\otimes V, which may also depend on a pair of arguments (u, v), the classical YBE is (suppressing parameters) _, r_+ _, r_+ _, r_= 0. This is quadratic in the r-matrix, unlike the usual quantum YBE which is cubic in R. This equation emerges from so called quasi-classical solutions to the quantum YBE, in which the R-matrix admits an asymptotic expansion in terms of an expansion parameter \hbar, R_\hbar = I + \hbar r + \mathcal(\hbar^2). The classical YBE then comes from reading off the \hbar^2 coefficient of the quantum YBE (and the equation trivially holds at orders \hbar^0, \hbar).


See also

*
Lie bialgebra In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi ...
* Yangian *
Reidemeister move In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. and, independently, , demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a seque ...
*
Quasitriangular Hopf algebra In mathematics, a Hopf algebra, ''H'', is quasitriangularMontgomery & Schneider (2002), p. 72 if there exists an invertible element, ''R'', of H \otimes H such that :*R \ \Delta(x)R^ = (T \circ \Delta)(x) for all x \in H, where \Delta is the cop ...
* Yang–Baxter operator


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 Vyjayanthi Chari (born 1958) is an Indian–American Distinguished Professor and the F. Burton Jones Endowed Chair for Pure Mathematics at the University of California, Riverside, known for her research in representation theory and quantum algebr ...
and Andrew Pressley, ''A Guide to Quantum Groups'', (1994), Cambridge University Press, Cambridge . * Jacques H.H. Perk and Helen Au-Yang, "Yang–Baxter Equations", (2006), .


External links

* {{DEFAULTSORT:Yang-Baxter equation Eponymous equations of physics Yang Chen-Ning Monoidal categories Statistical mechanics Exactly solvable models Conformal field theory