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Yang–Baxter equation

${\displaystyle ({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )=(\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}),}$

where ${\displaystyle {\check {R}}}$is${\displaystyle R}$ followed by a swap of the two objects. In one-dimensional quantum systems, ${\displaystyle R}$ 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 ${\displaystyle R}$ 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.

According to Jimbo,^[1] the Yang–Baxter equation (YBE) manifested itself in the works of J. B. McGuire^[2] in 1964 and C. N. Yang^[3] in 1967. They considered a quantum mechanical many-body problem on a line having ${\displaystyle c\sum _{i<j}\delta (x_{i}-x_{j})}$ as the potential. 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.

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

Another line of development was the theory of factorized S-matrix in two dimensional quantum field theory.^[6] Zamolodchikov pointed out^[7] 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 ${\displaystyle \mathbb {C} [S_{n}]}$ of the symmetric group in the work of A. A. Jucys^[8] in 1966.

Let ${\displaystyle A}$ be a unital associative algebra. In its most general form, the parameter-dependent Yang–Baxter equation is an equation for ${\displaystyle R(u,u')}$, a parameter-dependent element of the tensor product ${\displaystyle A\otimes A}$ (here, ${\displaystyle u}$ and ${\displaystyle u'}$ 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 ${\displaystyle R_{ij}(u,u')=\phi _{ij}(R(u,u'))}$ for ${\displaystyle 1\leq i<j\leq 3}$, with algebra homomorphisms ${\displaystyle \phi _{ij}:A\otimes A\to A\otimes A\otimes A}$ determined by

${\displaystyle \phi _{12}(a\otimes b)=a\otimes b\otimes 1,}$

${\displaystyle \phi _{13}(a\otimes b)=a\otimes 1\otimes b,}$

${\displaystyle \phi _{23}(a\otimes b)=1\otimes a\otimes b.}$

The general form of the Yang–Baxter equation is

${\displaystyle R_{12}(u_{1},u_{2})\ R_{13}(u_{1},u_{3})\ R_{23}(u_{2},u_{3})=R_{23}(u_{2},u_{3})\ R_{13}(u_{1},u_{3})\ R_{12}(u_{1},u_{2}),}$

for all values of ${\displaystyle u_{1}}$, ${\displaystyle u_{2}}$ and ${\displaystyle u_{3}}$.

Let ${\displaystyle A}$ be a unital associative algebra. The parameter-independent Yang–Baxter equation is an equation for ${\displaystyle R}$, an invertible element of the tensor product ${\displaystyle A\otimes A}$. The Yang–Baxter equation is

${\displaystyle R_{12}\ R_{13}\ R_{23}=R_{23}\ R_{13}\ R_{12},}$

where ${\displaystyle R_{12}=\phi _{12}($R)} , ${\displaystyle R_{13}=\phi _{13}($R)} , and ${\displaystyle R_{23}=\phi _{23}($R)} .

Often the unital associative algebra is the algebra of endomorphisms of a vector space ${\displaystyle V}$ over a field ${\displaystyle k}$, that is, ${\displaystyle A={\text{End}}($V)} . With respect to a basis ${\displaystyle \{e_{i}\}}$of${\displaystyle V}$, the components of the matrices ${\displaystyle R\in {\text{End}}($V)\otimes {\text{End}}(V)\cong {\text{End}}(V\otimes V)} are written ${\displaystyle R_{ij}^{kl}}$, which is the component associated to the map ${\displaystyle e_{i}\otimes e_{j}\mapsto e_{k}\otimes e_{l}}$. Omitting parameter dependence, the component of the Yang–Baxter equation associated to the map ${\displaystyle e_{a}\otimes e_{b}\otimes e_{c}\mapsto e_{d}\otimes e_{e}\otimes e_{f}}$ reads

${\displaystyle (R_{12})_{ij}^{de}(R_{13})_{ak}^{if}(R_{23})_{bc}^{jk}=(R_{23})_{jk}^{ef}(R_{13})_{ic}^{dk}(R_{12})_{ab}^{ij}.}$

Let ${\displaystyle V}$ be a moduleof${\displaystyle A}$, and ${\displaystyle P_{ij}=\phi _{ij}($P)} . Let ${\displaystyle P:V\otimes V\to V\otimes V}$ be the linear map satisfying ${\displaystyle P(x\otimes y)=y\otimes x}$ for all ${\displaystyle x,y\in V}$. The Yang–Baxter equation then has the following alternate form in terms of ${\displaystyle {\check {R}}(u,u')=P\circ R(u,u')}$on${\displaystyle V\otimes V}$.

${\displaystyle (\mathbf {1} \otimes {\check {R}}(u_{1},u_{2}))({\check {R}}(u_{1},u_{3})\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u_{2},u_{3}))=({\check {R}}(u_{2},u_{3})\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u_{1},u_{3}))({\check {R}}(u_{1},u_{2})\otimes \mathbf {1} )}$.

Alternatively, we can express it in the same notation as above, defining ${\displaystyle {\check {R}}_{ij}(u,u')=\phi _{ij}({\check {R}}(u,u'))}$ , in which case the alternate form is

${\displaystyle {\check {R}}_{23}(u_{1},u_{2})\ {\check {R}}_{12}(u_{1},u_{3})\ {\check {R}}_{23}(u_{2},u_{3})={\check {R}}_{12}(u_{2},u_{3})\ {\check {R}}_{23}(u_{1},u_{3})\ {\check {R}}_{12}(u_{1},u_{2}).}$

In the parameter-independent special case where ${\displaystyle {\check {R}}}$ does not depend on parameters, the equation reduces to

${\displaystyle (\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}})=({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )}$,

and (if${\displaystyle R}$ is invertible) a representation of the braid group, ${\displaystyle B_{n}}$, can be constructed on ${\displaystyle V^{\otimes n}}$by${\displaystyle \sigma _{i}=1^{\otimes i-1}\otimes {\check {R}}\otimes 1^{\otimes n-i-1}}$ for ${\displaystyle i=1,\dots ,n-1}$. This representation can be used to determine quasi-invariants of braids, knots and links.

Solutions to the Yang–Baxter equation are often constrained by requiring the ${\displaystyle R}$ matrix to be invariant under the action of a Lie group ${\displaystyle G}$. For example, in the case ${\displaystyle G=GL($V)} and ${\displaystyle R(u,u')\in {\text{End}}(V\otimes V)}$, the only ${\displaystyle G}$-invariant maps in ${\displaystyle {\text{End}}(V\otimes V)}$ are the identity ${\displaystyle I}$ and the permutation map ${\displaystyle P}$. The general form of the ${\displaystyle R}$-matrix is then ${\displaystyle R(u,u')=A(u,u')I+B(u,u')P}$ for scalar functions ${\displaystyle A,B}$.

The Yang–Baxter equation is homogeneous in parameter dependence in the sense that if one defines ${\displaystyle R'(u_{i},u_{j})=f(u_{i},u_{j})R(u_{i},u_{j})}$, where ${\displaystyle f}$ is a scalar function, then ${\displaystyle R'}$ also satisfies the Yang–Baxter equation.

The argument space itself may have symmetry. For example translation invariance enforces that the dependence on the arguments ${\displaystyle (u,u')}$ must be dependent only on the translation-invariant difference ${\displaystyle u-u'}$, while scale invariance enforces that ${\displaystyle R}$ is a function of the scale-invariant ratio ${\displaystyle u/u'}$.

A common ansatz for computing solutions is the difference property, ${\displaystyle R(u,u')=R(u-u')}$ , where R depends only on a single (additive) parameter. Equivalently, taking logarithms, we may choose the parametrization ${\displaystyle R(u,u')=R(u/u')}$ , 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:

${\displaystyle R_{12}($u)\ R_{13}(u+v)\ R_{23}(v)=R_{23}(v)\ R_{13}(u+v)\ R_{12}(u),}

for all values of ${\displaystyle u}$ and ${\displaystyle v}$. For a multiplicative parameter, the Yang–Baxter equation is

${\displaystyle R_{12}($u)\ R_{13}(uv)\ R_{23}(v)=R_{23}(v)\ R_{13}(uv)\ R_{12}(u),}

for all values of ${\displaystyle u}$ and ${\displaystyle v}$.

The braided forms read as:

${\displaystyle (\mathbf {1} \otimes {\check {R}}($u))({\check {R}}(u+v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(v))=({\check {R}}(v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u+v))({\check {R}}(u)\otimes \mathbf {1} )}

${\displaystyle (\mathbf {1} \otimes {\check {R}}($u))({\check {R}}(uv)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(v))=({\check {R}}(v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(uv))({\check {R}}(u)\otimes \mathbf {1} )}

In some cases, the determinant of ${\displaystyle R($u)} can vanish at specific values of the spectral parameter ${\displaystyle u=u_{0}}$. Some ${\displaystyle R}$ matrices turn into a one dimensional projector at ${\displaystyle u=u_{0}}$. In this case a quantum determinant can be defined ^{[clarification needed]}.

• A particularly simple class of parameter-dependent solutions can be obtained from solutions of the parameter-independent YBE satisfying ${\displaystyle {\check {R}}^{2}=\mathbf {1} }$, where the corresponding braid group representation is a permutation group representation. In this case, ${\displaystyle {\check {R}}($u)=\mathbf {1} +u{\check {R}}} (equivalently, ${\displaystyle R($u)=P+uP\circ {\check {R}}} ) is a solution of the (additive) parameter-dependent YBE. In the case where ${\displaystyle {\check {R}}=P}$ and ${\displaystyle R($u)=P+u\mathbf {1} } , this gives the scattering matrix of the Heisenberg XXX spin chain.

• The ${\displaystyle R}$-matrices of the evaluation modules of the quantum group ${\displaystyle U_{q}({\widehat {sl}}($2))} are given explicitly by the matrix

${\displaystyle {\check {R}}($z)={\begin{pmatrix}qz-q^{-1}z^{-1}&&&\\&q-q^{-1}&z-z^{-1}&\\&z-z^{-1}&q-q^{-1}&\\&&&qz-q^{-1}z^{-1}\end{pmatrix}}.}

Then the parametrized Yang-Baxter equation with the multiplicative parameter is satisfied:

${\displaystyle ({\check {R}}($z)\otimes \mathbf {1} )({\check {R}}(zz')\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(z'))=({\check {R}}(z')\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(zz'))({\check {R}}(z)\otimes \mathbf {1} )}

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 solutions were studied by Drinfeld.^[9] In this case, there is an ${\displaystyle R}$-matrix invariant basis ${\displaystyle X}$ for the vector space ${\displaystyle V}$ in the sense that the ${\displaystyle R}$-matrix maps the induced basis on ${\displaystyle V\otimes V}$ to itself. This then induces a map ${\displaystyle r:X\times X\rightarrow X\times X}$ given by restriction of the ${\displaystyle R}$-matrix to the basis. The set-theoretic Yang–Baxter equation is then defined using the 'twisted' alternate form above, asserting $${\displaystyle (id\times r)(r\times id)(id\times r)=(r\times id)(id\times r)(r\times id)}$$ as maps on ${\displaystyle X\times X\times X}$. The equation can then be considered purely as an equation in the category of sets.

• ${\displaystyle R=id}$.
• ${\displaystyle R=\tau }$ where ${\displaystyle \tau (u\otimes v)=v\otimes u}$, the transposition map.
• If${\displaystyle (X,\triangleleft )}$ is a (right) shelf, then ${\displaystyle r(x,y)=(y,x\triangleleft y)}$ is a set-theoretic solution to the YBE.

Solutions to the classical YBE were studied and to some extent classified by Belavin and Drinfeld.^[10] Given a 'classical ${\displaystyle r}$-matrix' ${\displaystyle r:V\otimes V\rightarrow V\otimes V}$, which may also depend on a pair of arguments ${\displaystyle (u,v)}$, the classical YBE is (suppressing parameters) $${\displaystyle [r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=0.}$$ This is quadratic in the ${\displaystyle r}$-matrix, unlike the usual quantum YBE which is cubic in ${\displaystyle R}$.

This equation emerges from so called quasi-classical solutions to the quantum YBE, in which the ${\displaystyle R}$-matrix admits an asymptotic expansion in terms of an expansion parameter ${\displaystyle \hbar ,}$ $${\displaystyle R_{\hbar }=I+\hbar r+{\mathcal {O}}(\hbar ^{2}).}$$ The classical YBE then comes from reading off the ${\displaystyle \hbar ^{2}}$ coefficient of the quantum YBE (and the equation trivially holds at orders ${\displaystyle \hbar ^{0},\hbar }$).

• Lie bialgebra
• Yangian
• Reidemeister move
• Quasitriangular Hopf algebra

• 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, ISBN 3-540-53503-9.
• Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0.
• Jacques H.H. Perk and Helen Au-Yang, "Yang–Baxter Equations", (2006), arXiv:math-ph/0606053.

• "Yang-Baxter equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]