Yang–Baxter equation

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 ${\displaystyle V}$ be a module of ${\displaystyle A}$, 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 dependence 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 .

Set-theoretic solutions were studied by Drinfeld.[1] 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.

Solutions to the classical YBE were studied and to some extent classified by Belavin and Drinfeld.[2] 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

• Jimbo, Michio (1989). "Introduction to the Yang-Baxter equation". International Journal of Modern Physics A. 4 (15): 3759–3777. Bibcode:1989IJMPA...4.3759J. doi:10.1142/S0217751X89001503. MR 1017340.
• 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]