# Yang-Baxter equation

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The Yang–Baxter equation (or star-triangle relation) is an equation which was first introduced in the field of statistical mechanics. It takes its name from independent work of C. N. Yang from 1968, and R. J. Baxter from 1971. It refers to a principle in integrable systems taking the form of local equivalence transformations which appear in a variety of contexts, such as electric networks, knot theory and braid groups, and spin systems, to name just a few.

## Parameter-dependent Yang–Baxter equation

Let $A$ be a unital associative algebra. The parameter-dependent Yang–Baxter equation is an equation for $R(u)$, a parameter-dependent invertible element of the tensor product $A \otimes A$ (here, $u$ is the parameter, which usually ranges over all real numbers in the case of an additive parameter, or over all positive real numbers in the case of a multiplicative parameter). The Yang–Baxter equation is

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

for all values of $u$ and $v$, in the case of an additive parameter. At some value of the parameter $R(u)$ can turn into one dimensional projector, this gives rise to quantum determinant. For multiplicative parameter Yang–Baxter equation is

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

for all values of $u$ and $v$, where $R_{12}(w) = \phi_{12}(R(w))$, $R_{13}(w) = \phi_{13}(R(w))$, and $R_{23}(w) = \phi_{23}(R(w))$, for all values of the parameter $w$, and $\phi_{12} : A \otimes A \to A \otimes A \otimes A$, $\phi_{13} : A \otimes A \to A \otimes A \otimes A$, and $\phi_{23} : A \otimes A \to A \otimes A \otimes A$ are algebra morphisms determined by

$\phi_{12}(a \otimes b) = a \otimes b \otimes 1,$
$\phi_{13}(a \otimes b) = a \otimes 1 \otimes b,$
$\phi_{23}(a \otimes b) = 1 \otimes a \otimes b.$

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

## Parameter-independent Yang–Baxter equation

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

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

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

Let $V$ be a module of $A$. Let $T : V \otimes V \to V \otimes V$ be the linear map satisfying $T(x \otimes y) = y \otimes x$ for all $x, y \in V$. Then a representation of the braid group, $B_n$, can be constructed on $V^{\otimes n}$ by $\sigma_i = 1^{\otimes i-1} \otimes \check{R} \otimes 1^{\otimes n-i-1}$ for $i = 1,\dots,n-1$, where $\check{R} = T \circ R$ on $V \otimes V$. This representation can be used to determine quasi-invariants of braids, knots and links.