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Yang–Baxter equation

The Yang–Baxter equation 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 1982.

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, and

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

for all values of u and v, in the case of a multiplicative parameter, where R12(w) = φ12(R(w)), R13(w) = φ13(R(w)), and R23(w) = φ23(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

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

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 R12 = φ12(R), R13 = φ13(R), and R23 = φ23(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, Bn, 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.