In the classical equation, one looks for $R\in\Lambda^2\mathfrak g$ such that $$[R,R]=0,$$ where the bracket is Schouten's bracker in $\Lambda^\bullet\mathfrak g$, the exterior algebra on a Lie algebra $\mathfrak g$. In the quantum one (in its non-parametric form...), one looks for endomorphisms $R:V\otimes V\to V\otimes V$ of tensor squares of vector spaces $V$ such that $$R_{12} \ R_{13} \ R_{23} = R_{23} \ R_{13} \ R_{12},$$
Let $A$ be a unital algebra unital associative Algebra over a field algebra. The parameter-dependent Yang–Baxter equation is an equation for $R(u)$, a parameter-dependent inverse element invertible element of the tensor product of algebras tensor product $A \otimes A$ (here, $u$ is the parameter, which usually ranges over Real numbers $\mathbb R$ in the case of an additive parameter, or over positive real numbers $\mathbb R^+$ 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
