A classical [$R$-matrix](https://en.wikipedia.org/wiki/R-matrix) $r\in \mathfrak{g}\otimes \mathfrak{g}$ is an element in the 2nd tensor power of an algebra $\mathfrak{g}$ (formally extending the algebra with a unit ${\bf 1}$). 

The [Sweedler notation](https://ncatlab.org/nlab/show/Sweedler+notation) $r_{k\ell}\in \mathfrak{g}\otimes \mathfrak{g}\otimes \mathfrak{g}$ for an element in the 3rd tensor power means that $r$ belongs to the $k$'th and $\ell$'th copy of the algebra $\mathfrak{g}$, and one should plug ${\bf 1}$ into the remaining copy. (If $k>\ell$ this involves a transposition.)

The notation for the quantum $R$-matrix $R$ and the [quantum Yang-Baxter equation](https://en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation) is similar.