A classical [$R$-matrix](https://en.wikipedia.org/wiki/R-matrix) $r\in {\rm End}(\mathfrak{g}\otimes \mathfrak{g})$ acts on 2 copies of a Lie algebra $\mathfrak{g}$. 

The [Sweedler notation](https://ncatlab.org/nlab/show/Sweedler+notation) $r_{k\ell}\in {\rm End}(\mathfrak{g}\otimes \mathfrak{g}\otimes \mathfrak{g})$ means that $r_{k\ell}$ acts on the triple tensor product $\mathfrak{g}\otimes \mathfrak{g}\otimes \mathfrak{g}$ by letting $r$ acting on the $k$'th and $\ell$'th copy of the Lie algebra $\mathfrak{g}$, and an identity ${\bf 1}_{\mathfrak{g}}$ acts on the remaining copy.

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.