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

 The notation $r_{k\ell}\in \mathfrak{g}\otimes \mathfrak{g}\otimes \mathfrak{g}$ for an element of 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.)

2. The notation for the quantum $R$-matrix 
$$R~=~{\bf 1}\otimes{\bf 1} +\hbar r +{\cal O}(\hbar^2)$$ 
and the [quantum Yang-Baxter equation](https://en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation) is similar. 

3. See also the related [Sweedler notation](https://ncatlab.org/nlab/show/Sweedler+notation).