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A classical $R$-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.)

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 is similar.

See also the related Sweedler notation.