Meaning of Yang-Baxter equation for classical $r$-matrix

Question

I'm reading this [math/9802054] paper on the structure of the phase space of Chern-Simons TQFT. I'm stuck at the definition of the classical $r$-matrix, which goes as follows:

This might sound dumb, but I don't understand what $r_{12}$, $r_{23}$, $r_{13}$, $r_{21}$ is. I could greatly benefit from an illustrating example.

Equation (15) suggests that $r_{12}$ and $r_{21}$ from equation (14) can be defined as $r$ and its transpose. But I suspect that $r_{12}$ from equation (13) is different, and probably has smth to do with higher tensor powers of $\mathfrak{g}$, but I simply couldn't come up with a meaningful definition. This is embarassingly confusing.

Answer 1

1. 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.)

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 is similar.

3. See also the related Sweedler notation.