How to obtain the classical Yang-Baxter equation from a related equation

Question

I have a question about the equation (1.24) in the paper about classical r-matrices.

It is said that when we put $\overline{r} = Pr$ in the equation (1.24): $$ \overline{r}_{23}\overline{r}_{12}P_{23} + \overline{r}_{23}P_{12}\overline{r}_{23} + P_{23}\overline{r}_{12}\overline{r}_{23} = \overline{r}_{12}\overline{r}_{23}P_{12} + \overline{r}_{12}P_{23}\overline{r}_{12} + P_{12}\overline{r}_{23}\overline{r}_{12}, $$ then we obtain the classical Yang-Baxter equation (1.16) $$ [r_{12}, r_{13}] + [r_{13}, r_{23}] + [r_{12}, r_{23}] = 0. $$ I try to prove this statement. But I encounter some problems.

I think that we have $$ (Pr)_{23} = P_{23} r_{23} P_{23} = P_{32}. $$

Therefore \begin{align} & \overline{r}_{23} \overline{r}_{12} P_{23} \\ & = P_{23} r_{23} P_{23} P_{12} r_{12} P_{12} P_{23} \\ & = r_{32} r_{21} P_{23}. \end{align}

But I don't know how to remove $P$ in $r_{32} r_{21} P_{23}$. How to prove the statement: when we put $\overline{r} = Pr$ in the equation (1.24), then we obtain the classical Yang-Baxter equation (1.16)? Thank you very much.

Answer 1

The index notation refers to the tensor factors these maps act on, i.e. $(Pr)_{23}=P_{23}r_{23}$ etc. To prove the claim, you collect all P factors on the left, where they give the total inversion permutation M on both sides of the equation. Multiplying by M from the left then gives the CYBE.