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The classical Yang-Baxter equation is \begin{align} [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0. \quad (1) \end{align} What are the differences between this equation in the case of Lie ...

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}...

Let $g= gl_2$. Suppose that $r \in g \otimes g$ satisfies the following properties: (1) $r_{12} + r_{21} \in g \otimes g$ is $g$-invariant, $r_{12} = r$, $r_{21} = \tau \ r_{12}$. (2) $[r_{12}, r_{...