For every classical r-matrix $r$, there is a Poisson bracket called Sklyanin bracket associated to $r$. It is defined in (3.3) of page 5 in (<http://arxiv.org/pdf/1101.0015v2.pdf>) as follows.
\begin{align}
\{f, g\} = \sum\_{ij} r\_{ij}( \partial\_i^L f \partial\_j^L g - \partial\_i^R f \partial\_j^R g ).
\end{align}
My question is: how to compute $\{f, g\}$ explicitly?


For example, in the case that $G = SL(2, \mathbb{C})$, $\mathfrak{g}=\mathfrak{sl}\_2$. We choose a basis of $\mathfrak{sl}\_2$ to be $e, f, h$. We choose the natural coordinate of $G = SL(2, \mathbb{C})$: $x\_{11}, x\_{12}, x\_{21}, x\_{22}$. How do we compute $\partial\_i^R x\_{kl}$ and $\partial\_i^L x\_{kl}$?


Any help will be greatly appreciated!



Questions about Sklyanin bracket
Perhaps you can use the method of section 4.3 (The second Russian formula (quadratic brackets)) in [Kosmann-Schwarzbach, Lie Bialgebras, Poisson Lie Groups, and Dressing Transformations](http://www.math.polytechnique.fr/cmat/kosmann/lnp2.pdf)? They calculate the bracket for SL(2,R) and SU(2) there.


See also example 4.9 in my [thesis](https://esc.fnwi.uva.nl/thesis/centraal/files/f973870249.pdf), where I have done the same calculation (based on the article of Kosmann-Schwarzbach).


