Questions about Sklyanin bracket

Question

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!

Answer 1

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? They calculate the bracket for SL(2,R) and SU(2) there.

See also example 4.9 in my thesis, where I have done the same calculation (based on the article of Kosmann-Schwarzbach).