On line 7 of page 61 of the book [a guide to quantum groups](https://books.google.de/books/about/A_Guide_to_Quantum_Groups.html?id=bn5GNkLfnsAC&redir_esc=y), a Poisson bracket is defined on $\mathbb{C}[GL\_n]$ for every classical $r$-matrix as follows.


Let $V$ be a vector space with a basis $v\_1, \ldots, v\_n$, and $r \in \mathfrak{g} \otimes \mathfrak{g}$ is a skew-symmetric classical $r$-matrix. Suppose that
\begin{align}
r(v\_i \otimes v\_j) = \sum\_{k,l} r\_{ij}^{(kl)} v\_k \otimes v\_l,
\end{align}
for some $r\_{ij}^{(kl)} \in \mathbb{C}$.


Define
\begin{align}
\{c\_{i j}, c\_{kl}\} = \sum\_{s,s'} r\_{s',s}^{(j l)} c\_{ks} c\_{i s'} - \sum\_{s, s' } r\_{ik}^{(s' s)} c\_{s' j} c\_{s l}, \ i,j,k,l = 1,2, \ldots, n. \quad (1)
\end{align}
Here $c\_{ij}$ are natural coordinates on $\mathbb{C}[GL\_n]$. By using classical $r$-matrix
\begin{align}
\sum\_{s} (r\_{sj}^{(s\_3s\_4)} r\_{ik}^{(ss\_2)} - r\_{sk}^{(s\_3s\_2)} r\_{ij}^{(ss\_4)} + r\_{is}^{(s\_3s\_4)} r\_{jk}^{(ss\_2)} - r\_{sk}^{(s\_4s\_2)} r\_{ij}^{(s\_3s)} + r\_{is}^{(s\_3s\_2)} r\_{jk}^{(s\_4s)} - r\_{js}^{(s\_4s\_2)} r\_{ik}^{(s\_3s)}) =0 
\end{align}
and 
\begin{align}
r\_{ij}^{(kl)} = - r\_{ji}^{(lk)},
\end{align}
we can show that the bracket defined by (1) is Poisson. Is it possible to modify (1) such that it is a super Poisson bracket on $\mathbb{C}[GL(m|n)]$? Thank you very much.



Construct super Poisson brackets on the coordinate rings of Lie super groups
I think that it is done in the paper of Andruskiewitsch "Lie superbialgebras and Poisson-Lie supergroups", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 63 (1993), 147-163, <http://link.springer.com/article/10.1007/BF02941339> (the key result is Proposition 3 in that paper). 


