TITLE: Construct super Poisson brackets on the coordinate rings of Lie super groups
QUESTION [1 upvotes]: On line 7 of page 61 of the book a guide to quantum groups, 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.

REPLY [1 votes]: 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).
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