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Super version of a formula for Poisson brackets

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I am reading a paper Lie superbialgebras and poisson-lie supergroups and trying to figure out how to compute a super Poisson bracket from a super $r$-matrix.

Let $G$ be a Lie supergroup and $\mathfrak{g}$ its Lie superalgebra. The formula (3) on page 158 of the paper Lie superbialgebras and poisson-lie supergroups is \begin{align} \{ \phi, \psi \} = \sum_{\mu, \nu \in B} (-1)^{|\phi||\nu|} r^{\mu \nu}( R_{\mu} \phi R_{\nu} \psi - L_{\mu} \phi L_{\nu} \psi ), \end{align} where $B$ is a homogeneous basis of $\mathfrak{g}$, $r = \sum_{\mu, \nu} r^{\mu \nu} \mu \otimes \nu$. This is the super Poisson bracket of $\mathcal{O}(G)$ which comes from $r$. This formula is very similar to the formula in the end of page 60 of a guide to quantum groups by Chari and Pressley.

Has the following formula \begin{align} \{T \overset{\otimes}{,} T\} = [T \otimes T, r] \end{align} on page 61 of "a guide to quantum groups" been translated to the super case? Are there some references about this? Thank you very much.


This post imported from StackExchange MathOverflow at 2016-10-02 10:49 (UTC), posted by SE-user Jianrong Li

asked Sep 5, 2016 in Theoretical Physics by Jianrong Li (30 points) [ revision history ]
edited Oct 2, 2016 by Dilaton

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