# Super version of a formula for Poisson brackets

+ 2 like - 0 dislike
408 views

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

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\varnothing$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.