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This is also proved in Lemma 2.6 of the paper: CO-POISSON COALGEBRA AND CO-POISSON HOPF ALGEBRA STRUCTURES ON $k[x_1, x_2, \ldots, x_d]$ .

In the book Algebras of Functions on Quantum Groups: Part I, Remark 3.1.4, we have the following result. Let $A$ be a Poisson Hopf algebra. That is, $A$ is both a Hopf algebra and a Poisson algebra …

Let $X$ be a Poisson variety. There is a concept "cluster algebra structure compatible with Poisson structure" introduced in the paper. Suppose that we construct a maximal independent set of functio …

Let $G$ be a Poisson-Lie group. Let $M$ be a symplectic manifold. In the paper, the third paragraph of page 1238, it is said that an action $G \times M \to M$ is called Poisson if $G \times M \to M$ …

This question was solved by Vladimir Dotsenko in the comments of the question.

Let $V$ be a finite dimensional vector space and $S(V)$ the corresponding symmetric algebra. Suppose that we have a Poisson bracket $\lambda = \{,\}: S(V) \otimes S(V) \to S(V)$. Let $V^*$ be the dual …

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{al …

On page 12 of the paper, there is a formula about super Poisson bracket on a Lie super group $G$: \begin{align} \{\phi, \psi\} = \sum_{\mu, \nu} (-1)^{|\phi||\nu|} r^{\mu \nu} ( R_{\mu} \phi R_{\nu} \ …

In this paper, page 149, the super Jacobi identity is given by \begin{align} J(x, y,z) := (-1)^{|x||z|}[[x, y],z] +(-1)^{|z||y|}[[z,x], y]+(-1)^{|y||x|}[[y,z],x] = 0. \end{align} But in this article, …

I am reading the lecture notes and trying to understand dressing actions. Let $G$ be a Poisson-Lie group and $G^*$ its dual Poisson-Lie group. In the lecture notes above, Proposition 5.22 on page 80 …

I have a question about decomposing elements in $SL_2$ as a pair of elements in $SL_2^*$. Here $SL_2^*$ is the dual Poisson Lie group of $SL_2$ which is defined as follows. Let $G$ be a Poisson-Lie …

Let $H$ be a Poisson algebra. A Poisson $H$-module is a vector space $V$ with two bilinear maps \begin{align} H \otimes V \to V \\ (h,v) \mapsto h.v, \end{align} \begin{align} H \otimes V \to V \\ (h, …

Let $A$ be a Poisson super algebra ($A$ is a super algebra and $A$ satisfies super Jacobi identity, super commutativity, super Leibniz rule). Super version of the product of two tensor products is \ …

What is the current status of the classifications of Lie bialgebras? In particular, has the following problem been solved? Let $gl_n$ be the general linear Lie algebra. Classify all Lie cobrackets $\d …

By a theorem of Drinfeld, there is a one to one correspondence between Lie bialgebras and Poisson Lie groups. Therefore given a Lie cobracket $\delta: g \to \Lambda^2 g$, there is a Poisson bracket on …