\ldots, y_n]$.

\[re:Class Poisson\] A classification of Poisson structures compatible with cluster algebras was obtained by Gekhtman, Shapiro and Vainshtein in [@GSV Theorem 1.4]. It is easy to see from their description that if $n$ is even, then the cluster algebra has an admissible Poisson structure of maximal rank.

We will refer to the cluster algebra $\AA$ defined by the initial seeed $({\bf x},B)$ together with the compatible Poisson structure defined by the coefficient matrix $\Lambda$ with respect to the cluster ${\bf x}$ as the [*Poisson cluster algebra*]{} defined by the [*Poisson seed*]{} $({\bf x},B,\Lambda)$.

It is not obvious under which conditions a Poisson seed $({\bf x},B,\Lambda)$ would yield a Poisson bracket $\{\cdot,\cdot\}_\Lambda$ on $\mathfrak{F}$ such that $\{\AA,\AA\}_\Lambda\subset \AA$. We have, however, the following fact.

Let $({\bf x},B,\Lambda)$ be a Poisson seed and $\AA$ the corresponding cluster algebra. Then $\Lambda$ defines a Poisson algebra structure on the upper bound $\UU_{{\bf x},B}(\AA)$ and the upper cluster algebra $\UU(\AA)$.

Denote as above by $\{\cdot,\cdot\}_\Lambda$ the Poisson bracket on $\mathfrak{F}$ by $\Lambda$. Observe that the algebras $\CC[x_1^{\pm 1},\ldots x_{i-1}^{\pm 1}, x_i,y_i, x_{i+1}^{\pm 1}, \ldots, x_n^{\pm 1}]$ are Poisson subalgebras of the Poisson algebra $\CC[x_1^{\pm 1},\ldots x_n^{\pm 1}]$ for each $1\le i\le m$, as 