$\{x_i,y_i\}_\Lambda=\{x_i,x_i^{-1}P_i\}_\Lambda\in \CC[x_1,\ldots, x_n]$. If $A$ is a Poisson algebra and $\{B_i\subset A:i\in I\}$ is a family of Poisson subalgebras, then $\bigcap_{i\in I} B_i$ is a Poisson algebra, as well. The assertion follows.

Compatible Pairs and Their Mutation {#se:Compatible Pairs and Mut}
-----------------------------------

Section \[se:Compatible Pairs and Mut\] is dedicated to compatible pairs and their mutation. Compatible pairs yield important examples of Poisson brackets which are compatible with a given cluster algebra structure, and as we shall see below, they are also integral in defining quantum cluster algebras. Note that our definition is slightly different from the original one in [@bz-qclust]. Let, as above, $m\le n$. Consider a pair consisting of a skew-symmetrizable $m\times n$-integer matrix $B$ with rows labeled by the interval $[1,m]=\{1,\ldots, m\}$ and columns labeled by $[1,n]$ together with a skew-symmetrizable $n\times n$-integer matrix $\Lambda$ with rows and columns labeled by $[1,n]$.

\[def:compa pair\] Let $B$ and $\Lambda$ be as above. We say that the pair $(B,\Lambda)$ is compatible if the coefficients $d_{ij}$ of the $m\times n$-matrix $D=B\cdot \Lambda$ satisfy $d_{ij}=d_i\delta_{ij}$ for some positive integers $d_i$ ($i\in [1,m]$).

This means that $D=B\cdot \Lambda$ is a $m\times n$ matrix where the only non-zero entries are positive integers on the 