$({\bf x}, B)$ where $B$ is a $m\times n$-matrix. If $(B,\Lambda)$ is a compatible pair, then $\Lambda$ defines a compatible Poisson bracket on $\mathfrak{F}$ and on $\UU(\AA)$.

\[ex:acyclic\] If $m=n$ (i.e. there are no coefficients/frozen variables) and $B$ has full rank, then $(B, \mu B^{-1})$ is a compatible pair for all $\mu\in \ZZ_{> 0}$ such that $\mu B^{-1}$ is an integer matrix. It follows from [@GSV Theorem 1.4] that in this case all compatible Poisson brackets arise in this way.

Recall that double Bruhat cells in complex semisimple connected and simply connected algebraic groups have a natural structure of an upper cluster algebra (see [@BFZ]). Berenstein and Zelevinsky showed that the standard Poisson structure is given by compatible pairs relative to this upper cluster algebra structure (see [@bz-qclust Section 8]).

Quantum Cluster Algebras {#se: q-cluster algebra spec}
------------------------

In this section we recall the the definition of a quantum cluster algebra, introduced by Berenstein and Zelevinsky in [@bz-qclust]. We define, for each skew-symmetric $n\times n$-integer matrix $\Lambda$, the skew-polynomial ring $\CC_\Lambda^t[x_1,\ldots, x_n]$ to be the $\CC[t^{\pm
1}]$-algebra generated by $x_1,\ldots, x_n$ subject to the relations $$x_ix_j=t^{\lambda_{ij}} x_jx_i\ .$$

Analogously, the quantum torus $H_\Lambda^t=\CC_\Lambda^t[x_1^{\pm 1},\ldots, x_n^{\pm 1} ]$ is defined as 