this case the corresponding homeomorphism is well known and easy to construct. The other extreme case are cluster algebras without coefficients and here the class that is usually easiest to study are the acyclic cluster algebras. For example, it is known that such a cluster algebra is always Noetherian and the coordinate ring of an affine variety (see [@BFZ] and [@bz-qclust] for the classical and quantum versions). It is our main objective to give evidence for Conjecture \[conj: homeo cluster\] by proving it in this very specific case. It is an immediate consequence of the following two main results.

\[th:Classical-intro\] Let $\AA$ be a cluster algebra with initial seed $({\bf x}, B)$ defined by an acyclic quiver where $B$ is invertible satisfying , and suppose that it is the coordinate ring of an affine variety $X$ and that $(B,\Lambda)$ is a compatible pairs. Then $X$ has the structure of a symplectic manifold, whose symplectic form is the corresponding Poisson bivector.

\[th:quantum-intro\] Let $\AA_q$ be a quantum cluster algebra with quantum seed $( {\bf x}, B,\Lambda)$ satisfying the assumptions of Theorem \[th:Classical-intro\]. Then $\{0\}$ is the only proper two sided prime ideal in $\AA_q$.

Our approach, is similar to that of 