[@ZW; @tpc], however all the proofs are self-contained and much easier, as our set-up is less general. The main idea is to study the intersection of ideals with the polynomial ring generated by a given cluster–in this case the acyclic seed. We are able to derive rather strong conditions that Poisson prime ideals–resp. two-sided prime ideals in the quantum case– must satisfy and are able to show that no non-trivial ideals satisfying them exist. A straightforward argument, then allows us to conclude that the variety $X$ is a symplectic manifold. We should also remark that we do not know whether any acyclic cluster algebras exist that do not satisfy the assumptions made in Theorem \[th:Classical-intro\].

The paper is organized as follows. We first briefly recall the definitions of cluster algebras and compatible Poisson structures, compatible pairs and quantum cluster algebras in Section \[se:Cluster Algebras\]. Thereafter, we continue with some technical key propositions (Section \[se:key propositions\]) and discuss in Section \[se:Poisson and symp geom\] the symplectic geometry of acyclic cluster algebras. The proof of Conjecture \[conj: homeo cluster\] in our specific case is completed in Section \[se:ideals in qca\] by proving Theorem \[th:quantum-intro\].

Cluster Algebras {#se:Cluster Algebras}
================

Cluster algebras {#se:Cluster 