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Let $X$ be a Poisson variety. There is a concept "cluster algebra structure compatible with Poisson structure" introduced in the paper.

Suppose that we construct a maximal independent set of functions $P_1, \ldots, P_n$ in the coordinate ring of $X$ which is log-canonical and we construct an initial quiver $Q$ such that the matrix $B$ of the quiver $Q$ and the matrix $\Omega$ of the Poisson brackets of $P_1, \ldots, P_n$ satisfy $B \Omega = [D \ 0]$, where $D$ is a non-degenerate diagonal matrix. Is the following true: after any sequence of mutations, the cluster variables are still log-canonical? Thank you very much.

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Yes. You might find the extended presentation of this work in the book by Gekhtman, Shapiro and Vainshtein more helpful. It is based on their papers in this area but with more examples.

(I also feel professionally obliged to observe that this question is closely related to the topic of quantum cluster algebras - see Berenstein-Zelevinsky's paper of this name for details.)

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