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Poisson geometry is the study of varieties endowed with a Poisson structure, which is a certain kind of 2-tensor. This is closely related to symplectic geometry.
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Cluster algebra structure compatible with Poisson brackets
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.
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