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From triangulated categories to cluster algebras

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The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra \(\mathcal{A}\) of finite type can be realized as a Hall algebra, called exceptional Hall algebra, of the cluster category. This realization provides a natural basis for \(\mathcal{A}\). We prove new results and formulate conjectures on ‘good basis’ properties, positivity, denominator theorems and toric degenerations.

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Correspondence to Philippe Caldero.

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Caldero, P., Keller, B. From triangulated categories to cluster algebras. Invent. math. 172, 169–211 (2008). https://doi.org/10.1007/s00222-008-0111-4

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