2015
DOI: 10.1007/s00209-015-1524-6
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Cluster algebras of infinite rank as colimits

Abstract: We formalize the way in which one can think about cluster algebras of infinite rank by showing that every rooted cluster algebra of infinite rank can be written as a colimit of rooted cluster algebras of finite rank. Relying on the proof of the posivity conjecture for skew-symmetric cluster algebras (of finite rank) by Lee and Schiffler, it follows as a direct consequence that the positivity conjecture holds for cluster algebras of infinite rank. Furthermore, we give a sufficient and necessary condition for a … Show more

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Cited by 7 publications
(15 citation statements)
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“…In general, a rooted cluster morphism is not ideal, we give such an example (Example 4), which clarifies a doubt in [1] (compare Problem 2.12 in [1]). Note that Gratz also obtains this result in [20]. Then we prove in Theorem 2.9 that three kinds of important rooted cluster morphisms are ideal.…”
Section: Introductionmentioning
confidence: 53%
“…In general, a rooted cluster morphism is not ideal, we give such an example (Example 4), which clarifies a doubt in [1] (compare Problem 2.12 in [1]). Note that Gratz also obtains this result in [20]. Then we prove in Theorem 2.9 that three kinds of important rooted cluster morphisms are ideal.…”
Section: Introductionmentioning
confidence: 53%
“…The idea to consider triangulations and mutations of infinitely marked surfaces is not new and has been executed in the context of cluster categories for example in [9] and [11] and in the context of cluster algebras in [6] and [7]. By introducing infinitely many marked points, interesting phenomena occur which do not appear in the finite setting.…”
Section: Introductionmentioning
confidence: 99%
“…The rank of the rooted cluster algebra (C(Σ), Σ) is defined as the cardinality of var. 2 We will use C for the cluster algebra, rather than A as in [1] and [11], and reserve A for quantum affine spaces. Remark 2.13.…”
Section: Rooted Cluster Algebrasmentioning
confidence: 99%
“…Remark 2.13. Here we follow [11] in our definition of rank. Traditionally, the rank of a cluster algebra C(Σ) is defined as the cardinality of the set of exchangeable variables of Σ, while we define it as the cardinality of the cluster of Σ.…”
Section: Rooted Cluster Algebrasmentioning
confidence: 99%