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. Our framework is the category of rooted cluster algebras as introduced by Assem, Dupont and Schiffler. 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 true for cluster algebras of infinite rank. Furthermore, we give a sufficient and necessary condition for a ring homomorphism between cluster algebras to give rise to a rooted cluster morphism without specializations. Assem, Dupont and Schiffler proposed the problem of a classification of ideal rooted cluster morphisms. We provide a partial solution by showing that every rooted cluster morphism without specializations is ideal, but in general rooted cluster morphisms are not ideal.
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Acknowledgments
The author thanks her supervisor Thorsten Holm as well as David Ploog and Adam-Christiaan van Roosmalen for helpful discussions. The author would also like to thank Wen Chang and Bin Zhu for pointing out some relations between our work on ideal rooted cluster morphisms and their paper [3].
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This work has been carried out in the framework of the research priority programme SPP 1388 Darstellungstheorie of the Deutsche Forschungsgemeinschaft (DFG). The author gratefully acknowledges financial support through the Grant HO 1880/5-1.
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Gratz, S. Cluster algebras of infinite rank as colimits. Math. Z. 281, 1137–1169 (2015). https://doi.org/10.1007/s00209-015-1524-6
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DOI: https://doi.org/10.1007/s00209-015-1524-6