Continuous quivers of type \(A\) (III) embeddings of cluster theories. (English) Zbl 1502.13051
Cluster algebras were introduced by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)]. K. Igusa and G. Todorov introduced a continuous version of a cluster category in [Algebr. Represent. Theory 18, No. 1, 65–101 (2015; Zbl 1329.18012)]. In part (I) [Rend. Circ. Mat. Palermo (2), (2022; doi:10.1007/s12215-021-00691-x)], the authors defined continuous quivers of type \(A\), generalizing quivers of type \(A\). Next, J. D. Rock in part (II) [“Continuous quivers of type A (II). The Auslander-Reiten space”, Preprint, arXiv:1910.04140] introduced the Auslander-Reiten \((AR)\) space, a continuous analog to the \(AR\) quiver.
This paper is part (III) of this series; the purpose is to form a continuous generalization of clusters and mutations. The authors classify which continuous quivers of type \(A\) are derived equivalent. They show that the original continuous cluster category of Igusa and Todorov [loc. cit.] is a localization of this new weak continuous cluster category. Next, cluster theories to be appropriate groupoids are defined and shown that cluster structures satisfy the conditions for cluster theories. The relationship between different cluster theories is described: some new and some obtained from cluster structures. The notion of continuous mutation which appears in cluster theories (but not in cluster structures) appears in the next paper by J. D. Rock, “Continuous quivers of type A (IV) continuous mutation and geometric models of E-clusters”, Preprint, arXiv:2004.11341].
This paper is part (III) of this series; the purpose is to form a continuous generalization of clusters and mutations. The authors classify which continuous quivers of type \(A\) are derived equivalent. They show that the original continuous cluster category of Igusa and Todorov [loc. cit.] is a localization of this new weak continuous cluster category. Next, cluster theories to be appropriate groupoids are defined and shown that cluster structures satisfy the conditions for cluster theories. The relationship between different cluster theories is described: some new and some obtained from cluster structures. The notion of continuous mutation which appears in cluster theories (but not in cluster structures) appears in the next paper by J. D. Rock, “Continuous quivers of type A (IV) continuous mutation and geometric models of E-clusters”, Preprint, arXiv:2004.11341].
Reviewer: Marek Golasiński (Olsztyn)
Keywords:
Auslander-Reiten \((AR)\) space; cluster algebra; continuous quiver; derived category; Krull-Schmidt categoryReferences:
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