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Continuous Quivers of Type A (IV)

Continuous Mutation and Geometric Models of E-clusters

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Abstract

This if the final paper in the series Continuous Quivers of Type A. In this part, we generalize existing geometric models of type A cluster structures for the new E-clusters introduced in part (III). We also introduce an isomorphism of cluster theories and a weak equivalence of cluster theories. Examples of both are given. We use these geometric models and isomorphisms of cluster theories to begin classifying continuous type A cluster theories. We also introduce a continuous generalization of mutation. This encompasses mutation and (infinite) sequences of mutation. Then we link continuous mutation to our earlier geometric models. Finally, we introduce the space of mutations which generalizes the exchange graph of a cluster structure, and show that paths in this space are continuous mutations.

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Acknowledgements

The author was partly supported by Brandeis University during their graduate studies and partly supported by UGent BOF grant BOF/STA/201909/038 and FWO grants G023721N and G0F5921N. The author thanks Kiyoshi Igusa and Gordana Todorov for their guidance and support, Ralf Schiffler for organizing the Cluster Algebra Summer School in 2017 where this series was conceived, and Eric J. Hanson for helpful discussions. Finally, the author thanks the anonymous reviewer for careful reading and numberous helpful suggestions, especially regarding the clarity of figures.

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Authors and Affiliations

  1. Department of Mathematics, Brandeis University, MS 050, 415 South Street, Waltham, MA, 02453, USA

    Job Daisie Rock

  2. Department W16, Ghent University, Krijgslaan 281, S8, Ghent, 9000, East Flanders, Belgium

    Job Daisie Rock

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Correspondence to Job Daisie Rock.

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Rock, J.D. Continuous Quivers of Type A (IV). Algebr Represent Theor 26, 2255–2288 (2023). https://doi.org/10.1007/s10468-022-10175-w

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