Continuous Quivers of Type A (III) Embeddings of Cluster Theories

Abstract

We continue the work started in parts (I) and (II). In this part we classify which continuous type A quivers are derived equivalent and introduce the new continuous cluster category with E-clusters, which are a generalization of clusters. In the middle we provide a rigorous connection between the previous construction of the continuous cluster category and the new construction. We conclude with the introduction of a cluster theory, generalizing the notion of a cluster structure. Using this new notion, we demonstrate how one embeds known type A cluster theories into the new E-cluster theory in a way compatible with mutation. This is part (III) in a series of work that will conclude with a continuous generalization of mutation for cluster theories.

arXiv comment37 pages. This is the version before the accepted version, per journal requirement

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Cited by in corpus (2)

• Continuous Quivers of Type A (IV) Continuous Mutation and Geometric Models of $\mathbf{E}$-clusters
• A continuous associahedron of type A