Abstract Cluster Structures
Jan E. Grabowski, Sira Gratz
TL;DR
This work introduces a comprehensive categorical framework for cluster combinatorics by defining abstract cluster structures on a directed exchange graph via two free Abelian presheaves and a nondegenerate pairing, then builds a category ecosystem (ACS,AQCS) of such structures and analyzes their universal properties. It links these abstract objects to concrete realizations—cluster algebras, cluster varieties, cluster categories, and surface models—through tropicalization, exponentiation, and Poisson/quantum data, while preserving mutation through morphisms that intertwine representations. The second part develops linear representations with quantum tori, toric frames, and quantum cluster algebras, and shows how abstract structures reproduce or recover these familiar algebras via tropicalization and retraction data; it further connects to geometry via cluster varieties and Poisson structures. The framework permits morphisms across representations, enabling principled comparisons (e.g., Grassmannian and hexagon models) and suggesting pathways to classify and translate cluster-theoretic phenomena across algebraic, geometric, and categorical contexts with a unified, mutation-respecting language.
Abstract
We describe a framework for encoding cluster combinatorics using categorical methods. We give a definition of an abstract cluster structure, which captures the essence of cluster mutation at a tropical level and show that cluster algebras, cluster varieties, cluster categories and surface models all have associated abstract cluster structures. For the first two classes, we also show that they can be constructed from abstract cluster structures. By defining a suitable notion of morphism of abstract cluster structures, we introduce a category of these and show that it has several desirable properties, such as initial and terminal objects and finite products and coproducts. We also prove that rooted cluster morphisms of cluster algebras give rise to morphisms of the associated abstract cluster structures, so that our framework includes a version of the extant category of cluster algebras. We can do more, however, because we can relate different types of representation of abstract cluster structures (cluster algebra, varieties, categories) directly via morphisms of their associated abstract cluster structures, even though no direct map from e.g. a cluster category to the associated cluster algebra is possible. In fact, we do much of the above in the setting of abstract quantum cluster structures, with some analysis of the difference between the category of these and that of the unquantized version. In order to show the relationship between abstract quantum cluster structures and quantum cluster algebras, we reformulate the usual construction of the latter in a way that is more amenable to our purposes and which we expect will be of independent interest and use.