Cluster algebras via cluster categories with infinite-dimensional morphism spaces

TL;DR

This paper connects cluster categories with infinite-dimensional morphism spaces to cluster algebras, providing new proofs of key conjectures, a bijection with decorated representations, and a categorical interpretation of invariants.

Contribution

It introduces a new categorical approach to cluster algebras, proving conjectures and establishing a bijection with decorated representations, extending the theory to infinite-dimensional settings.

Findings

Proved several conjectures of Fomin and Zelevinsky for skew-symmetric cluster algebras.

Constructed an explicit bijection between cluster category objects and decorated representations.

Provided a categorical interpretation of the E-invariant and g-vectors.

Abstract

We apply our previous work on cluster characters for Hom-infinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky's Cluster algebras IV for skew-symmetric cluster algebras. We also construct an explicit bijection sending certain objects of the cluster category to the decorated representations of Derksen, Weyman and Zelevinsky, and show that it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of the E-invariant and show that an arbitrary decorated representation with vanishing E-invariant is characterized by its g-vector. Finally, we obtain a substitution formula for cluster characters of not necessarily rigid objects.