From triangulated categories to cluster algebras

Abstract

The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of the cluster category. This realization provides a natural basis for A. We prove new results and formulate conjectures on `good basis' properties, positivity, denominator theorems and toric degenerations.

31 pages, typos corrected

References in corpus (6)

• On triangulated orbit categories
• Tilting theory and cluster combinatorics
• Cluster mutation via quiver representations
• Cluster-tilted algebras are Gorenstein and stably Calabi-Yau
• Quivers with relations and cluster tilted algebras
• Clusters and seeds in acyclic cluster algebras

Cited by in corpus (6)

• Quiver Grassmannians associated with string modules
• Denominators of cluster variables
• Quiver grassmannians, quiver varieties and the preprojective algebra
• Applications of BGP-reflection functors: isomorphisms of cluster algebras
• Minimal mutation-infinite quivers
• Cluster algebras as Hall algebras of quiver representations