Applications of BGP-reflection functors: isomorphisms of cluster algebras

Abstract

Given a symmetrizable generalized Cartan matrix $A$, for any index $k$, one can define an automorphism associated with $A,$ of the field $\mathbf{Q}(u_1,>...,u_n)$ of rational functions of $n$ independent indeterminates $u_1,...,u_n.$ It is an isomorphism between two cluster algebras associated to the matrix $A$ (see section 4 for precise meaning). When $A$ is of finite type, these isomorphisms behave nicely, they are compatible with the BGP-reflection functors of cluster categories defined in [Z1, Z2] if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined in [FZ2, FZ3]. Using the construction of preprojective or preinjective modules of hereditary algebras by Dlab-Ringel [DR] and the Coxeter automorphisms (i.e., a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.

revised version

References in corpus (8)

• Cluster algebras II: Finite type classification
• On triangulated orbit categories
• Quivers with relations arising from clusters (A_n case)
• From triangulated categories to cluster algebras
• Cluster-tilted algebras
• BGP-reflection functors and cluster combinatorics
• Preprojective cluster variables of acyclic cluster algebras
• Equivalences between cluster categories

Cited by in corpus (2)

• Cluster automorphism groups of cluster algebras of finite type
• Cluster automorphism groups of cluster algebras with coefficients