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 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 sec. 4 for the 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 a previous work 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 by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by Dlab-Ringel 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.