Extending Upper Cluster Algebras

2017/07/14 by Jiarui Fei, Jerzy Weyman, Fei, Jiarui +1
Mathematics · #16G20 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13F60 #Representation Theory (math.RT) #Secondary 13A50

paper · pdf · doi:10.48550/arxiv.1707.04661

openalex publication_date 2017/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S be an upper cluster algebra, which is a subalgebra of R. Suppose that there is some cluster variable x_e such that R_x_e = S[x_e^{± 1}]. We try to understand under which conditions R is an upper cluster algebra, and how the quiver of R relates to that of S. Moreover, if the restriction of (Δ,W) to some subquiver is a cluster model, we give a sufficient condition for (Δ,W) itself being a cluster model. As an application, we show that the semi-invariant ring of any complete m-tuple flags is an upper cluster algebra whose quiver is explicitly given. Moreover, the quiver with its rigid potential is a polyhedral cluster model.

Related

• Cluster Algebras and Semi-invariant Rings II. Projections
• Crystal Structure of Upper Cluster Algebras
• Cluster algebras as Hall algebras of quiver representations
• Quivers with relations and cluster tilted algebras
• Generic bases for cluster algebras from the cluster category