2024/02/19 by Zengqiang Lin, Lin, Zengqiang, Xiuping Su +1
Mathematics · Physics and Astronomy · #Functional Equations Stability Results #Advanced Topics in Algebra #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2402.11946
Associated to a symmetrisable Cartan matrix C, Geiss-Lerclerc-Schröer constructed and studied a class of Iwanaga-Gorenstein algebras H. They proved a generalised version of Gabriel's Theorem, that is, the rank vectors of τ-locally free H-modules are the positive roots of type C when C is of finite type, and conjectured that this is true for any C. In this paper, we look into this conjecture when C is of affine type. We construct explicitly stable tubes, some of which have rigid mouth modules, while others not. We deduce that any positive root of type C is the rank vector of some τ-locally free H-module. However, the converse is not true in general. Our construction shows that there are τ-locally free H-modules whose rank vectors are not roots, when C is of type \widetilde\mathbbBn, \widetilde\mathbbCDn, \widetilde\mathbbF41 and \widetilde\mathbbG21, and so the conjecture fails in these four types.