2014/08/09 by ̛İlke Çanakçı, Sibylle Schroll, Canakci, Ilke +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1408.2074
openalex publication_date 2014/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the context of representation theory of finite dimensional algebras,\nstring algebras have been extensively studied and most aspects of their\nrepresentation theory are well-understood. One exception to this is the\nclassification of extensions between indecomposable modules. In this paper we\nexplicitly describe such extensions for a class of string algebras, namely\ngentle algebras associated to surface triangulations. These algebras arise as\nJacobian algebras of unpunctured surfaces. We relate the extension spaces of\nindecomposable modules to crossings of arcs in the surface and give explicit\nbases of the extension spaces for indecomposable modules in almost all cases.\nWe show that the dimensions of these extension spaces are given in terms of\ncrossing arcs in the surface.\n Our approach is new and consists of interpreting snake graphs as\nindecomposable modules. In order to show that our basis is a spanning set, we\nneed to work in the associated cluster category where we explicitly calculate\nthe middle terms of extensions and give bases of their extension spaces. We\nnote that not all extensions in the cluster category give rise to extensions\nfor the Jacobian algebra.\n