2023/07/25 by Emre Sen, Sen, Emre · 1 citation
Mathematics · #16E05 #16E10 #16E35 #16G20 #16G70 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2307.13262
openalex publication_date 2023/07/25 · openalex created_date 2023/07/27 · openalex updated_date 2026/07/28
In the derived category of mod-KQ for Dynkin quiver Q, we construct a full subcategory in a canonical way, so that its endomorphism algebra is a higher Auslander algebra of global dimension 3k+2 for any k≥ 1. Furthermore, we extend this construction for higher analogues of representation finite and hereditary algebras. Specifically, if M is an n-cluster tilting object in the bounded derived category of n-representation finite and n-hereditary algebra, then we construct a full subcategory in a canonical way, so that its endomorphism algebra is a higher Auslander algebra of global dimension (n+2)k+n+1 for any k≥ 1. As an application, we revisit the higher Auslander correspondence. Firstly, we describe the corresponding module categories that have higher cluster-tilting objects, and then we discuss their relationship with certain full subcategories of the derived category. Consequently, we obtain a vast list of n-representation finite n-hereditary algebras whose n-cluster tilting objects are always minimal generator-cogenerator. Moreover, resulting algebras can be realized as endomorphism algebras of certain full subcategories of (higher) cluster categories.