Completions of discrete cluster categories of type A. (25th January 2021)

• Record Type: Journal Article
• Authors: Paquette, Charles; Yıldırım, Emine
• Is Part Of: Transactions of the London Mathematical Society. Volume 8:Number 1(2021)
• Page Start: 35
• Page End: 64
• Publication Date: 2021-01-25
• Subjects: 18G80 -- 16G20 (primary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/tlm3.12025
• Languages: English
• ISSNs: 2052-4986

Abstract

Abstract: We complete the discrete cluster categories of type A as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a Hom‐finite Krull–Schmidt triangulated category containing the discrete cluster category as a full subcategory. The objects and Hom‐spaces in this new category can be described geometrically, even though the category is not 2‐Calabi–Yau and Ext‐spaces are not always symmetric. We describe all cluster‐tilting subcategories. Given such a subcategory, we define a cluster character that takes values in a ring with infinitely many indeterminates. Our cluster character is new in that it takes into account infinite‐dimensional subrepresentations of infinite‐dimensional ones. We show that it satisfies the multiplication formula and also the exchange formula, provided that the objects being exchanged satisfy some local Calabi–Yau conditions.