Completions of discrete cluster categories of type A
Transactions of the London Mathematical Society, vol.8, no.1, pp.35-64, 2021 (ESCI, Scopus)
- Publication Type: Article / Article
- Volume: 8 Issue: 1
- Publication Date: 2021
- Doi Number: 10.1112/tlm3.12025
- Journal Name: Transactions of the London Mathematical Society
- Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus
- Page Numbers: pp.35-64
- Galatasaray University Affiliated: Yes
Abstract
We complete the discrete cluster categories of type (Formula presented.) 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.