Completions of discrete cluster categories of type $\mathbb{A}$

TL;DR

This paper extends discrete cluster categories of type A by embedding them into larger triangulated categories, describing their structure geometrically, and introducing a novel cluster character that accounts for infinite-dimensional representations.

Contribution

It constructs a new triangulated category containing the discrete cluster category and develops a novel cluster character for infinite-dimensional sub-representations.

Findings

Embedded discrete cluster categories into larger triangulated categories.

Described objects and Hom-spaces geometrically.

Introduced a new cluster character for infinite-dimensional representations.

Abstract

We complete the discrete cluster categories of type $\mathbb{A}$ 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 sub-representations 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.