Freezing rooted cluster morphisms and pro-cluster algebras

Wierzbicki, Damian (2022) Freezing rooted cluster morphisms and pro-cluster algebras. PhD thesis, University of Glasgow.

Abstract

We provide a new tool for studying cluster algebras by introducing a new category fClus of rooted cluster algebras. We characterize isomorphisms in our new category and show that it is neither complete nor cocomplete. We give a recipe for constructing morphisms in fClus with an interesting geometric interpretation and study the corresponding inverse systems.

We define and study a new family of algebras, called pro-cluster algebras, with clusterlike combinatorics. The pro-cluster algebras are generated inside inverse limits of inverse systems in the category fClus. Initially, the generators of a pro-cluster algebra are grouped into certain subsets, called pro-clusters, of an inverse limit. In this new setting pro-clusters take the role of clusters and we construct pro-cluster algebras which are modelled by the combinatorics of infinitely marked surfaces and prove that all triangulations of those surfaces arise as pro-clusters.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Supported by funding from the Engineering and Physical Sciences Research Council (EPSRC).
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Engineering
• Supervisor's Name: Gratz, Professor Sira and Korff, Professor Christian
• Date of Award: 2022
• Depositing User: Theses Team
• Unique ID: glathesis:2022-83342
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 09 Jan 2023 16:04
• Last Modified: 10 Jan 2023 09:04
• Thesis DOI: 10.5525/gla.thesis.83342
• URI: https://theses.gla.ac.uk/id/eprint/83342