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# Representation theory via subcategories

**Alternative title:** Representasjonsteori via underkategorier

#### Awarded: NOK 7.0 mill.

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Project Manager:

Project Number:

250056

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Project Period:

2016 - 2021

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The principal challenge in representation theory is the following:
Given a ring, find all modules over this ring.
Many classification problems for abelian groups or vector spaces with some extra structure can be translated to this language. For instance, finding normal forms for square matrices can be regarded as finding all finite dimensional modules over a polynomial ring. In this case, if we consider complex scalars, the answer is given by the Jordan canonical form.
Unfortunately, it turns out that many similar problems involving several matrices cannot be solved is such a convincing way: In fact, it can be proven that polynomial rings in two (or more) variables are "wild", which is being understood to mean that it is impossible to obtain a complete classification of (finite dimensional) modules.
The idea to be pursued within this project is to study certain nice subcategories of a module category, which on the one hand are structural enough to give a picture of how the entire category looks, but on the other hand are simple enough to understand.
Within this project the most progress has been made in a representation theoretic setup closely related to geometry: In joint work with Herschend, Iyama, and Minamoto, the project leader has introduced the notion of "Geigle Lenzing projective spaces". (A new and vastly extended version of our preprint has appeared on the arXiv this fall.) For these spaces (and certain associated algebras), we explain how the categories of coherent sheaves contain nice, structural subcategories. For the nicest cases, the reader may imagine these to consist of all line bundles. Studying these subcategories also leads to for instance finding derived equivalences, and finding non-commutative crepant resolutions.
A more categorical approach to the same subject has been taken in Lerner-Oppermann, "A recollement approach to Geigle-Lenzing weighted projective varieties".
A different type of "nice subcategories" is studied in Oppermann-Psaroudakis-Stai "Change of rings and singularity categories". Here we observe that some objects, which we call "0-cocompact" often control triangulated categories in some way. In the follow-up paper "Partial Serre duality and cocompact objects" we investigated this phenomenon more systematically.
Finally, the most recent type of subcategories studied came from a practical application: Toplogical data analysis. The idea of that field is to turn a given set of data into a representation, and by analysing this representation infer the structure of a topological space underlying the data. When studying connected components it turns out that one is studying representations in which certain structure maps are epimorphisms. In Bauer-Botnan-Oppermann-Steen "Cotorsion torsion triples and the representation theory of filtered hierarchical clustering", we study the subcategory given by such representations. Unfortunately, it turns out that the subcategory can only be described fully explicitly in a few more cases that the base category, while it in general will also be "wild". There are further natural subcategories in this context (for instance interval decomposable modules), and we are continuing working on understanding their structure after the end of the project.

De to postdocene i prosjektet har utviklet seg fra phd-studenter til selvstendige forskere i løpet av prosjektet. En av dem jobber på et eget forskningsprosjekt nå.
Samarbeid med det internasjonale miljøet i høyere representasjonsteori, spesielt Osamu Iyama, ble forsterket i løpet av prosjektet.
Ny tverfaglig samarbeid mellom representasjonsteori og topologisk data-analyse har oppstått under prosjektet. Vi håper og forventer at det samarbeidet vil føre til mange nye resultater i framtida.

In many situations in representation theory it is not possible to classify all (indecomposable) representations. Even when this is possible, the Auslander-Reiten quiver (usually the strongest tool available for getting an intuitive picture of a module category) will typically only carry limited information in infinite situations.
The idea to be pursued in the project here is to study certain nice subcategories of a module category, which on the one hand are structural enough to give a picture of how the entire category looks, but on the other hand are simple enough to understand. Motivating examples include the cluster tilting module for a 2-representation finite algebra, or (for people with a more geometric mind set) the subcategory of line bundles inside all vector bundles on projective d-space.
One (semi-classical) example that is both inspiration for and integral part of the project is the concept of cluster tilting: For 2-Calabi-Yau triangulated categories this concept gives a way to understand "most" of the category by knowing only (the endomorphism ring of) a single object.
Starting from classical cluster tilting theory, the project aims to explore various facets of "higher dimensional representation theory": This concept, first developed by and with Iyama has seen rising interest the last years. The idea is that one studies categories which, instead of containing short exact sequences (or triangles) have a structure determined by longer exact sequences (or (d+2)-angles). Often interesting examples arise as cluster tilting subcategories of usual abelian (triangulated) categories, and there are various instances where this higher representation theory of a subcategory draws a clearer picture of what is happening than the study of the entire category.

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3.1BILL. NOKtotal funding in the programme period586PROJECTShave received funding in the programme period3SOURCEShave financed the programme