# Interpretation functors and infinite-dimensional representations of finite-dimensional algebras

Lead Research Organisation:
University of Manchester

Department Name: Mathematics

### Abstract

In the 1980s unexpected applications of the model theory of modules to the representation theory of finite-dimensional algebras were discovered and since then there has been further, sometimes deep, interaction between these areas. Model theory uses ideas and results from mathematical logic to investigate general questions about mathematical structure and also to obtain new results in other parts of mathematics. It provides a particular perspective which often gives new insights into other parts of mathematics. Almost always model theory makes heavy use of the Compactness Theorem of mathematical logic and, for that, one needs to be working in a context within which there is room to make infinitary constructions. In the specific context of the representation theory of finite-dimensional algebras, where interest is typically focussed on finite-dimensional representations, that means that we have to extend our interest to at least some of the infinite-dimensional representations, even if our eventual applications are back in the context of the finite-dimensional ones. This particular project will deep the interaction of model theory and representation theory.

The question underlying the project is "How complex is a particular collection of representations?"; various ways of answering this question have been investigated already and the principal aim is to show that the most standard algebraic answer - which is given in terms of certain embeddings of one collection in another - fits well with the model-theoretic one. The latter is in terms of the notion of interpretation, which is essentially a translation from one language (associated to a collection of representations) to another. That has already been shown to be equivalent to a particularly nice kind of embedding but it is not known how to close the gap between that and kind of embedding which is the standard algebraic answer to the above question. Closing that gap is one of the aims of the project. Going beyond that, the project has as an aim a substantial refinement of the existing rather broad algebraic classification of complexity classes into tame and wild (with further refinements of tame).

The project will combine very general methods, some being inspired by algebraic geometry and abstract category theory, with very specific investigations of the representations of particular algebras where entirely explicit descriptions are the aim. It will draw on two well-developed subjects; the model theory of modules and the representation theory of finite-dimensional algebras, and will use techniques from homological algebra and additive functor category theory. In view of that breadth of necessary input as well as on account of the number and nature of the aims of the project, two PDRAs, working together with the PI, all sharing their expertise, will form the research team.

The question underlying the project is "How complex is a particular collection of representations?"; various ways of answering this question have been investigated already and the principal aim is to show that the most standard algebraic answer - which is given in terms of certain embeddings of one collection in another - fits well with the model-theoretic one. The latter is in terms of the notion of interpretation, which is essentially a translation from one language (associated to a collection of representations) to another. That has already been shown to be equivalent to a particularly nice kind of embedding but it is not known how to close the gap between that and kind of embedding which is the standard algebraic answer to the above question. Closing that gap is one of the aims of the project. Going beyond that, the project has as an aim a substantial refinement of the existing rather broad algebraic classification of complexity classes into tame and wild (with further refinements of tame).

The project will combine very general methods, some being inspired by algebraic geometry and abstract category theory, with very specific investigations of the representations of particular algebras where entirely explicit descriptions are the aim. It will draw on two well-developed subjects; the model theory of modules and the representation theory of finite-dimensional algebras, and will use techniques from homological algebra and additive functor category theory. In view of that breadth of necessary input as well as on account of the number and nature of the aims of the project, two PDRAs, working together with the PI, all sharing their expertise, will form the research team.

### Planned Impact

The most `concrete' aims are precise descriptions of the indecomposable pure-injective modules over non-domestic string algebras and over canonical tubular algebras. The description of the former has been an explicit open problem for about 20 years; the lack of knowledge about the latter has been recognised for even longer but has seemed more possibly within reach in the light of advances in the past decade.

The question of undecidability of the theory of modules over a wild algebra mixes algebra and model theory. It has been open for over thirty years and serves as a well-known test case in the context of answers to the question "How complex is a particular collection of representations?".

As regards model-theoretic impact, ``the model theory of modules" has been developing a new style of model theory, somewhere between classical and categorical model theory, in its enthusiastic adoption of category-theoretic techniques. This has been natural in the additive context because of the pervasiveness of homological techniques but also because of the success of the Auslander-Reiten philosophy of using functor categories to study representations. It is unlikely that in other model-theoretic contexts the category-theoretic approach will be quite so natural and successful but this approach does hold general lessons: an example is to take full advantage of the category-theoretic structure of the collection of imaginary sorts and definable functions. In the context of the current proposal, the aspects which relate to interpretability and to wider notions of inter-interpretability make sense, and raise new questions, within the general model-theoretic context.

It is this last aspect which might also have impact in the, overlapping, accessible category and topos theory communities since the picture in the algebraic context parallels one in the Set-based context. The algebraic picture is becoming increasingly refined and this project will extend that. The structure being revealed in the algebraic context is not paralleled in the Set-based one (for instance refinements in the topos context tend to be inspired by logic or geometry) but it may well be that there can be transfer at some level of detail from the algebraic to the Set-context.

To facilitate the project and its impact two workshops will be run: one aimed at algebraists and concentrating on how model-theoretic and functor-category-theoretic techniques are used in the infinite-dimensional representation theory of finite-dimensional algebras; the other will be aimed at model-theorists and will concentrate on the new types of question raised by this work, especially at questions involving (inter-)interpretability.

The question of undecidability of the theory of modules over a wild algebra mixes algebra and model theory. It has been open for over thirty years and serves as a well-known test case in the context of answers to the question "How complex is a particular collection of representations?".

As regards model-theoretic impact, ``the model theory of modules" has been developing a new style of model theory, somewhere between classical and categorical model theory, in its enthusiastic adoption of category-theoretic techniques. This has been natural in the additive context because of the pervasiveness of homological techniques but also because of the success of the Auslander-Reiten philosophy of using functor categories to study representations. It is unlikely that in other model-theoretic contexts the category-theoretic approach will be quite so natural and successful but this approach does hold general lessons: an example is to take full advantage of the category-theoretic structure of the collection of imaginary sorts and definable functions. In the context of the current proposal, the aspects which relate to interpretability and to wider notions of inter-interpretability make sense, and raise new questions, within the general model-theoretic context.

It is this last aspect which might also have impact in the, overlapping, accessible category and topos theory communities since the picture in the algebraic context parallels one in the Set-based context. The algebraic picture is becoming increasingly refined and this project will extend that. The structure being revealed in the algebraic context is not paralleled in the Set-based one (for instance refinements in the topos context tend to be inspired by logic or geometry) but it may well be that there can be transfer at some level of detail from the algebraic to the Set-context.

To facilitate the project and its impact two workshops will be run: one aimed at algebraists and concentrating on how model-theoretic and functor-category-theoretic techniques are used in the infinite-dimensional representation theory of finite-dimensional algebras; the other will be aimed at model-theorists and will concentrate on the new types of question raised by this work, especially at questions involving (inter-)interpretability.

### Publications

Arnesen K
(2017)

*The Ziegler spectrum for derived-discrete algebras*in Advances in Mathematics
Arnesen K
(2016)

*Morphisms between indecomposable complexes in the bounded derived category of a gentle algebra*in Journal of Algebra
Arnesen K K
(2017)

*The Ziegler spectrum for derived-discrete algebras*in Advances in Mathematics
Barbieri-Viale L
(2018)

*Definable categories and T-motives*in Rendiconti del Seminario Matematico della UniversitĂ di Padova
Broomhead N
(2016)

*Discrete derived categories I: homomorphisms, autoequivalences and t-structures*in Mathematische Zeitschrift
Broomhead N
(2018)

*Discrete triangulated categories DISCRETE TRIANGULATED CATEGORIES*in Bulletin of the London Mathematical Society
Broomhead N
(2016)

*Discrete derived categories II: the silting pairs CW complex and the stability manifold*in Journal of the London Mathematical Society
Coelho SimĂµes R
(2016)

*Torsion pairs in a triangulated category generated by a spherical object*in Journal of Algebra
Gregory L
(2016)

*Representation embeddings, interpretation functors and controlled wild algebras*in Journal of the London Mathematical Society
Prest, M

*Modules as exact functors*Description | It was shown that controlled-wild algebras have undecidable theory of modules. Since all known wild algebras are controlled-wild this is potentially a (positive) resolution of the conjecture that algebras of wild representation type have algorithmically undecidable theory of modules. Additionally, a number of results were obtained that algebraically-defined complexity classes do not interpret algebras of higher complexity. The first examples of algebras of tame, non-domestic representation type and with decidable theory of modules were established. This shows that the boundary between decidable and undecidable theory of modules lies beyond the domestic/tame boundary. Ringel's conjecture as to the infinite-dimensional points of the Ziegler spectrum of domestic string algebras was proved. Using the resulting classification of infinite-dimensional pure-injective the conjecture that domestic string algebras have finite Krull-Gabriel dimension was verified. However the Ziegler spectrum over non-domestic string algebras is still mysterious and its description remains a major goal of the broad research programme. The relevance of questions about triangulated categories rapidly became apparent during the project. The description of the Ziegler spectra of the homotopy categories of derived-discrete algebras was obtained and shown to include all indecomposable objects. This was achieved following the description of all morphisms between indecomposables in such categories. In a more geometric direction, the space of Bridgeland stability conditions of derived-discrete categories was shown to be contractible. Surprising connections to arithmetic/algebraic geometry, specifically the category of Nori motives, fit into our framework and are now the subject of further research, along with connections to other areas, as well as those within representation theory. |

Exploitation Route | The findings fit into an extensive research landscape, specifically the areas of representations of algebras and of model theory. There, the results will be used and extended, will suggest new approaches to existing questions and lead to new questions and lines of research. |

Sectors | Digital/Communication/Information Technologies (including Software),Other |

URL | http://www.maths.manchester.ac.uk/~mprest/publications.html |

Description | various collaborations |

Organisation | University of Verona |

Country | Italy |

Sector | Academic/University |

PI Contribution | There were many collaborations between team members and researchers from other universities (just one is listed above), aided by visits of team members to other institutions and conferences and visits by researchers from other universities to the team at Manchester. Typically these resulted in joint authorship of papers and/or continuing collaboration, as well as currently developing grant applications. It would take quite some time to list and detail these; if it would be useful to do so then I will happily do that. |

Collaborator Contribution | see above |

Impact | see above These were all within pure mathematics, spanning representation theory, model theory, category theory and algebraic geometry. |

Start Year | 2014 |