2-representation theory and categorification
Lead Research Organisation:
University of East Anglia
Department Name: Mathematics
Abstract
While in mathematics it is very often helpful to break complicated problems down into less complicated ones by simplifying and forgetting data, the converse has often proved useful. The term categorification refers to the process of finding more complicated structures which, upon forgetting some information, reproduce the original problem that one wants to study. The more complicated structures often allow us to deduce useful information that was previously inaccessible. For example, an integer number solving a certain equation might a priori be anything, but if we then discover that this certain number, in fact, describes the number of elements in a set (e.g. cats in a household), it cannot be negative.
Categorification in representation theory is usually formulated in terms of certain structures called 2-categories encoding generalised symmetries of other categories, or in other words, 2-representations of 2-categories. To develop the theory of 2-representations of 2-categories (with certain nice properties abstracted from interesting examples) and to apply them to some of the original problems that inspired their definition is the aim of this proposal.
Categorification in representation theory is usually formulated in terms of certain structures called 2-categories encoding generalised symmetries of other categories, or in other words, 2-representations of 2-categories. To develop the theory of 2-representations of 2-categories (with certain nice properties abstracted from interesting examples) and to apply them to some of the original problems that inspired their definition is the aim of this proposal.
Planned Impact
The impact of this project will first and foremost be of academic nature, through knowledge transfer and training the next generation of researchers.
Scientific advances in 2-representation theory will have impact on surrounding areas like geometry, topology and number theory. Quick dissemination through uploading articles on the arXiv upon completion before publishing them in leading peer-reviewed journals, and presenting results in seminars and at conferences will guarantee fast transfer of the newly acquired knowledge. The summer school envisioned as part of this project will provide training for young mathematicians in the techniques required to successfully apply those advances. Similarly, the training of the PDRA will have significant impact on his or her career development.
Scientific advances in 2-representation theory will have impact on surrounding areas like geometry, topology and number theory. Quick dissemination through uploading articles on the arXiv upon completion before publishing them in leading peer-reviewed journals, and presenting results in seminars and at conferences will guarantee fast transfer of the newly acquired knowledge. The summer school envisioned as part of this project will provide training for young mathematicians in the techniques required to successfully apply those advances. Similarly, the training of the PDRA will have significant impact on his or her career development.
Organisations
People |
ORCID iD |
Vanessa Miemietz (Principal Investigator) |
Publications
Mackaay M
(2024)
Evaluation birepresentations of affine type A Soergel bimodules
in Advances in Mathematics
Mackaay M
(2023)
Kostant's problem for fully commutative permutations
in Revista Matemática Iberoamericana
Hristova K
(2023)
Basic Hopf algebras and symmetric bimodules
in Journal of Pure and Applied Algebra
Mackaay M
(2023)
Simple transitive 2-representations of Soergel bimodules for finite Coxeter types
in Proceedings of the London Mathematical Society
Hristova K
(2022)
Basic Hopf algebras and symmetric bimodules
Laugwitz R
(2022)
Pretriangulated 2-representations via dg algebra 1-morphisms
Mackaay M
(2022)
Kostant's problem for fully commutative permutations
Mackaay M
(2022)
Evaluation birepresentations of affine type A Soergel bimodules
Mackaay M
(2021)
Finitary birepresentations of finitary bicategories
in Forum Mathematicum
Mazorchuk V
(2020)
2-categories of symmetric bimodules and their 2-representations
in Pacific Journal of Mathematics
Mazorchuk V
(2019)
2-categories of symmetric bimodules and their 2-representations
Description | In joint work with Marco Mackaay, Volodymyr Mazorchuk, Daniel Tubbenhauer, Xiaoting Zhang, we recently proved the conjecture we formulated last year on the classification of simple transitive 2-representations for Soergel bimodules in finite Coxeter type in characteristic zero. This answers Objective 3 of the proposal but has only recently been submitted for peer-review, and is hence not yet published. It was uploaded to arxiv.org on 8th January 2021 and can be found on https://arxiv.org/abs/1906.11468 |
Exploitation Route | The findings are relevant to other research in pure mathematics, and are expected to inspire new research both in various branches of representation theory and at the interfaces to mathematical physics and how-dimensional topology. |
Sectors | Other |
Description | Workshop Representations of monoidal categories and 2-categories |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Other audiences |
Results and Impact | Workshop on Representations of monoidal categories and 2-categories |
Year(s) Of Engagement Activity | 2019 |
URL | https://archive.uea.ac.uk/~byr09xgu/workshop.html |