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.

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.

Publications

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Mackaay M (2024) Evaluation birepresentations of affine type A Soergel bimodules in Advances in Mathematics

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Mackaay M (2021) Finitary birepresentations of finitary bicategories in Forum Mathematicum

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Hristova K (2023) Basic Hopf algebras and symmetric bimodules in Journal of Pure and Applied Algebra

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Mazorchuk V (2020) 2-categories of symmetric bimodules and their 2-representations in Pacific Journal of Mathematics

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Mackaay M (2023) Simple transitive 2-representations of Soergel bimodules for finite Coxeter types in Proceedings of the London Mathematical Society

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Mackaay M (2023) Kostant's problem for fully commutative permutations in Revista Matemática Iberoamericana

 
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 and is published in Proceedings of the London Mathematical Society (joint with MacKaay, Mazorchuk, Tubbenhauer, Zhang). Objective 2 turned out to be the stepping stone for this and was achieved in the paper Finitary birepresentations of finitary bicategories (joint with MacKaay, Mazorchuk, Tubbenhauer, Zhang). Objective 1 was deliberately open-ended, and some progress towards it was made in the paper Basic Hopf algebras and symmetric bimodules (with Katerina Hristova).
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 Research School on Bicategories, Categorification and Quantum Theory 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact Research School for graduate students and postdocs in Leeds, organised with funding by the London Mathematical Society
Year(s) Of Engagement Activity 2023
URL https://conferences.leeds.ac.uk/bcqt2022/
 
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