2-representation Theory of Soergel Bimodules

Categorification is the idea of taking a mathematical object and replacing it with a higher categorical one, that is, one with an extra layer of structure. This has led to major advances in representation theory, as well as many other areas, in the last 20-years, e.g. through the categorification of quantum groups and Hecke algebras. It is nowadays phrased as a 2-category acting on other categories via a 2-representation.

This has inspired an abstract theory of 2-representations of so-called "fiat" 2-categories, modelled on the relevant examples.

The project aims to advance the general subject of 2-representation theory, led by the example of categorified affine Hecke algebras, that is, 2-categories of affine Soergel bimodules. These satisfy less stringent finiteness conditions (only being "wide fiat") than those 2-categories previously studied. Moreover, they exhibit, for the first time, interesting triangulated 2-representations, which cannot be obtained as bounded homotopy categories of additive ones.

The main objectives of the project are

- to develop the 2-representation theory of wide fiat 2-representatons with the aim of classifying additive simple 2-representations of Soergel bimodules; - to develop a (pre-)triangulated 2-representation theory, led by the example of affine Soergel bimodules.

Expected academic impacts include insights into certain categories of Lie algebra representations via the actions of so-called Gaitsgory central sheaves, categories of tilting objects for quantum groups as categorifications of antispherical modules, new invariants for links on an annulus, as well as applications of the theory to other wide fiat 2-categories, such as Heisenberg, affine partition or Deligne categories.