Cell 2-representations of finitary 2-categories

TL;DR

This paper investigates 2-representations of finitary 2-categories, introducing cell 2-representations inspired by Kazhdan-Lusztig modules, and proves their uniqueness and simplicity under certain conditions.

Contribution

It defines and analyzes cell 2-representations for finitary 2-categories, extending known categorification results for Hecke algebras of type A.

Findings

Cell 2-representations are strongly simple.

They do not depend on the choice of a right cell within a two-sided cell.

The results extend previous categorification uniqueness theorems.

Abstract

We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2-representations inspired by Kazhdan-Lusztig cell modules for Hecke algebras. Under some natural assumptions we show that cell 2-representations are strongly simple and do not depend on the choice of a right cell inside a two-sided cell. This reproves and extends the uniqueness result on categorification of Kazhdan-Lusztig cell modules for Hecke algebras of type $A$ from \cite{MS2}.