Simple transitive $2$-representations of left cell $2$-subcategories of projective functors for star algebras

TL;DR

This paper investigates simple transitive 2-representations of certain 2-subcategories of projective functors over star algebras, showing classification in the simplest case and proposing a conjecture for the general case.

Contribution

It classifies simple transitive 2-representations for the Dynkin type A2 case and conjectures the existence of non-cell 2-representations in the general case.

Findings

Classification of simple transitive 2-representations for type A2.

Conjecture on the existence of non-cell 2-representations in general.

Evidence supporting the conjecture.

Abstract

In this paper we study simple transitive $2$-representations of certain $2$-subcategories of the $2$-category of projective functors over a star algebra. We show that in the simplest case, which is associated to the Dynkin type $A_2$, simple transitive $2$-representations are classified by cell $2$-representations. In the general case we conjecture that there exist many simple transitive $2$-representations which are not cell $2$-representations and provide some evidence for our conjecture.