Isotypic faithful 2-representations of $${\mathcal {J}}$$ J -simple fiat 2-categories

Classification problems in 2-representation theory

“…If a similar structure exists for a not necessarily involutive anti-autoequivalence , the 2-category C is called weakly fiat, see [41,Subsection 7.3] and [45,Appendix]. In many situations, involutions in 2-categories change the direction of 1-morphisms but preserve the direction of 2-morphisms, see e.g.…”

“…Indeed, the Grothendieck decategorification of a finitary 2-category C gives a finite dimensional k-algebra, call it A. For each 1-morphism F in C , we thus have the minimal polynomial g In many papers, for instance, in [44,45,47,48,59], the classification problem was approached in two steps. The first step addressed classification of all possibilities for matrices [F] M .…”

“…We will say that M is isotypic provided that all weak Jordan-Hölder subquotients of M are equivalent, see [51,Subsection 4.3.4] and [45,Subsection 3.6].…”

“…Despite of the fact that the 2-category of 2-representations of an abstract finitary 2-category is much more complicated than the category of modules over a finite dimensional algebra (in particular, it is not abelian), it turned out that, in many cases, it is possible to construct, compare and even classify various classes of 2-representations. These kinds of problems were recently studied in a number of papers, see [32,[37][38][39][44][45][46][47][48]58]. The aim of the present article is to give an overview of these results with a particular emphasis on open problems and future challenges.…”

Coidempotent Subcoalgebras and Short Exact Sequences of Finitary 2-Representations

“…The subject of 2-representation theory originated from [CR, KL, Ro] and is the higher categorical analogue of the classical representation theory of algebras. The articles [MM1]- [MM6] develop the 2-categorical analogue of finite-dimensional algebras and their finite-dimensional modules, by defining and studying finitary 2-categories and their finitary 2-representations. One of the fundamental questions in representation theory is to find the simple representations of a given algebra.…”